Thermodynamic bounds on current
fluctuations
Von der Fakultät Mathematik und Physik der Universität Stuttgart
zur Erlangung der Würde eines Doktors der
Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung
vorgelegt von
Patrick Pietzonka
aus Stuttgart
Hauptberichter: Prof. Dr. Udo Seifert
Erster Mitberichter: Prof. Dr. Siegfried Dietrich
Zweiter Mitberichter: Prof. Dr. Andreas Engel
Vorsitzender: Prof. Dr. Tilman Pfau
Tag der Einreichung: 26. März 2018
Tag der mündlichen Prüfung: 11. Juni 2018
II. Institut für Theoretische Physik der Universität Stuttgart
2018

Contents
Publications 7
Zusammenfassung 9
Summary 15
1 Introduction 19
2 Stochastic thermodynamics on discrete state spaces 23
2.1 Markovian dynamics in closed systems . . . . . . . . . . . . . . . . . . 23
2.2 Markovian dynamics in open systems . . . . . . . . . . . . . . . . . . 26
2.3 Energetics of open systems . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Entropy production . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Time extensivity and time reversal . . . . . . . . . . . . . . . . . . . . 33
2.6 Fluctuation relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.7 Unicyclic molecular motor as paradigmatic example . . . . . . . . . . . 38
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Fluctuations in stationary Markov processes 43
3.1 Cumulant generating function for time-additive observables . . . . . . . 43
3.2 Calculation of low-order cumulants . . . . . . . . . . . . . . . . . . . . 49
3.3 Large deviation functions . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Level 2.5 large deviation theory . . . . . . . . . . . . . . . . . . . . . . 58
3.5 Continuous state spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Dissipation-dependent bound on current fluctuations 67
4.1 General setup for bounds on current fluctuations . . . . . . . . . . . . . 67
4.2 Linear response regime . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Beyond linear response: Unicyclic case . . . . . . . . . . . . . . . . . 73
4.4 Beyond linear response: multicyclic case . . . . . . . . . . . . . . . . . 75
4.5 The thermodynamic uncertainty relation . . . . . . . . . . . . . . . . . 77
4.6 Dissipation-dependent bound in driven diffusive one dimensional systems 78
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3
Contents
5 Application: Bounds on the efficiency of molecular motors 81
5.1 Thermodynamically consistent motor model . . . . . . . . . . . . . . . 81
5.2 The bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.3 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.4 Numerical case study . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Affinity- and topology-dependent bound on current fluctuations 89
6.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.3 Numerical illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.4 Uniform cycle decomposition . . . . . . . . . . . . . . . . . . . . . . . 95
6.5 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.6 Vanishing currents and equilibrium . . . . . . . . . . . . . . . . . . . . 101
6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Appendices to chapter 6 103
6.A Proof of the upper bound on the generating function for unicyclic networks103
6.B Proof of the concavity of ψ(ζ, s) in s . . . . . . . . . . . . . . . . . . . 105
6.C Proof of the monotonic decrease of ψ(ζ, s)/s in s . . . . . . . . . . . . 106
7 Activity-dependent and asymptotic bound 107
7.1 Activity-dependent bound . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2 Asymptotic bound for unicyclic networks . . . . . . . . . . . . . . . . 110
7.3 Asymptotic bound for multicyclic networks . . . . . . . . . . . . . . . 112
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8 Finite-time generalizations 117
8.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.2 Experimental illustration . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.3 Bound on the generating function . . . . . . . . . . . . . . . . . . . . . 121
8.4 Short-time and linear response limit . . . . . . . . . . . . . . . . . . . 123
8.5 Illustration for unicyclic networks . . . . . . . . . . . . . . . . . . . . 124
8.6 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.7 Uncertainty relation for transient non-equilibrium states . . . . . . . . . 129
8.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
9 Concluding perspective 133
References 137
Danksagungen 151
4
List of Figures
1.1 Uncertainty associated with a molecular motor walking along a filament 21
2.1 Illustration of the network representation of an open system . . . . . . . 28
2.2 Models for unicyclic motors . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Generic spectrum of the tilted operator . . . . . . . . . . . . . . . . . . 46
3.2 Large deviation function for the coin-toss experiment . . . . . . . . . . 53
3.3 Large deviation function for the asymmetric random walk . . . . . . . . 55
4.1 Unicyclic and multicyclic networks . . . . . . . . . . . . . . . . . . . . 68
4.2 Unicyclic illustration of the dissipation-dependent bound. . . . . . . . . 74
4.3 Multicyclic illustration of the dissipation-dependent bound. . . . . . . . 76
4.4 Schematic illustrations of lattice transport models . . . . . . . . . . . . 79
4.5 Dissipation dependent bound for lattice transport models . . . . . . . . 80
5.1 Molecular motor walking along a periodic track . . . . . . . . . . . . . 82
5.2 Efficiency bound applied to experimental data for kinesin . . . . . . . . 85
5.3 Numerical illustration for a motor-cargo complex . . . . . . . . . . . . 87
6.1 Bounds on the generating function for a unicyclic network with high
affinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.2 Bounds on the generating function for unicyclic network with various
affinities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3 Affinity- and topology-dependent bound for a multicyclic network . . . 94
6.4 Illustration of the uniform cycle decomposition . . . . . . . . . . . . . 96
6.5 Illustration of the existence of uniform cycles . . . . . . . . . . . . . . 97
6.6 Properties of the function ψ(ζ, s) . . . . . . . . . . . . . . . . . . . . . 99
6.A.1Characteristic polynomial for unicyclic networks. . . . . . . . . . . . . 104
7.1 Example for the activity-dependent bound in a unicyclic network . . . . 109
7.1 Illustration of the asymptotic bound for a multicyclic network . . . . . . 112
7.1 Summary of bounds for a unicyclic network . . . . . . . . . . . . . . . 115
8.1 Experimental setup and data for the dichotomously switching optical trap 119
8.2 Work distributions for the dichotomously switching trap . . . . . . . . . 120
5
List of Figures
8.1 Illustration for the finite-time bound on the generating function for a
unicyclic network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.2 Illustration for the finite-time bound on the generating function for a
multicyclic network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6
Publications
Parts of this thesis have been published in:
• P. Pietzonka, A. C. Barato, and U. Seifert,
“Universal bounds on current fluctuations”,
Phys. Rev. E 93, 052145 (2016).
© 2016 American Physical Society. Chapter 4 and parts of Chapters 6 and 7 are
based on this publication.
• P. Pietzonka, A. C. Barato, and U. Seifert,
“Affinity- and topology-dependent bound on current fluctuations”,
J. Phys. A: Math. Theor. 49, 34LT01 (2016).
© 2016 IOP Publishing Ltd. Chapter 6 is based on this publication.
• P. Pietzonka, A. C. Barato, and U. Seifert,
“Universal bound on the efficiency of molecular motors”,
J. Stat. Mech.: Theor. Exp. 124004 (2016).
© 2016 IOP Publishing Ltd and SISSA Medialab srl. Chapter 5 is based on this
publication.
• P. Pietzonka, F. Ritort, and U. Seifert,
“Finite-time generalization of the thermodynamic uncertainty relation”,
Phys. Rev. E 96, 012101 (2017).
© 2017 American Physical Society. Chapter 8 is based on this publication.
Further publications I have co-authored:
• P. Pietzonka, E. Zimmermann, and U. Seifert,
“Fine-structured large deviations and the fluctuation theorem: Molecular motors
and beyond”,
EPL 107, 20002 (2014).
• P. Pietzonka, K. Kleinbeck, and U. Seifert,
“Extreme fluctuations of active Brownian motion”,
New J. Phys. 18, 052001 (2016).
7
Publications
• P. Pietzonka andU. Seifert, “Entropy production of active particles and for particles
in active baths”, J. Phys. A: Math. Theor. 51, 01LT01 (2018).
• M. Uhl, P. Pietzonka, and U. Seifert,
“Fluctuations of apparent entropy production in networkswith hidden slowdegrees
of freedom”,
J. Stat. Mech.: Theor. Exp. 023203 (2018).
• L. P. Fischer, P. Pietzonka, and U. Seifert,
“Large deviation function for a driven underdamped particle in a periodic poten-
tial”,
Phys. Rev. E 97, 022143 (2018).
• P. Pietzonka and U. Seifert,
“Universal trade-off between power, efficiency, and constancy in steady-state heat
engines”,
Phys. Rev. Lett. 120, 190602 (2018).
8
Zusammenfassung
Prozesse in lebendigen Systemen und nützlichen künstlichen Maschinen laufen unter
Nichtgleichgewichtsbedingungen ab. Das bedeutet, dass diese Systeme sich im Kontakt
mit mehreren Reservoiren befinden, die nicht in wechselseitigem thermodynamischem
Gleichgewicht sind. Die Reservoire stellen Ressourcen wie Nahrung oder Treibstoff
bereit oder fungieren als thermische Umgebung die Wärme absorbiert. Unter der Vor-
aussetzung, dass das betrachtete System hinreichend klein gegenüber den Reservoiren ist
kann man davon ausgehen, dass der Zustand der Reservoire sich auf der relevanten Zeit-
skala nur unwesentlich ändert. Wenn außerdem das System nicht von außen manipuliert
wird stellt sich eine zeitunabhängige Dynamik ein, die als stationärer Nichtgleichge-
wichtszustand bezeichnet wird.
Durch zufällige thermische Einflüsse aus der Umgebung ist der künftige Zustand des
Systems unvorhersehbar, sodass dessen Dynamik als stochastischer Prozess modelliert
werden kann. Besonderes Augenmerk liegt in dieser Dissertation auf den Strömen, die
zwischen den Reservoiren und dem System im stationären Nichtgleichgewicht fließen.
Dazu gehören z.B. der Verbrauch oder die Produktion einer chemischen Spezies oder die
mit demAnheben eines Gewichtes verknüpfte mechanische Arbeit. Von besonderer ther-
modynamischer Bedeutung ist der Strom der mit der gesamten Entropieproduktion des
Systems verknüpft ist und damit dessen Nichtgleichgewichtscharakter quantifiziert. Im
deutlichem Gegensatz zu Systemen im thermodynamischen Gleichgewicht sind Nicht-
gleichgewichtssysteme in der Lage, Ströme aufrechtzuerhalten, die im Mittel von Null
verschieden sind. Insbesondere die Rate der Entropieproduktion ist dabei, wie vom zwei-
ten Hauptsatz der Thermodynamik gefordert, immer größer als Null. Jedoch unterliegen
auch diese Ströme dem thermischen Einfluss der Umgebung, sodass ihre zeitliche Ent-
wicklung von Fluktuationen überlagert wird. Das bedeutet, dass auf kurzen Zeitskalen
Ströme von ihren Mittelwerten abweichen können und sogar die Entropieproduktion
negativ werden kann.
Das Ziel dieser Arbeit ist, eine umfassende Charakterisierung der Statistik von Strom-
fluktuationen zu entwickeln. Die exakte Berechnung dieser Statistik ist zwar möglich,
hängt aber ab von allen mikroskopischen Details des Systems und den von den Reservoi-
ren aufgebrachten treibenden Kräften. Da derart detaillierte Informationen in der Praxis
nicht verfügbar und auch nicht von Interesse sind, setzen wir uns zum Ziel, Schran-
ken an die Statistik von Stromfluktuationen herzuleiten, die von wenigen, möglichst
aussagekräftigen thermodynamischen Eigenschaften des Systems abhängen.
Ausgangspunkt für diese Arbeit is eine bemerkenswerte Ungleichung, die als “thermo-
9
Zusammenfassung
dynamische Unschärferelation” bekannt ist [A.C. Barato und U. Seifert, Phys. Rev. Lett.
114, 158101 (2015)]. In diese Relation gehen zwei wichtige Kenngrößen ein. Die eine
ist eine statistische Kenngröße, nämlich die Unschärfe des Stromes, welche sich aus der
Amplitude von Fluktuationen im Vergleich zum Mittelwert ergibt. Zweitens betrach-
ten wir die mittlere Entropieproduktionsrate als thermodynamische Kenngröße. Das
Produkt dieser zwei Größen muss immer größer als zwei sein, was eine fundamentale
Abwägung zwischen Präzision und den thermodynamischen Kosten für einen Nicht-
gleichgewichtsprozess zum Ausdruck bringt. Diese Abwägung gilt für jeden Strom und
für die große Klasse an Systemen die sich durch Markovprozesse beschreiben lassen.
Wir stellen dieses Ergebnis in einen breiteren mathematischen Kontext, indem wir
es als Konsequenz aus einer ebenso universellen Schranke an das gesamte Spektrum
der Stromfluktuationen herleiten. Dazu verwenden wir als mathematisches Werkzeug
die sogenannte “large deviation theory”, welche die Statistik großer Abweichungen
vomErwartungswert einer geeigneten Zufallsgröße beschreibt. Diese Herangehensweise
erlaubt es, etliche Verfeinerungen und Verallgemeinerungen dieser Schranke herzuleiten
und liefert neue, ergänzende Schranken an Stromfluktuationen.
Kapitel 2: Stochastische Thermodynamik in diskreten Zustandsräumen. In die-
sem Kapitel erläutern wir die Formulierung einer thermodynamisch konsistenten sto-
chastischen Dynamik in einem diskreten Zustandsraum. Dessen Herleitung beginnt mit
der Identifikation von Mesozuständen in einem geschlossenen Gesamtsystem, welches
die Reservoire beinhaltet. Innerhalb dieser Mesozustände erreicht das System ein loka-
les Gleichgewicht, was dazu führt, dass die Übergänge zwischen den Mesozuständen
einemMarkovprozess entsprechen. Da das Gesamtsystem langfristig ein thermodynami-
sches Gleichgewicht erreichen muss, können wir detailliertes Gleichgewicht (“detailed
balance”) für die Übergangsraten zwischen den Mesozuständen voraussetzen. Bei der
Einschränkung auf ein kleines offenes System im Kontakt mit Reservoiren, kann diese
Bedingung als ein lokales detailliertes Gleichgewicht ausgedrückt werden. Maßgeb-
lich für diese Relation sind dann die Entropien der Mesozustände des kleinen Systems
(bzw. deren freie Energien in einem isothermen Kontext) und die mit einem Übergang
einhergehenden Änderungen in den Reservoiren. Die resultierende stochastische Dyna-
mik beschreibt den stationären Nichtgleichgewichtszustand, welchen das kleine System
erreicht. Den Ideen der stochastischen Thermodynamik folgend definieren wir dann
thermodynamische Observablen wie Wärme, Arbeit uns insbesondere die Entropiepro-
duktion entlang einzelner Trajektorien des Systems. Diese Observablen erweisen sich als
Zeit-additiv und als antisymmetrisch unter Zeitumkehr, was sie als Ströme im oben ein-
geführten Sinn identifiziert. Einige dieser Observablen erfüllen Fluktuationsrelationen,
die bislang bekanntesten universellen Eigenschaften der Statistik von Stromfluktuatio-
nen. Zum Abschluss werden die in diesem Kapitel eingeführten Konzepte anhand einer
Auswahl einfacher, unizyklischer Modelle für molekulare Motoren illustriert.
10
Zusammenfassung
Kapitel 3: Fluktuationen in stationären Markovprozessen. In diesem Kapitel erläu-
tern wir die mathematischen Werkzeuge die nötig sind um Fluktuationen in stationären
Markovprozessen zu beschreiben.Wir beginnenmit der allgemeinenDefinition von Zeit-
additiven Observablen und definieren die kumulantenerzeugende Funktion, welche die
Fluktuationen solcher Observablen charakterisiert. Im Grenzfall großer Zeitintervalle
wird die Skalierung der kumulantenerzeugenden Funktion durch den größten Eigenwert
einer modifizierten Ratenmatrix festgelegt. Kumulanten niedriger Ordnung der Fluk-
tuationen, insbesondere deren Mittelwert und die Varianz, können jedoch auch ohne
explizite Kenntnis dieses Eigenwerts berechnet werden. Hierfür stellen wir zwei ver-
schiedene Methoden vor.
Die “large deviation theory”, welche sowohl typische als auch extreme Fluktuationen
von Strömen charakterisiert, stellt eine weitere Möglichkeit zur mathematischen Be-
schreibung dar. Wir stellen kurz die Grundkonzepte dieser Theorie vor, deren zentrales
Element die sogenannte Ratenfunktion ist. Sie erfasst den exponentiellen Zerfall der
Wahrscheinlichkeit untypischer Fluktuationen und steht in engem Zusammenhang mit
der kumulantenerzeugenden Funktion. Als wesentliche Methode für später Beweise von
Schranken an Stromfluktuationen, stellen wir die “level 2.5 large deviation theory” vor,
die die gemeinsamen Fluktuationen empirischer Ströme und Dichten in Markovnetzwer-
ken beschreibt. Zum Abschluss zeigen wir durch einen geeigneten Kontinuumslimes,
dass der zunächst für diskrete Zustände entwickelte Formalismus sich auch auf über-
dämpfte Langevindynamik in einem kontinuierlichen Zustandsraum anwenden lässt.
Kapitel 4: Dissipationsabhängige Schranke an Stromfluktuationen. In diesem Ka-
pitel leiten wir eine dissipationsabhängige Schranke an Stromfluktuationen her, welche
unser am vielseitigsten anwendbares Ergebnis darstellt. Es handelt sich dabei um eine
parabolische obere Schranke an die Ratenfunktion die einen beliebigen fluktuierenden
Strom beschreibt bzw. um eine parabolische untere Schranke an die entsprechende ku-
mulantenerzeugende Funktion. Entscheidend ist dabei, dass die Entropieproduktionsrate
des Gesamtsystems als einziger Parameter in diese Schranke eingeht. Nahe dem ther-
modynamischen Gleichgewicht kann diese Schranke aus der positiven Definitheit der
Onsagermatrix abgeleitet werden. Fernab vom Gleichgewicht kann die Schranke in-
tuitiv anhand von unizyklischen Netzwerken erklärt werden, den multizyklischen Fall
illustrieren wir numerisch. Bei der Auswertung für typische Fluktuationen geht unsere
allgemeine Schranke über in die thermodynamische Unschärferelation, deren wichtigs-
ten Implikationen wir kurz diskutieren. Abschließend illustrieren wir die Gültigkeit der
dissipationsabhängigen Schranke anhand analytischer Lösungen für eindimensionale
getriebene diffusive Transportmodelle.
Kapitel 5: Anwendung: Schranken an die Effizienz molekularer Motoren. Dieses
Kapitel ist als Einschub gedacht, in dem eine wichtige Implikation der thermodyna-
11
Zusammenfassung
mischen Unschärferelation für die Effizienz molekulare Motoren hergeleitet wird. In
dieser Anwendung ist der relevante Strom die von einem Motor zurückgelegt Strecke,
die man typischerweise experimentell beobachten kann. Die Messung der Fluktuatio-
nen in diesem Strom ermöglicht eine Abschätzung der Dissipationsrate, die wiederum
dazu verwendet werden kann eine obere Schranke an die thermodynamische Effizienz
des Motors anzugeben. Diese Schranke gilt für beliebige thermodynamisch konsistente
Modelle für molekulare Motoren und benötigt ausschließlich experimentell leicht zu-
gängliche Messgrößen. Eine ähnliche Schranke erhalten wir für die Stokes-Effizienz, die
einen Motor ohne externe mechanische Last charakterisiert.
Kapitel 6: Affinitäts- und topologieabhängige Schranke anStromfluktuationen. Mit-
hilfe der durch die level 2.5 large deviation theory bereitgestellten Werkzeuge leiten
wir eine Verfeinerung der dissipationsabhängigen Schranke her. Das Aufstellen dieser
Schranke benötigt zusätzlich Kenntnis über die Affinitäten die das System antreiben und
über die Topologie des zugrunde liegenden Markov-Netzwerkes. Entscheidend ist dabei
der Zyklus mit der kleinsten Affinität pro Anzahl der Knoten. Ausgedrückt als unte-
re Abschätzung für die kumulantenerzeugende Funktion hat diese Schranke die Form
eines Kosinus hyperbolicus, ähnlich der kumulantenerzeugenden Funktion des asym-
metrischen Random Walks. Wir illustrieren diese Schranke numerisch. Als wesentliche
Voraussetzung für den Beweis entwickeln wir eine Zerlegung der Netzwerkströme in
“gleichförmige” Zyklen, die so definiert sind, dass der mit einem einzelnen Zyklus ver-
knüpfte Strom entlang jeder Kante parallel zum gesamten Strom ist. Die Affinitäts- und
topologieabhängige Schranke kann auch auf Systeme im thermodynamischen Gleich-
gewicht angewandt werden, wobei die Anzahl der Zustände im gesamten Netzwerk die
Entropieproduktion als entscheidende Kenngröße ersetzt.
Kapitel 7: Aktivitätsabhängige und asymptotische Schranke. In diesem Kapitel
führen wir weitere Schranken an Stromfluktuationen ein, die von anderen Kenngrö-
ßen des Systems als dessen Entropieproduktionsrate abhängen. Zunächst leiten wir eine
Schranke her die nur von der Aktivität des Systems abhängt, d.h. von der Anzahl der
Übergänge pro Zeiteinheit. Ausgedrückt als Schranke an die kumulantenerzeugende
Funktion hat diese Schranke die Form einer Exponentialfunktion, die die tatsächliche
Funktion an einer Seite berührt. Mithilfe des Fluktuationstheorems für Ströme kann
diese Schranke symmetrisiert werden, sodass sie für beide Seiten der kumulantener-
zeugenden Funktion relevant wird. Diese Schranke ist fernab vom thermodynamischen
Gleichgewicht oftmals stärker als die dissipationsabhängige Schranke. Ihr Beweis ba-
siert auf einem einfachen linearen Ansatz für den empirischen Fluss in der level 2.5
Ratenfunktion.
Das zweite Hauptergebnis in diesem Kapitel ist eine Schranke die eher schwach ist
für typische Fluktuationen, die aber asymptotisch exakt wird für extreme Fluktuationen
12
Zusammenfassung
und damit in einem Bereich in dem die anderen Schranken typischerweise eher schwach
sind. Diese Schranke benötigt im Wesentlichen Information über die Topologie des
dem Modell zugrunde liegenden Netzwerks, in dem derjenige Zyklus identifiziert wird
der für eine bestimmte extreme Fluktuation am relevantesten ist. Zusammen mit den
anderen Schranken erhält man so eine umfassende Charakterisierung des Spektrums der
Stromfluktuationen.
Kapitel 8: Verallgemeinerung für endliche Zeitintervalle. Alle so weit diskutierten
Schranken beziehen sich auf Stromfluktuationen die im Grenzfall langer Zeitintervalle
auftreten. Die stochastische Thermodynamik hat sich jedoch besonders dann als nütz-
lich herausgestellt, wenn relevante Ergebnisse bereits auf endlichen Zeitskalen gewonnen
werden können. In diesem Kapitel stellen wir eine Verallgemeinerung der dissipations-
abhängigen Schranke für endliche Zeitintervalle vor, die wiederum auch die thermodyna-
mische Unschärferelation für endliche Zeiten verallgemeinert. Diese Verallgemeinerung
illustrieren wir anhand experimenteller Daten für die verrichtete Arbeit an einemKolloid
das sich in einer stochastisch zwischen zwei Positionen springenden optischen Falle be-
findet. Überraschenderweise kann dieMessung für endliche Zeitintervalle sogar stärkere
Schranken an die Entropieproduktionsrate liefern als der entsprechende Grenzfall langer
Zeiten.
Wir besprechen eine Beweismethode für diese Verallgemeinerung der thermodyna-
mischen Unschärferelation, die auf der level 2.5 large deviation theory für ein Ensemble
von unabhängigen Kopien des Systems basiert. Dieser Beweis gilt zunächst für Anfangs-
bedingungen die der stationären Verteilung entstammen. Wir entwickeln eine weitere
Verallgemeinerung, die sich auch auf transiente Prozesse in endlichen Zeitintervallen
mit beliebigen Anfangsverteilungen anwenden lässt.
13

Summary
Living systems, as well as useful artificial machines, operate under non-equilibrium
conditions. This means that they are in contact with several reservoirs that are not in
mutual thermodynamic equilibrium. These reservoirs provide resources such as food
or fuel, or act as a thermal environment absorbing heat. Provided that the system
under consideration is sufficiently small compared to the reservoirs, the state of the
reservoirs will change negligibly on relevant time scales. If additionally the system
is not manipulated externally, its dynamics becomes time-invariant, which is called a
non-equilibrium steady state (NESS).
Due to the thermal influence from its environment, the state of the system becomes
erratic, which allows us to model its dynamics as a stochastic process, for which one
can define several thermodynamic observables. Of particular interest throughout this
Thesis are the input- and output-currents associated with a NESS. Examples for such
currents include the consumption or production of a specific chemical species or thework
associated with lifting a weight. A current of particular thermodynamic importance is
the production of entropy in the total system, which quantifies its non-equilibrium
character. In stark contrast to equilibrium systems, non-equilibrium systems are capable
of maintaining non-zero average currents. In particular, the rate of entropy production is,
due to the second law of thermodynamics, always greater than zero on average. However,
again due to the thermal influence from the environment, the temporal evolution of these
currents is superimposed by fluctuations. This means, that on short time scales, currents
can deviate from the average intensity, and the entropy production can even become
negative.
The main objective of the work documented in this Thesis is to provide a comprehen-
sive characterization of the statistics of current fluctuations. While an exact calculation
of these statistical properties is possible, the results typically depend on all microscopic
details of the system and on the driving forces associated with the reservoirs. Since
such detailed information is practically neither available nor relevant, we focus on the
derivation of bounds on the statistics of current fluctuations, which ideally depend on
only a few thermodynamic properties of the system.
Starting point for our work is a prominent inequality known as the “thermodynamic
uncertainty relation” [A.C. Barato and U. Seifert, Phys. Rev. Lett. 114, 158101 (2015)].
It considers the uncertainty of a current, comparing the amplitude of its fluctuations to its
mean, as a statisticalmeasure and on the other hand the average rate of entropy production
as a thermodynamic measure. The product of these two key quantities must always be
15
Summary
greater than two, expressing a trade-off between precision and the thermodynamic cost
for a non-equilibrium process. It holds for any current and for the huge class of systems
that can be described in terms of Markov processes.
We put this relation in a wider mathematical context, employing large deviation theory
to derive it as a result of an equally general bound on the whole spectrum of current
fluctuations. Our formalism allows for several refinements and generalizations of that
bound and yields complementary, novel bounds on current fluctuations.
Chapter 2: Stochastic thermodynamics on discrete state spaces. We review a for-
malism for thermodynamically consistent stochastic dynamics on a discrete set of states.
Its derivation starts with an identification of mesostates in the total, closed system in-
cluding the reservoirs. Within these mesostates, the system equilibrates locally, such that
the transitions between mesostates become Markovian. Since the total system must ulti-
mately reach thermodynamic equilibrium, we can stipulate a detailed balance condition
for the transition rates between mesostates. Focusing on a small system in contact with
reservoirs, this detailed balance condition gets transformed to a local detailed balance
condition involving the entropies of the mesostates of the small systems (or their free
energies in an isothermal setting) and the changes in the reservoirs upon a transition. The
resulting stochastic dynamics describes the NESS associated with the small system. Fol-
lowing the ideas of stochastic thermodynamics, we define thermodynamic observables,
such as heat, work and in particular the entropy production along individual trajectories
of the system. These observables turn out to be time-additive and antisymmetric under
time reversal, which qualifies them as currents in the above defined sense. Fluctuation
relations, as the most prominent constraints on current fluctuations so far, are briefly
reviewed. Finally, we illustrate the concepts introduced in this Chapter for a selection of
simple, unicyclic models for molecular motors.
Chapter 3: Fluctuations in stationaryMarkov processes. In thisChapter, we present
the mathematical toolbox that is necessary for the description of fluctuations in station-
ary Markov processes. We start with a general definition of time-additive observables
and define the cumulant generating function that characterizes the fluctuations of such
an observable. In the long-time limit, the scaling of the cumulant generating function is
given by the largest eigenvalue of a modified transition matrix. Low order cumulants of
the fluctuations, in particular their mean and variance, can be calculated without explicit
knowledge of this eigenvalue. We present two different methods for carrying out this
calculation.
Large deviation theory characterizes both typical and extreme current fluctuations,
which is complementary to the description via the scaled cumulant generating function.
We briefly review the basic concepts of this theory and work out the connection to
the algebraic approach using generating functions. A key element of this theory is the
16
Summary
so-called large deviation function, or rate function, which captures the exponential decay
of the probability of unlikely fluctuations. As the central method for proving bounds on
current fluctuations later on, we re-derive the level 2.5 large deviation function, which
describes the joint fluctuations of empirical currents and densities in a Markovian net-
work. Finally, we show through a suitable continuum limit that the formalism developed
so far for discrete state-spaces applies also to overdamped Langevin dynamics on a
continuous state space.
Chapter 4: Dissipation-dependent bound on current fluctuations. In this Chapter
we derive a dissipation-dependent bound on current fluctuations as our most universally
applicable result. It can be formulated as a quadratic upper bound on the large deviation
function or, equivalently, as a quadratic lower bound on the generating function associ-
ated with any current. Crucially, the only parameter entering in this bound is the average
rate of entropy production in the total system. In linear response, this bound can be
derived from the positive definiteness of the Onsager matrix. Beyond linear response,
we give an intuitive explanation for the bound for unicyclic networks and illustrate the
general, multicyclic case numerically. When evaluated for typical current fluctuations,
our general bound implies the thermodynamic uncertainty relation, whose major impli-
cations we briefly review. Finally, we illustrate the validity of the dissipation-dependent
bound for analytic solutions for one-dimensional driven diffusive transport-models.
Chapter 5: Application: Bounds on the efficiency ofmolecularmotors. ThisChap-
ter is intended as an intermezzo, deriving an important implication of the thermodynamic
uncertainty relation for the efficiency of molecular motors. The current of interest in
this example is the displacement of a motor, which can typically be observed experi-
mentally. The measurements of its fluctuations allows for an estimate of the dissipation
rate, which can, in turn, be used to derive an upper bound on the thermodynamic effi-
ciency of the motor. This bound holds for any thermodynamically consistent model for
a molecular motor and uses only experimentally accessible quantities. A similar bound
can be derived for the Stokes efficiency that characterizes a motor in the absence of a
mechanical load. We illustrate the bound numerically for a motor-cargo complex and
for experimental data for kinesin.
Chapter 6: Affinity- and topology-dependent bound on current fluctuations. We
employ the toolbox provided by level 2.5 large deviation theory to derive a refinement
of the dissipation-dependent bound. It requires additional knowledge of the driving
affinities and the network topology of the Markov model underlying the system of
interest. In particular, the bound is characterized by the cycle with the minimal affinity
per length. Formulated as a bound on the generating function, this bound has the shape
of a hyperbolic cosine, akin to the generating function for the asymmetric random walk,
17
Summary
which we illustrate numerically. As an essential ingredient to the proof of the bound, we
derive a decomposition of the network currents into what we call uniform cycles. Along
every edge of such a cycle, the cycle current is parallel to the total current. The affinity-
and topology-dependent bound can also be applied to current fluctuations in equilibrium
systems, where the total number of states enters instead of the entropy production rate.
Chapter 7: Activity-dependent and asymptotic bound. In this Chapter, we introduce
further bounds on current fluctuations, which depend on other characteristic features of
the system than its rate of entropy production. First, we derive a bound that depends only
on the activity of the system, i.e., the total number of transitions per unit time. Expressed
as a bound on the generating function, this bound has the shape of an exponential function
that touches the actual generating function at one flank. Using the fluctuation theorem
for currents, this bound can be symmetrized so that it becomes relevant for both flanks of
the generating function. This bound is often stronger than the dissipation-dependent one
in systems that are driven far away from equilibrium. The proof of the activity-dependent
bound is based on a simple, linear ansatz for the empirical flow entering the level 2.5.
large deviation function.
The second main result of this Chapter is a bound that is rather loose for the typical
current fluctuations, but which becomes asymptotically tight for extreme fluctuations,
where the other bounds are typically weak. This bound essentially requires information
about the topology of the underlying network of states, from which one identifies the
cycle that is most relevant for producing a specific extreme fluctuation. Together with the
other bounds one obtains a comprehensive characterization of the spectrum of current
fluctuations.
Chapter 8: Finite-time generalizations. The bounds that have been discussed so
far characterize the fluctuations that occur in the limit of long time intervals. However,
stochastic thermodynamics usually provesmost useful in cases where relevant results can
be obtained on finite time-scales. In this Chapter, we present a finite time-generalization
of the dissipation-dependent bound, which likewise generalizes the thermodynamic
uncertainty relation to finite times. This generalization is illustrated using experimental
data for the work performed on a colloid in a stochastically switching optical trap.
Surprisingly, a measurement on finite time intervals can yield stronger bounds on the
rate of entropy production than the corresponding long-time limit.
Using level 2.5 large-deviation theory for an ensemble of independent copies of the
system, we review a rigorous proof for the finite-time uncertainty relation in its original
formulation applying to initial conditions sampled from the steady-state distribution.
Moreover, this method allows us to derive a further generalization to transient finite-
time processes with initial conditions sampled from an arbitrary initial distribution.
18
1 Introduction
The statistical mechanics for systems in thermodynamic equilibrium, as developed in
the 19th century mainly by Gibbs, Maxwell and Boltzmann, is arguably one of the most
successful concepts in physics. It provides universally consistent formal definitions for
such intuitive quantities as temperature or pressure, which are based on the microscopic
mechanics of a system. In doing so, one needs to make only minimal assumptions
about the microscopic mechanics, namely that it conserves energy and that it leads to
every microscopic state of the system being equally likely. Thus, statistical mechanics
can be applied to virtually every closed many-particle system, including large systems
comprising a heat reservoir and an open system of interest.
A distinctive feature of a system in thermodynamic equilibrium is its time-reversal
symmetry: Any realization of a dynamic process the system might undergo must occur
equally likely in a time-reversed fashion. However, this property is in conflict to what
one would expect from a useful machine or a living organism. To name a few examples,
an engine burns fuel to perform mechanical work, a fridge uses electricity to cool down
its content, and a living organism processes food in order to maintain a multitude of life-
supporting cycles. For all of these examples, the reverse process would be impossible
or at least undesirable. Indeed, all of these systems operate under non-equilibrium
conditions. This means that the total system, including the thermal environment and any
fuel-, electricity-, or food-reservoirs, has not yet reached thermodynamic equilibrium.
In this Thesis, we focus on autonomous systems that show persistent non-equilibrium
behavior. Such a persistence requires the system proper to be small compared to the
reservoirs, such that the supply of resources does not vary on time-scales relevant for
the observation. If this condition is met, the system will reach what is called a non-
equilibrium steady state (NESS).
How can one quantify the non-equilibrium character of such a NESS? While the
macroscopic state of an equilibrium system is fully characterized by a few thermody-
namic observables such as temperature, entropy or internal energy, the situation is far
less clear in non-equilibrium. Two important characteristics will be recurringly con-
sidered in this Thesis. One of these characteristics is a thermodynamic one, namely
the production of entropy. It describes the effect of the consumption of resources and
of dissipation into the environment transducing the total system from a prepared initial
state with low entropy to the final equilibrium state with high entropy. In the long run,
the entropy saturates towards its final equilibrium value, but on the time scale relevant
for the small system, one observes a constant rate of increase of the entropy. Due to
19
Introduction
the second law of thermodynamics, this rate must always be positive. In general, one
expects that a high rate of entropy production corresponds to a strongly driven system
operating far away from equilibrium.
The second characteristic is a statistical one and is less concerned with the driving
of the system than with its output. It is based on observables, called “currents”, that
are antisymmetric under time reversal. This means, that when a movie of the system is
played backwards, the value of such an observable reverses its sign. For the example
of a molecular motor walking along a filament, the velocity would qualify as a current
observable, which can be integrated to yield the traveled distance. In equilibrium, the
time-reversal symmetry stipulates that any current must be zero on average. Nonetheless,
the current exhibits unbiased fluctuations. In the example, a non-driven molecular motor
would perform a randomwalk taking equally likely forward- and backward-steps, so that
it remains on average at the same position. Hence, we can identify a non-vanishing
average current as a further signature of a non-equilibrium system. Accordingly, a bias
in the motion of a molecular motor is a clear indication of the presence of a driving
mechanism. However, the average value of the current has an arbitrary dimension and
is therefore not suitable for a universal quantification of the non-equilibrium character.
We will therefore compare the average current to the amplitude of its fluctuations, thus
defining a dimensionless measure of precision or, as its inverse, the uncertainty.
For driven systems very close to equilibrium any current must be rather small on
average, such that the current fluctuations, which are present already in equilibrium,
prevail. This leads to low precision or, equivalently, high uncertainty. In contrast,
strongly driven systems can be expected to produce substantial currents whose averages
exceed the respective fluctuations, yielding high precision. These considerations suggest
that precision in a current generally requires a system to be driven sufficiently far from
equilibrium, which should be reflected in its entropy production as well.
Indeed, in a seminal paper [1], Barato and Seifert have reported an inequality that es-
tablishes a universal relation between the uncertainty in any current of a non-equilibrium
system and the total entropy production. This relation, which has been dubbed the ther-
modynamic uncertainty relation, can be stated as
ε2C ≥ 2.
The squared uncertainty ε2 is defined via the variance and mean of a time-integrated
current, and C is the on average produced entropy in units of the Boltzmann constant,
characterizing the thermodynamic “cost” for the driving.
The thermodynamic uncertainty relation can be interpreted at least in two ways. The
first one is from the perspective of a system that needs to achieve a certain degree of
precision. For example, the distance traveled by the molecular motor shown in Fig. 1.1
in a fixed time interval fluctuates around an average value. Smaller fluctuations require a
higher consumption of fuel that comes with an increased rate of dissipation. As another
20
Introduction
START
+-
heat
ε
Figure 1.1: A molecular motor walking along a filament. The chemical driving due to the
hydrolysis of ATP leads to a bias in the direction of motion and ultimately to
dissipation of heat. The motor is repeatedly let run from a fixed starting position
and for a fixed time interval. In each run it reaches a slightly different final position,
as indicated by the overlaid snapshots and characterized by the uncertainty ε.
example, circadian clocks of organisms have evolutionarily adapted to anticipate the
end of nighttime even under deprivation of any external stimuli [2]. According to the
uncertainty relation, the organism has to expend the more energy, the more precise this
prediction is required to be.
Second, the thermodynamic uncertainty relation is also promising from an experimen-
tal point of view. With the advent of new experimental techniques for the manipulation
and observation of small non-equilibrium systems, measuring the fluctuations of an
accessible degree of freedom has become a fairly simple procedure [3]. However,
measuring the rate of dissipation that characterizes such a system remains a major chal-
lenge [4, 5], especially when chemical reactions are involved. The uncertainty relation
yields lower bounds on the rate of entropy production based on the visible fluctuations
of any current observable, without requiring any knowledge of the microscopic details
of the system. Such bounds will potentially prove useful for testing hypotheses one may
have for the yet unknown functioning of, e.g., biomolecular systems.
In the work documented in this Thesis, we put the thermodynamic uncertainty relation
on a broad theoretical basis, integrating it into the framework of stochastic thermody-
namics [6]. In doing so, we derive various generalizations, refinements and implications
of the uncertainty relation. Moreover, we explore which other properties of a system in
a steady state provide bounds on its current fluctuations.
As key result, we introduce a universal, dissipation-dependent bound on the whole
spectrum of current fluctuations [7]. While the mean and variance of a fluctuating
current, forming the uncertainty ε, characterize only typical fluctuations, we will use
a description that applies also to untypical, extreme fluctuations. It is based on the
so-called large deviation function that captures the exponential contributions to the
distribution of current fluctuations. This function turns out to be bounded by a quadratic
function, in which the total dissipation enters as the only parameter. This bound implies
21
Introduction
the thermodynamic uncertainty relation as the special case of typical fluctuations. This
augmented level of description in terms of large deviation theory has indeed proven
crucial for a general proof of the uncertainty relation [8].
The dissipation-dependent bound is particularly strong for weak driving of the system
close to equilibrium. Far away from equilibrium the bound is still valid, but presents a
rather loose estimate of the actual current fluctuations. Quite in general, more details
about a system will lead to stronger bounds on its fluctuations. For instance, we present
a major refinement of the dissipation-dependent bound that depends additionally on the
topology of the Markovian network that models the system and on the strengths of the
individual driving forces. Two other bounds that are unrelated to dissipation complement
the dissipation-dependent bound. One of them depends on the overall rate of transitions
in the system and is often stronger far from equilibrium, the other one depends essentially
on the topology of the network and becomes strong for the asymptotics of extremely
untypical fluctuations.
The initial formulation of the thermodynamic uncertainty relation considers exclu-
sively current fluctuations that occur on large time scales. On these time scales cor-
relations vanish, such that the central limit theorem requires the distribution of typical
current fluctuations to be Gaussian. In contrast, current fluctuations on finite time scales
are distributed in a rather different, non-Gaussian fashion. Despite these differences, the
uncertainty relation along with the bounds on the fluctuation spectra turn out to be valid
on finite time scales. For experimental applications, this insight not only reduces the
amount of required data, it can also lead to stronger bounds on the dissipation rate.
All these bounds taken together provide a comprehensive picture of how only a few
crucial thermodynamic properties coin the extend and shape of current fluctuations of a
system in a NESS.
22
2 Stochastic thermodynamics on
discrete state spaces
The past few decades have seen a rise in new experimental techniques to observe and
manipulate micro- and mesoscopically small systems like colloids, biomolecules or
nanoelectronic devices [3]. For such systems, the ubiquitous thermal noise plays a
non-negligible role. Stochastic thermodynamics provides a theoretical framework for
the description of such systems, that merges concepts from classical thermodynamics,
non-equilibrium statistical mechanics and stochastic processes [6, 9–11].
In this Chapter, we review the concepts of stochastic thermodynamics on the basis of
a universally applicable description of thermal systems using Markovian processes on
discrete state spaces.
2.1 Markovian dynamics in closed systems
On the most detailed level of description, we consider a closed system with microstates
ξ that undergo Hamiltonian dynamics with some Hamiltonian function H (ξ) and a total
energy Etot. Here, the microstate ξ represents a high dimensional vector that contains
the configuration and momenta of all N particles within the system. For example, for a
biomolecular system, this would comprise all atoms of the biomolecules, of the solvent,
and of dissolved reactants. In equilibrium, the microcanonical distributions for these
microstates is
peq(ξ) = δ(H (ξ) − Etot)/(h3NΩ(Etot)), (2.1)
with the normalization constant
Ω(Etot) ≡
∫
dξ
h3N
δ(H (ξ) − Etot) ≡ eS/δE, (2.2)
where the integration is carried out over the whole phase space. The quantities h and δE
are arbitrarily chosen constants with dimensions of an action and an energy, respectively,
which allows for a dimensionless definition of themicrocanonical entropy S. Throughout
this Thesis, we set Boltzmann’s constant kB = 1, such that the conventional definition of
entropy becomes equivalent to S.
23
2 Stochastic thermodynamics on discrete state spaces
Since a full solution of the microscopic equations of motion for ξ is neither feasible
nor desirable, we aim for a coarse-grained description of the system and its dynamics
by grouping the microstates into mesostates x. The mapping has to be such that every
microstates belongs to exactly one mesostate, moreover, we group microstates with the
same spacial configuration of particles but different momenta into the same mesostate.
This way, in equilibrium there is no net flow between two mesostates x and y, i.e.,
p[ξ (0) ∈ x, ξ (t) ∈ y] = p[ξ (0) ∈ y, ξ (t) ∈ x] (2.3)
for arbitrary times t, as can be checked via time reversal andLiouville’s theorem. Notably,
Eq. (2.3) already implies micro-reversibility, stating that whenever a transition from x
to y is possible, also the reverse transition from y to x must be possible. The probability
to observe the microstate x in equilibrium is given by
peqx =
∫
ξ∈x
dξ peq(ξ) =
∫
ξ∈x
dξ
h3N
δ(H (ξ) − Etot)/Ω(Etot) ≡ eSx−S, (2.4)
which defines the constrained entropies Sx .
The sequence of mesostates visited by a trajectory ξ (t) defines a coarse-grained
trajectory x(t). Due to the large number of microstates underlying one mesostate
the dynamics of x(t) becomes effectively stochastic even if, for a closed system, the
dynamics of ξ (t) is a fully deterministic solution of Hamilton’s equations. In order
to describe the dynamics of the coarse grained trajectory using stochastic methods,
a further assumption is necessary. Since the knowledge of the sequence of visited
mesostates allows one in principle to narrow down the set of possible microstates ξ (t),
the probability of transitions from one mesostate x to another one y depends a priori on
the whole history of previously visited mesostates. However, we will assume that within
a mesostate the distribution of microstates and states of the heat bath relax into a local
equilibrium distribution on a time scale that is much smaller than the typical time scale
for transitions between mesostates. Such an assumption is justified if the mesostates are
chosen such that they are separated by sufficiently high potential or entropic barriers. As
a consequence, we canmodel the dynamics of x(t) as a “memoryless” stochastic process,
i.e., conditional probabilities for trajectories with x(ti) = xi at times t1 > t2 > . . . satisfy
the Markov property [12]
p(x1, t1 |x2, t2; x3, t3; . . .) = p(x1, t1 |x2, t2) = p(x1, t1 − t2 |x2, 0), (2.5)
where the second equality reflects the time-homogeneity of the process, which is ulti-
mately a consequence of the explicit time-independence of the Hamiltonian H (ξ). The
propagator p(y, t |x, 0) thus fully characterizes the dynamics of the mesostates, however,
this characterization is not yet minimal. Since the propagator for a large time interval
can be calculated from the propagators for smaller time intervals via the consistency
24
2.1 Markovian dynamics in closed systems
condition, known as Chapman-Kolmogorov equation,
p(y, t |x, 0) =
∑
x′
p(x, t |x′, t′)p(x′, t′|y, 0) (2.6)
with arbitrary 0 < t′ < t, we can focus on the transition rates kxy that are defined from
the propagator in the small t limit as
kxy ≡ lim
t→0
1
t
p(y, t |x, 0). (2.7)
Throughout this Thesis, we denote in the index of transition rates first the old state and
then the new state. With this definition, Eq. (2.6) can be written in a differential form,
leading to the so-called Master equation
∂tpx (t) =
∑
y
[py (t)kyx − px (t)kxy] (2.8)
for the temporal evolution of a probability distribution px (t) on the set of mesostates
{x}. Assuming that the full system is ergodic, the dynamics (2.8) transforms for large t
any initial distribution into the equilibrium distribution peqx in Eq. (2.4). Its stationarity
requires
0 =
∑
y
[peqy kyx − peqx kxy]. (2.9)
Considering Eq. (2.3) for small times t, we see that every term of the sum in Eq. (2.9)
vanishes individually, which can be expressed as the detailed balance condition
peqy kyx = p
eq
x kxy . (2.10)
Inserting the microcanonical distribution (2.4) for peqx leads to
kxy
kyx
= eSy−Sx . (2.11)
Thus, the mere existence of an equilibrium distribution, as postulated by statistical
mechanics, yields a thermodynamic constraint on the transition rates kxy, that is itself
independent of the actual probability distribution.
The main concern of this Thesis is on systems that are not in an equilibrium state. For
a closed system as described above, a non-equilibrium state is any distribution px that
does not satisfy Eq. (2.10), leading to non-zero probability currents
jxy ≡ pxkxy − pykyx, (2.12)
25
2 Stochastic thermodynamics on discrete state spaces
while the transition rates still satisfy the constraint (2.11). Such a distribution may arise
in various ways.
First, the distribution may represent the knowledge of an external observer who has
performed a measurement on the system in equilibrium. Without further measurements
ormanipulations of the system, the distribution ofmesostates conditioned on the outcome
of the measurement is governed by the master equation (2.8) and relaxes back into the
equilibrium distribution.
Second, the system may be prepared in a non-equilibrium state. For example, such
a preparation can be achieved by letting the system equilibrate for some Hamiltonian
function which is then suddenly changed. After this quench, the system is in a transient
non-equilibrium state that relaxes towards the Boltzmann distribution corresponding to
the new Hamiltonian.
Third, the Hamiltonian function H (ξ, λ) with a control parameter λ can be subject to
a time-dependent protocol λ(t) that is performed by an external agent. Provided that the
decomposition into time-scale separated mesostates is still independent of λ, one obtains
time-dependent entropies Sx (λ) and S(λ) in Eq. (2.4) and subsequently time-dependent
transition rates in Eq. (2.11) and the master equation (2.8). If the external protocol
modulates λ(t) in a time-periodic fashion, the dynamics of the master equation will
lead to an equally time-periodic probability distribution px (t), as realized in so-called
stochastic pumps, see, e.g., [13, 14]. If the control parameter λ(t) itself undergoes an
athermal, time-homogeneous Markov process, as realized experimentally for example
in [15, 16], the dynamics in the joint state space of x and λ becomes a non-equilibrium
stationary state (NESS), which has a unique stationary distribution with non-vanishing
probability currents.
2.2 Markovian dynamics in open systems
Without external manipulation, closed systems (or systems in contact with a single
heat bath) can only be in transient non-equilibrium states that are bound to relax into
equilibrium after some equilibration time. In contrast, open systems are often observed
to be in a steady state and yet maintain steady fluxes, which is only possible out of
equilibrium. Examples include an electric circuit with a bulb that steadily produces light
or biological systems like chloroplasts producing organic molecules. Common to these
systems is the contact to various reservoirs providing mechanical work, heat, chemicals,
or electrons.
Following the ideas put forward in Refs. [17–20], we deduce the (thermo-)dynamics
of such an open system by embedding it along with all the reservoirs in a large closed
system. The reservoirs (labeled by r) are fully characterized by their entropies Sr , other
extensive quantities Xr , and an equation of state that gives the energy Er = Er (Sr,Xr ).
Associated with these extensive observables are the temperature Tr ≡ β−1r ≡ ∂Er/∂Sr
26
2.2 Markovian dynamics in open systems
and the generalized force Kr ≡ ∂Er/∂Xr . For instance, for a heat and particle reservoir
one would have the number of particles as extensive quantity Xr = N and the chemical
potential as intensive quantity Kr = µ. A pure work reservoir could be a solid lifted
to height Xr = h against the gravitational force mg. Such pure work reservoirs with a
single degree of freedom are formally assigned zero entropy and temperature. Likewise,
pure heat reservoirs are assigned zero Xr and Kr . For reservoirs that provide more than
one type of resource, the quantities Xr and Kr are understood as vectors with implied
dot product multiplication.
The reservoirs are weakly coupled to the open system, with which they exchange heat,
work, particles or other resources. Thus, in the long run, the full system will relax into
an equilibrium state characterized by all temperatures Tr and all forces Kr becoming
pairwise equal. However, we assume that the reservoirs are sufficiently large, so that
that there is a clear time scale separation between this slow relaxation process and the
time scale relevant for the open system. In particular, the intensive quantities Tr and Kr
can be assumed to be constant on the time scale of interest. Under these conditions, the
open system can effectively be described by a dynamics that leads locally to a NESS.
Mesostates of the full system can now be identified as x = (i, {Er }, {Xr }), where i
denotes a mesostate of the open system alone. The constrained entropy (2.4) of such a
state is then composed of an intrinsic entropy Si and the entropy of the reservoirs,
Sx = Si +
∑
r
Sr (Er,Xr ). (2.13)
We assume Markovian transitions with rate kxy between two states
x = (i, {Er }, {Xr }) → y = ( j, {Er + ∆Er }, {Xr + ∆Xr }), (2.14)
which requires a sufficiently fast local equilibration of the degrees of freedom of both
the open system and the reservoirs. The detailed balance condition (2.11) now reads
with the entropies (2.13)
ln
(
kxy/kyx
)
= Sj − Si +
∑
r
(βr ∆Er − βrKr ∆Xr ). (2.15)
Since the reservoirs are assumed to be large, the rates kxy will be independent of the
absolute values of Er and Xr during the relevant observation times. This fact allows us
to reduce the state space necessary for the description from the mesostates {x} of the
full system to only the mesostates {i} of the open system, while still keeping track of the
changes occurring in the reservoirs. In this reduced description, the transition rates kxy
are denoted as kai j , where the upper index
a ≡ ({∆Ear }, {∆Xar }) (2.16)
27
2 Stochastic thermodynamics on discrete state spaces
Figure 2.1: Illustration of the network representation of an open system. If multiple transi-
tions between two states or within a single state are possible (colored arrows), the
corresponding transition rates are labeled with upper indices.
denotes the changes in the reservoirs associated with a particular transition. This way,
we arrive at a network description of the open system, in which each vertex represents
a state i and each directed edge represents a non-zero transition rate kai j , as illustrated in
Fig. 2.1. As a consequence of micro-reversibility, each transition with rate kai j comes in
pair with a reversed transition with rate k a¯ji, where a¯ ≡ ({−∆Ear }, {−∆Xar }). For this pair
of rates, the constraint (2.15) appears as the so-called local detailed balance condition
ln
(
kai j/k
a¯
ji
)
= Sj − Si +
∑
r
(βr ∆Ear − βrKr ∆Xar ). (2.17)
If the changes in the reservoirs are a unique function of the states i and j, we will drop
the upper index in the transition rates for notational convenience.
A further constraint on the rates in Eq. (2.17) is imposed by the conservation of the
energy in the total system, leading for every transition to
E j − Ei +
∑
r
∆Ear = 0, (2.18)
where Ei is the intrinsic energy associated with state i of the open system.
We can now formulate, analogously to Eq. (2.8), the master equation for the distribu-
tion pi (t) for the mesostates of the open system only as
∂tpi (t) =
∑
j,a
[p j (t)kaji − pi (t)kai j] ≡
∑
i,a
jai j (t), (2.19)
which can be regarded as a conservation law for the probability currents jai j (t). Using
the Perron-Frobenius theorem [21], one can prove that in the long-time limit, pi (t) tends
28
2.3 Energetics of open systems
to a stationary distribution psi . It can be obtained as the unique solution of the linear
system of equations
0 =
∑
j,a
[psj k
a
ji − psi kai j] ≡
∑
i,a
ja,si j , (2.20)
which can be solved either directly using linear algebra or using a diagram method for
cycle fluxes [22]. Characteristically for a NESS, the stationary probability currents ja,si j
are typically non-vanishing. What vanishes instead is the sum of all net influx currents
jai j into a state j, which can be regarded as an instance of Kirchhoff’s law.
For isothermal systems with all reservoir temperatures βr ≡ β, the constraint (2.18)
allows one to restate the local detailed balance condition (2.17) as
ln
(
kai j/k
a¯
ji
)
= −β(Fj − Fi +
∑
r
Kr ∆Xar ), (2.21)
where we have defined the intrinsic free energy Fi ≡ Ei − Si/β. In the presence of only
a heat reservoir, the stationary distribution solving Eq. (2.20) is given by the Boltzmann
distribution
peqi ≡ exp
(−βFi)/Z (β), (2.22)
which is normalized by the canonical partition function Z (β). For this equilibrium
distribution, all probability currents vanish. Thus, in order to generate a genuine NESS,
at least one further reservoir that is not in equilibrium with the existing one is required.
2.3 Energetics of open systems
The basic concept of stochastic thermodynamics is to define thermodynamic quanti-
ties like heat, work, entropy, or internal energy not only macroscopically, but also as
trajectory-dependent and hence fluctuating quantities [6, 9].
The internal energy of the open system can straightforwardly be defined as a trajectory-
dependent observable by evaluating the intrinsic energy Ei along the trajectory. Upon
an individual transition i → j of type a, the change of this energy can be expressed with
the conservation law (2.18) as
∆Ei j ≡ E j − Ei = −
∑
r
∆Ear = −
∑
r
(Tr ∆Sar + Kr ∆Xar ). (2.23)
In between the transitions, the internal energy and the reservoir observables stay constant,
since we assume that there is no time-dependent external manipulation of the system or
the reservoirs.
29
2 Stochastic thermodynamics on discrete state spaces
The identification of work and heat in stochastic thermodynamics is highly context-
dependent. In systems that are externally manipulated one typically defines a “protocol
work” along a microscopic trajectory ξ (t) as the change of the Hamiltonian H (ξ, λ)
of the total system for a time-dependent protocol λ(t). For instance, this definition of
work is the one occurring in the Jarzynski relation [23], and it can be identified along
trajectories of mesostates i(t) through appropriate coarse-graining techniques [24]. In
contact with a heat reservoir, heat is identified through the first law of thermodynamics
as the difference between the protocol work and the change of internal energy. Since we
focus in this Thesis on autonomously operating systems, we refer for a discussion of this
definition of work and heat to the literature [6, 9].
In the context of open systems that are time-independently driven by several reser-
voirs, we consider both work and heat as energies transferred to or from the reservoirs.
Tentatively stating the first law for work and heat associated with a transition i → j of
type a as
∆Ei j = −W a −Qa =
∑
r
(−W ar −Qar ) (2.24)
(with the sign convention of work performed by the system and heat dissipated into the
reservoirs), leads to the question of an appropriate splitting of the terms on the right-
hand side of Eq. (2.23) to the amounts of work and heat exchanged with the respective
reservoirs. For the case of a pure heat reservoir (which does not have a variableXr), it is
obvious to attribute the term Qar ≡ Tr ∆Sar fully to the heat. Conversely, for a pure work
reservoir consisting of a single mechanical degree of freedom X one readily identifies
the mechanical workW ar ≡ Kr ∆Xar .
One might be tempted to use the same identification for other types of reservoirs,
e.g., for particle reservoirs that unavoidably act as heat reservoirs as well. However, this
identification can be misleading. For instance, consider a reservoir providing particles
that can have two different states A and B that affect neither the internal energies of the
particles nor their interactions. Now suppose that the open system catalyzes a reaction
A ⇌ B without changing its own state. Such a reaction step alters the configurational
entropy of the reservoir but leaves its energy unaffected. Since this process obviously
involves neither heat nor mechanical work, the Tr ∆Sar cannot be the heat.
In order to arrive at a consistent and unambiguous definition of heat, we consider each
such mixed reservoir as consisting of a finite container of the resource Xr , which is still
large enough for the thermodynamic limit to be applicable, and which is embedded in
a larger, purely thermal heat bath at equal temperature Tr . Denoting the small part of
the reservoir as r′ and the large part as r′′, the total change of energy in the combined
reservoir r upon a transition of type a becomes
∆Ear = ∆Ear ′ + ∆Ear ′′ = Tr ∆Sar ′ + Kr ∆Xar + Tr ∆Sar ′′ . (2.25)
30
2.4 Entropy production
We now define the heat associated with this transition and this reservoir as the energy
Qar ≡ ∆Ear ′′ = Tr ∆Sar ′′ (2.26)
that is dissipated into the heat reservoir r′′. Using the free energy Fr ′ of the reservoir
r′, we find for its change of entropy in the isothermal environment thermostatted by the
reservoir r′′
∂Sr ′
∂Xr
Tr = − ∂
2Fr ′
∂Tr∂Xr
= −∂Kr
∂Tr
. (2.27)
Therefore, the heatQar can be identified without explicit reference to the formal splitting
of the reservoir as
Qar = ∆Ear +
(
Tr
∂Kr
∂Tr
− Kr
)
∆Xar . (2.28)
For the example considered above with the reservoir containing two species with particle
numbersX = (NA, NB), the chemical potentials derived from the configurational entropy
are K = (µA, µB) = (T ln NA,T ln NB). A transition does not change the reservoir
energy and the term in parentheses in Eq. (2.28) vanishes, therefore the associated heat
Qa vanishes as expected.
Mechanical work is a well defined concept only for athermal systems with one or a
few controllable mechanical degrees of freedom. Using the then obvious definitions
W ar = Kr ∆Xar andW a =
∑
r W ar , the first law of thermodynamics for a transition i → j
of type a can be stated generally as
∆Ei j +
∑
r ′
∆Ear ′ = W a −Qa . (2.29)
Here, the left-hand side of this equation then subsumes the internal energies of the open
system and of all finite parts of combined reservoirs.
Occasionally, we will also consider non-mechanical types of work, in particular a
“chemical work” that is associated with the change of free energy ∆µ in a particle
reservoir. It should be noted, however, that this type of work involves also a change of
entropy and should therefore not be interpreted in the sense of the first law (2.29).
2.4 Entropy production
For the change of entropy along a trajectory, one can identify three contributions. First,
there is the intrinsic entropy Si(t) of the current state i, as introduced in Eq. (2.13). As
the intrinsic energy, this entropy is constant while the system remains in a mesostate
31
2 Stochastic thermodynamics on discrete state spaces
and changes discontinuously with increment Sj − Si upon a transition i → j. A sec-
ond contribution is the stochastic entropy − ln[pi (t)], where pi (t) is a solution of the
Master equation (2.19) for a given initial distribution evaluated for the current state
i [25]. This entropy is defined such that its ensemble average matches the Gibbs entropy
−∑i pi (t) ln pi (t) of the distribution pi. The intrinsic and the stochastic contributions to
the entropy can be subsumed as the system entropy
Ssys(t) ≡ Si − ln[pi (t)] (2.30)
while the system is in state i. The third contribution is the entropy production S(t) in
the reservoirs. It changes upon a transition i → j of type a by
∆Sa =
∑
r
∆Sar =
∑
r
(βr ∆Ear − βrKr ∆Xar ) = ln
(
kai j/k
a¯
ji
)
− (Sj − Si), (2.31)
where we have used the local detailed balance condition (2.17). Provided that there are
only pure work and pure heat reservoirs, this change of entropy is connected to the heat
Qa =
∑
r ∆Sar /βr exchanged with the reservoirs.
The total entropy production follows as the sum
Stot(t) ≡ Ssys(t) + S(t) (2.32)
of the system entropy and the entropy of the reservoirs. Its change upon a transition
i → j of type a at time t is
∆Satot = ∆Sa + ln
pi (t)
p j (t)
= ln
pi (t)kai j
p j (t)k a¯ji
. (2.33)
If the probability distribution pi (t) is not yet the stationary distribution, Stot(t) changes
also continuously in between transitions due to the contribution from the stochastic
entropy. Importantly, the total entropy production can be identified solely through the
transition rates and the corresponding solution of the Master equation and is thus inde-
pendent of the identification of heat and work. Eq. (2.33) could therefore be regarded as
a premise that defines entropy production for any Markovian jump process. This “math-
ematical” entropy production becomes the physical one through the thermodynamic
consistency imposed by local detailed balance, provided that the Markovian model fol-
lows the principles detailed in Secs. 2.1 and 2.2. In particular, at times in between
transitions, the reservoirs must not exchange (or “leak”) energy or other resources, ex-
cept for the small fluctuations with zero mean of the local equilibrium associated with a
mesostate.
32
2.5 Time extensivity and time reversal
2.5 Time extensivity and time reversal
Building on the concepts developed above for open systems in contact with reservoirs,
we consider a trajectory as a sequence of transitions between mesostates. At times
{tn}, the trajectory jumps from state i−n to state i+n with the concomitant changes an =
({∆Eanr }, {∆Xanr }) in the reservoirs. In order to keep the notation simple, we assume
that these changes in a transition i → j are a unique functions ∆Ei jr , ∆Xi jr , allowing
us to replace the upper index a by i j and a¯ by ji. Under this assumption, trajectories
can simply be referred to as functions i(τ), where τ parameterizes the time. The
generalization to trajectories in networks with several transition pathways between two
states labeled additionally by the sequence of indices an is obvious.
In total, we identify two types of observables in stochastic thermodynamics. Observ-
ables like the internal energy, intrinsic entropy or stochastic entropy are state functions
depending only on the mesostate and possibly explicitly on time. Their change along
a stochastic trajectory i(τ) with 0 ≤ τ ≤ t can therefore be stated using only boundary
terms, e.g., the change of system entropy is given by
∆Ssys[i(τ)] = Si(t) − Si(0) − ln pi(t) (t)pi(0) (0) . (2.34)
In the steady state, these observables fluctuate around an average value and do not scale
with time. In contrast, observables like heat, work, reservoir entropy and the exchange
of energy or other resources with an individual reservoir are typically time-extensive
observables (see also [26]). They depend not only on the boundary but on the whole
course of a trajectory, as they are accumulated with every transition. The total (or
“integrated”) work along a trajectory is given by the sum over all transitions
W [i(τ)] =
∑
n
W i
+
ni
−
n (2.35)
and likewise for the other time-extensive observables defined through their respective
changes upon a transition. The average rate of change of such an observable in the steady
state is given by⟨
W˙ (t)
⟩
=
∑
i j
psi ki jW
i j =
∑
i> j
jsi jW
i j, (2.36)
where psi ki j gives the average number of transitions from i to j per unit time. Using
the antisymmetry of W i j , this average can also be represented through the stationary
currents jsi j and a sum over unique pairs of states (indicated by i > j, such that pairs are
not counted twice). Throughout this Thesis, we use the notation ⟨. . .⟩ for averages over
the ensemble of trajectories under consideration, in this case the ensemble of possible
trajectories in the steady state.
33
2 Stochastic thermodynamics on discrete state spaces
The total entropy production (2.32) along a trajectory is given by
Stot[i(τ)] = − ln pi(t) (t)pi(0) (0) +
∑
n
ln
ki+ni−n
ki−ni+n
. (2.37)
Since the contributions from the intrinsic and stochastic entropy are not time-extensive,
the rate of entropy production in the steady state is the same for the total entropy and the
reservoir entropy, leading to [27]
σ = ⟨S˙(t)⟩ = ⟨S˙tot(t)⟩ =
∑
i> j
jsi j ln
ki j
k ji
=
∑
i> j
jsi j ln
psi ki j
psj k ji
≥ 0. (2.38)
The non-negativity of σ follows from the non-negativity of the individual terms in the
sum of the last equality with the stationary current jsi j = p
s
i ki j − psj k ji. Using Eq. (2.31),
the rate of entropy production can be written as a sum over contributions from the
individual reservoirs
σ =
∑
r
βr (⟨E˙r⟩ − Kr⟨X˙r⟩) (2.39)
with the rates of energy transfer ⟨E˙r⟩ and the turnover rates of the resources ⟨X˙r⟩.
Crucially, the observables we have discussed so far are odd under time-reversal, i.e.,
the sign of an increment is flipped for the opposite transition, e.g., for the increment
of the work, W i j = −W ji must hold. Expressed in a time-integrated fashion, the time
antisymmetry reads
W [i(τ)] = −W [i˜(τ)], (2.40)
where we denote the time-reversed counterpart of a trajectory of length t as
i˜(τ) ≡ i(t − τ). (2.41)
Analogous relations hold for the heat, the integrated changes in the reservoirs and
essentially all intrinsic observables of the state of the open system. The only exception
is the stochastic entropy, which has a non-symmetric explicit time dependence when the
system is not in the steady state.
Three other types of time-extensive observables are conceivable and have indeed
been considered is various contexts. First, one can define time-symmetric observables
like the “dynamical activity”, “traffic” or “frenecy” that count a number of transitions
irrespective of the direction, see, e.g., [28, 29]. A second time-symmetric observable is
the time spent in a state or set of states [30, 31]. Third, a “directed current” or “flow”
counting transitions between states only in a specific direction is neither odd nor even
under time reversal. Despite their technical relevance for Markov processes in general,
such types of observables play a subordinate role in stochastic thermodynamics, since
they have no physical counterpart in classical thermodynamics.
34
2.6 Fluctuation relations
2.6 Fluctuation relations
The central role that entropy production plays throughout stochastic thermodynamics is
largely due to its quantification of the concept of irreversibility in a series of prominent
results that are known as fluctuation relations [6].
A straightforward derivation of these results for systems in a steady state starts with the
consideration of path weights. The statistical weight of a trajectory i(τ) underMarkovian
dynamicswith transition rates ki j andwith the initial state from the stationary distribution
psi is given by
p[i(τ)] = psi(0) *,
N∏
n=1
exp
[
−(τn − τn−1)ri−n
]
ki−ni+n+- exp
[
−(t − τN )ri+N
]
, (2.42)
for a trajectory of length t performing jumps at times {τ1, . . . , τ2}. We use the same
notation for the visited states as in Sec. 2.5, set τ0 ≡ 0, and define the exit rate
ri ≡
∑
j
ki j . (2.43)
This exit rate describes in Eq. (2.42) the exponential decay of the probability to stay
in an individual state. The full path weight is made up of the probability of the initial
state psi(0) drawn from the steady state and the product of the probabilities for the times
spent in states and the transition rates along the path. For the time reversed trajectory
i˜(τ) = i(t − τ), the time spent in the individual states is the same as for i(τ). This
contribution therefore cancels in the ratio of weights of the forward and time-reversed
path, leading to
ln
p[i(τ)]
p[i˜(τ)]
= ln
⎡⎢⎢⎢⎢⎣
psi(0)
psi(t)
N∏
n=1
ki−ni+n
ki+ni−n
⎤⎥⎥⎥⎥⎦ = ∆Stot[i(τ)], (2.44)
where we identify the total entropy production of the forward trajectory from Eq. (2.37)
(for the case of a unique type a of a transition considered here). Hence, the total
entropy production of a trajectory quantifies its degree of irreversibility by comparing
its likelihood to that of the time-reversed trajectory. With this property, the integral
fluctuation theorem for the total entropy production [25] is readily derived as
⟨
e−∆Stot
⟩
=
∑
i(τ)
p[i(τ)]e−∆Stot[i(τ)] =
∑
i(τ)
p[i(τ)]
p[i˜(τ)]
p[i(τ)]
=
∑
i˜(τ)
p[i˜(τ)] = 1. (2.45)
Here, we have represented the ensemble average using a path integral, denoted as a
sum over trajectories i(τ) weighted with the path weight (2.42). Since forward and
35
2 Stochastic thermodynamics on discrete state spaces
time-reversed trajectories must have the same integration measure in the path integral,
we can switch to an integration over time-reversed trajectories. The normalization of
the path weights finally allows for a derivation of Eq. (2.45) without need for an explicit
specification of the measure underlying the path integrals. While we have focused on
the case of systems in a steady state, the integral fluctuation theorem (2.45) and variants
thereof (e.g., the Jarzynski relation [23]) can be derived along similar lines for time-
dependently driven systems and relaxations from non-stationary distributions [6, 25].
Since the exponential is a convex function, one can apply Jensen’s inequality to
Eq. (2.45), which yields the positivity of entropy production
⟨∆Stot⟩ = σt ≥ 0 (2.46)
as required by the second law of thermodynamics (see also Eq. (2.38)). In this sense, the
fluctuation theorem (2.45) can be regarded as a refinement of the second law, replacing
the inequality by an equality.
Other fluctuation relations relevant for systems in the steady can be derived from
a “master fluctuation theorem”, as has been discussed in a more general setting in
Ref. [6]. We consider an arbitrary time-antisymmetric and possibly vectorial functional
X[i(τ)] = −X[i˜(τ)] of the trajectory and an arbitrary function Γ (X ) thereof. For the
average of this function we then derive the relation
⟨Γ (X )⟩ =
∑
i(τ)
p[i(τ)] Γ (X[i(τ)])
=
∑
i(τ)
p[i˜(τ)] e−∆Stot[i˜(τ)] Γ (−X[i˜(τ)]) =
⟨
e−∆StotΓ (−X )
⟩
. (2.47)
Setting X = ∆Stot and Γ (X ) = δ(X + s) yields the so-called detailed fluctuation theorem
p(−∆Stot)
p(∆Stot)
= e−∆Stot, (2.48)
which expresses that the probability to observe a negative entropy production during
the time t is exponentially less likely than to observe the corresponding positive entropy
production. Similarly, one derives for the joint fluctuations of the entropy production
and of a vector of time-antisymmetric observables X the relation
p(−∆Stot,−X )
p(∆Stot, X )
= e−∆Stot (2.49)
by setting X = (∆Stot, X ) and using a delta-like function Γ (X ).
The fluctuation relations we have discussed so far all hold for arbitrary time intervals t.
In contrast, fluctuation relations that do not directly involve the total entropy production
36
2.6 Fluctuation relations
often hold only asymptotically in the limit of large time intervals. Consider an observable
of the type X = ∆(Stot(τ) + ϕi(τ)), which is the change of the total entropy up to a state
function ϕi (the notation using ∆ implies the difference of the argument evaluated at
τ = t and τ = 0). The general idea is that on long time scales the values of X and ∆Stot
can scale with t, whereas the contribution from ϕi remains of order O(1). The most
prominent example of such an observable is the entropy production ∆S in the reservoirs
(2.33), for which we set ϕi = ln psi − Si. With Γ (X ) = δ(X + x) we then obtain the
relation [6]
p(−X )
p(X )
= e−X ⟨eϕi (t )−ϕi (0) |X⟩, (2.50)
where ⟨. . . |X⟩ denotes an average conditioned on the observation of X . Since this
average does not scale with t, the asymptotic fluctuation relation
ln
p(−X )
p(X )
≈ −X (2.51)
holds in linear order for large t. For X = ∆S, this relation is known as Gallavotti-
Cohen symmetry, which was found numerically [32] and has been proven for chaotic
dynamics [33] and stochastic dynamics [34, 35]. It should be noted, however, that
for values of X which do not scale with t, the relation (2.51) is generally not exact
even in the limit t → ∞. For instance, for one-dimensional periodic systems, the
sub-exponential contributions lead to a periodic modulation or “fine-structure” in the
fluctuation relation [36]. Fluctuation relations of the type (2.51) may also hold for
observables that are unrelated to the entropy production [37].
A similar asymptotic fluctuation relation is the fluctuation theorem for currents [38–
41], which follows for a vector of currents X that determine the total entropy production
as a linear form
∆Stot =A · X + ∆ϕi (2.52)
with a vector of affinities A up to a state function ϕi. Such a linear form can be
identified in Eq. (2.31) with the currents X = ({∆Er }, {∆Xr }) denoting the exchange
with the reservoirs along a trajectory and the affinitiesA = ({βr }, {−βrKr }). Along the
same lines as for Eq. (2.51), one obtains the fluctuation relation
ln
p(−X )
p(X )
≈ −A · X (2.53)
in linear order for large t.
37
2 Stochastic thermodynamics on discrete state spaces
2.7 Unicyclic molecular motor as paradigmatic example
In order to illustrate the concepts of open systems and stochastic thermodynamics intro-
duced above, we discuss simple models of a molecular motor that is fueled by chemical
energy and that produces mechanical work. These models are based on considerations
of thermodynamic consistency, which are also the basis of more elaborate models mo-
tivated by experimental observations, e.g., widely used models for kinesin [42] or the
F1-ATPase [43, 44]. Unicyclic models, such as the ones presented here, have recently
been used to infer design principles for molecular machines that optimize the output
under some natural constraints [45].
We consider an isothermal setup thermostatted by a heat reservoir at temperature T .
Embedded in this reservoir is a particle reservoir for substrate and product molecules.
To be specific, we take ATP as substrate, which is hydrolyzed to ADP and inorganic
phosphate (Pi). The particle reservoir is a dilute solution with concentrations cT, cD, cP
of ATP, ADP, and phosphate, respectively. The corresponding chemical potentials are
then given by
µρ = µ
0
ρ + T ln
(
cρ/c0
)
(2.54)
with ρ ∈ {T,D, P} and where µ0ρ denotes the chemical potential of species ρ at a
concentration of reference c0. In addition, the enzyme is coupled to an athermal work
reservoir containing a single degree of freedom X that can be displaced against a
mechanical force f . In the formalism introduced in Sec. 2.2, we identify the resources
(Xr ) = (NT, ND, NP, X ) with the particle numbers Nρ in the reservoir and the generalized
forces (Kr ) = (µT, µD, µP, f ).
As the first version of a model for the enzyme proper, we consider a set of three
mesostates i ∈ {0,T,D}, comprising the bare enzyme (0), a state with ATP bound to
the enzyme (T), and a state with ADP bound to the enzyme (D), as shown in Fig. 2.2a.
Each of these states has an intrinsic energy Ei, entropy Si and free energy Fi = Ei −TSi.
We allow for six different transitions with rates ki j that are fully identified by the initial
state i and the final state j and that form a cycle in the network of states. For simplicity,
only the transitions between D and 0 come with a transfer of work to or from the work
reservoir. The transition rates for the transition between 0 and T must satisfy the local
detailed balance condition (2.21)
ln(k0T/kT0) = −(FT − F0 − µT)/T, (2.55)
which involves the free energy change of the enzyme and of the particle reservoir
providing one ATP molecule, i.e., ∆N0TT = −1. In the step 0 → T, the energy ∆E0T =
E0 − ET is transferred to the joint heat and particle reservoirs. Subtraction of the
energy that is transferred to the particle reservoir leads according to Eq. (2.28) and
with Eq. (2.54) to the heat Q0T = E0 − ET + µ0T that is dissipated into the isothermal
38
2.7 Unicyclic molecular motor as paradigmatic example
(a) 0
TD
(b) 0
D
(c)
0
Figure 2.2:Models for unicyclic motors. (a) Three-state model, (b) two-state model, (c) one-
state model. Transitions are indicated by black arrows, along which the particle and
work exchanges are illustrated. ATP is colored red, ADP and Pi are colored pink,
and work transfered to the work reservoir is indicated as a lifted weight.
environment. The transition from T to D involves the release of a phosphate molecule.
The local detailed balance condition (2.21) therefore reads
ln(kTD/kDT) = −(FD − FT + µP)/T (2.56)
and we identify the heat QTD = ET − ED − µ0P. Finally, upon a transition between D
and 0, an ADP molecule is released into the reservoir and there is an exchange of work
displacing the mechanical degree of freedom by ∆XD0 ≡ d against the force f . The
local detailed balance condition (2.21) for this transition is
ln(kD0/k0D) = −(F0 − FD + µD + f d)/T (2.57)
and work and heat follow as WT0 = f d and QT0 = ED − E0 − µ0P + f d, respectively.
Thus, for given chemical potentials and intrinsic free energies, we arrive at a model for
which the six transition rates are reduced through the constraints (2.55), (2.56), (2.57)
to a set of three model parameters.
The steady state of this model system is determined through the stationary master
equation (2.20), leading to a stationary cycle current
Js ≡ js0T = jsTD = jsD0 (2.58)
39
2 Stochastic thermodynamics on discrete state spaces
that flows, according to Kirchhoff’s law in Eq. (2.20), through all links. Due to the
coupling of the transitions to the exchange with the reservoirs, this current is connected
to the chemical turnover rates ⟨N˙ρ⟩ and the mean velocity v ≡ ⟨X˙⟩ of the mechanical
degree of freedom as
Js = v/d = −⟨N˙T⟩ = ⟨N˙D⟩ = ⟨N˙P⟩. (2.59)
The entropy production rate follows from Eq. (2.38) as
σ = Js
(
ln
k0T
kT0
+ ln
kTD
kDT
+ ln
kD0
k0D
)
= Js(∆µ − f d)/T ≡ JsA, (2.60)
where we define ∆µ ≡ µT − µD − µP and identify the cycle affinity A as the amount
of entropy that is produced with every completion of the cycle (see Chap. 6 for a more
general treatment of cycle affinities and fluxes). The sign of A therefore determines
the direction of the cycle current Js: For ∆µ > f d, ATP is hydrolyzed and work is
performed by the system, whereas for the case ∆µ < f d, work is performed on the
system resulting in the synthesis of ATP. In the special case A = 0, i.e., ∆µ = f d,
there is a balance between the mechanical and the chemical driving, leading to vanishing
current Js and entropy production σ. Even for non-equilibrium concentrations of the
chemicals in the reservoir, i.e. ∆µ , 0, the total system is then in equilibrium.
The mean work current or mechanical output power of the motor is given by
P ≡ ⟨W˙ ⟩ = Js f d = f v (2.61)
and the rate of heat dissipation reads
⟨Q˙⟩ = Js(µ0T − µ0D − µ0P − f d), (2.62)
which is different from the entropy production rate σ comprising also the entropy
production in the particle reservoir. The rate of change of the free energy of the solution
is described by the chemical work
⟨W˙ chem⟩ = Js∆µ. (2.63)
The three-state model is not yet the simplest model for a unicyclic molecular motor. If
one of the three states is only short-lived, say, the state T with bound ATP, it is possible
to use a description employing only the remaining long-lived mesostates 0 and D, as
shown in Fig. 2.2b. The microstates belonging to the state T are then considered as part
of the barrier between 0 and D and as such it is irrelevant to which of the two mesostates
they are formally attributed. For this model, transitions are no longer uniquely identified
by the states of the open system, it is therefore necessary to distinguish them by the
40
2.7 Unicyclic molecular motor as paradigmatic example
concomitant changes in the reservoir. The transition 0 → D via the former mesostate T
comes with the change
a = (∆Ea = E0 − ED, ∆NaT = −1, ∆NaD = 0, ∆NaP = 1, ∆X a = 0) (2.64)
and the direct transition from D to 0 is associated with
b = (∆Eb = ET − E0 − f d, NbT = 0, ∆NbD = 1, ∆NbP = 0, ∆X b = d), (2.65)
using the notation from Eq. (2.16) with the corresponding reverse transitions a¯ and b¯.
In total, this model has four transition rates ka0D, k
a¯
D0, k
b
D0, k
b¯
0D, which satisfy the local
detailed balance conditions
ln
(
ka0D/k
a¯
D0
)
= −(F0−FT− µT+ µP)/T, ln
(
kbD0/k
b¯
0D
)
= −(FT−F0+ µD+ f d)/T,
(2.66)
leaving two free model parameters. A connection between the rates of the original
three-state model and the simplified two-state model can be made through appropriate
methods of coarse-graining, see, e.g., [44, 46, 47]. The stationary distribution does not
depend on the multiplicity of the transition pathways and is therefore determined by the
combined rates k0D = ka0D + k
b¯
0D and kD0 = k
b
D0 + k
a¯
D0, leading to
ps0 =
kD0
k0D + kD0
, psT =
k0D
k0D + kD0
. (2.67)
The cycle current follows as
Js = ja,s0D = j
b,s
D0 =
kD0kaD0 − k0Dk a¯0D
k0D + kD0
(2.68)
and determines the average entropy, heat, and work currents in the same way as in
Eqs. (2.60)-(2.63).
As the simplest version for a unicyclic molecular motor, for which both mesostates
D and T are shortlived, one can reduce the state space of the enzyme to only a single
mesostate 0, see Fig. 2.2c. For this model, only one type of transition is possible, which
involves the change
c = (∆Ec = − f d, N cT = −1, ∆N cD = 1, ∆N cP = 1, ∆X c = d). (2.69)
The forward and backward rates for this transition then satisfy the detailed balance
relation
ln
(
kc00/k
c¯
00
)
= (∆µ − f d)/T = A. (2.70)
Eqs. (2.60)-(2.63) hold as before with the cycle current Js = kc,s00 − k c¯,s00 .
41
2 Stochastic thermodynamics on discrete state spaces
2.8 Conclusion
In this Chapter we have recalled the physical foundations for the stochastic description
of systems out of equilibrium. Key concept is the detailed balance condition for transi-
tions between mesostates of the total systems, which includes reservoirs. Focusing on
transitions in the system proper leads to the local detailed balance relation (2.17). On
this level of description, the reservoirs provide constant forces that drive the system into
a non-equilibrium steady state. We have illustrated these concepts for minimal models
of molecular motors, which are driven by chemical and mechanical forces.
Stochastic thermodynamics allows one to identify thermodynamic quantities, such as
heat, work, energy and, most notably, the entropy along individual fluctuating trajecto-
ries. The key role that the entropy production plays is due to the fact that it quantifies
the time-reversibility of trajectories. This property is expressed in terms of a set of fluc-
tuation relations, which provide first constraints on the spectrum of current fluctuations.
These constraints will turn out to be crucial for the design of new universal bounds later
on.
42
3 Fluctuations in stationary Markov
processes
The objective of this Chapter is to introduce the general mathematical formalism that
is used to quantify the fluctuations of the thermodynamic observables introduced in
Chap. 2 for non-equilibrium steady states. Two main approaches are discussed: A
direct computation of cumulants using algebraic methods [48, 49] and a description
using large deviation theory [35, 50–53]. While the former is particularly useful for the
exact computation of current fluctuations, the latter provides a powerful tool for proving
universal bounds.
3.1 Cumulant generating function for time-additive
observables
We consider a Markovian network with a discrete and finite state-space {i}. Transitions
occur at continuous times with transition rates ki j ≥ 0 (with kii ≡ 0 for all i), where i
denotes the initial state and j the final state, giving rise to an ensemble of trajectories
i(τ) with τ ≥ 0 on the state space. The time-evolution of the occupation probabilities
pi (t) of the states i at some time τ = t is given by the Master equation (2.19), reading
∂tpi (t) =
∑
j
Li jp j (t) (3.1)
with the matrix
Li j ≡ k ji − δi jri (3.2)
and the exit rate ri ≡ ∑ j ki j as before in Eq. (2.43). The normalized stationary distribu-
tion psi follows as the solution of
0 =
∑
j
Li jpsj . (3.3)
Along each trajectory i(τ), we define an observable X (t) as
X (t) ≡
∑
i
aiTi (t) +
∑
i, j
di jmi j (t), (3.4)
43
3 Fluctuations in stationary Markov processes
where Ti (t) denotes the time the trajectory has spent in state i up to the time t and mi j (t)
counts the number of transitions from i to j that have occurred up to the time t. The
vector ai and the “distance-matrix” di j (with dii ≡ 0 for all i) have real entries that
specify the observable. This type of observable is called “time-additive”, since for a
concatenation of two trajectories (with matching final and initial state), X (t) is the sum
of the values of X evaluated along the individual parts. In the mathematical literature,
such processes are known as “Markov additive processes” [54–56]. The fluctuating
currents associated with heat, work, entropy, etc., as introduced in Chap. 2 form an
important subclass of time-additive observables, for which di j = −d ji and ai = 0 hold
due to time-antisymmetry.
Ultimately, we are interested in the statistics of the fluctuations of the observable X .
Since X (t) itself is not Markovian, we start the calculation for the joint process of X (t)
and i(t). The evolution of the joint probability p(i, X, t) of i and X in an infinitesimal
time interval [t, t + dt] is
p(i, X, t + dt) =
∑
j
p(i, t + dt | j, t) p( j, X − δX ji, t), (3.5)
with the propagator p(i, t + dt | j, t) = δi j + Li jdt for the dynamics of the states {i}. The
change of the observable X during the time interval is δXii = aidt for the case that there
is no transition and δXi j = di j + O (dt) for a transition from i to j. We thus obtain
p(i, X, t + dt) = (1 − ridt)p(i, X − aidt, t) +
∑
i, j
k jidt p( j, X − d ji, t)
= p(i, X, t) + dt
[
− (ri + ai∂X )p(i, X, t) +
∑
j
k jie−d j i∂X p( j, X, t)
]
,
(3.6)
where the operator exp(d∂X ) conveys a Taylor series shifting the argument X by d. With
this, the partial differential equation governing the evolution of the joint probability
becomes
∂tp(i, X, t) =
∑
j
(
Li je−d j i∂X − aiδi j∂X
)
p( j, X, t) ≡
∑
j
Li j (X ) p( j, X, t), (3.7)
in which we identify the operator Li j (X ) acting on both j and X .
Due to the time-independent and additive character of the Markov process for i and
X , the propagator p(i, X ′′, t′′| j, X ′, t′) depends only on the differences X = X ′′ − X ′ and
t = t′′ − t′ and can thus be written as
p(i, X, t | j) = ⟨i |etL(X )δ(X ) | j⟩ , (3.8)
using the matrix exponential and a bra-ket notation |i⟩ for the distribution corresponding
to the system being localized in state i, with the usual rules for multiplication. The
44
3.1 Cumulant generating function for time-additive observables
distribution of X for an ensemble of trajectories of length t sampled from the steady
state follows by using the stationary distribution psj for the initial state and integrating
out the initial and final state. This leads to
p(X, t) =
∑
i, j
p(i, X, t | j)psj =
⟨−etL(X )δ(X )ps⟩ , (3.9)
where ⟨−| ≡ ∑i ⟨i | is the vector containing 1 in every entry and |ps⟩ ≡ ∑ j psj | j⟩. An
explicit calculation of p(X, t) using Eq. (3.9) is typically not possible. Nonetheless,
this equation can be used to calculate the cumulants of the distribution of X . For this
purpose, we define the moment generating function
g(z, t) ≡
⟨
ezX (t)
⟩
=
∫
dX p(X, t) ezX (3.10)
with a typically real parameter z. This function yields the cumulants of the distribution
of X by taking the logarithm and repeated derivation at z = 0,
cn(t) ≡ ∂nz ln g(z, t) |z=0. (3.11)
In particular, themean is given by c1 = ⟨X (t)⟩ and the variance by c2 = ⟨X (t)2⟩−⟨X (t)⟩2.
The normalization of p(X, t) is expressed in c0 = ln g(0) = 0. Using Eq. (3.9) and
integrating by parts, the cumulant generating function follows as
g(z, t) =
⟨−etL(z) ps⟩ (3.12)
with the so-called “tilted” matrix
Li j (z) ≡ Li jed j i z + aiδi j z = k jied j i z + δi j (−ri + zai). (3.13)
In contrast to L(X ), the entries of this matrix are not operators but simple algebraic
functions.
For small time intervals t, the matrix exponential (3.12) can be evaluated in a Taylor
series. For arbitrary t, we instead rely on an expansion of the vectors |ps⟩ and ⟨−| in
eigenvectors of L(z). Since L(z) is generally not Hermitian, one has to distinguish the
right eigenvector |qν (z)⟩ and the left eigenvector ⟨q˜ν (z) | corresponding to an eigenvalue
λν (z) of L(z). A generic spectrum λν (z) as a function of the parameter z is shown
in Fig. 3.1. Only at singular branching points some eigenvalues are degenerate1, for
all other values of z we can exploit the orthogonality of left and right eigenvectors
corresponding to different eigenvalues, yielding
etL(z) =
∑
ν
etλν (z) qν (z)⟩ ⟨q˜ν (z) , (3.14)
1At the singular values of z where some eigenvalues are degenerate, one can use the Jordan canonical
form of L(z) in order to evaluate the matrix exponential. Alternatively, one can evaluate the matrix
exponential for non-singular z and perform a limit for singular z.
45
3 Fluctuations in stationary Markov processes
✲ ✲✁ ✲✂ ✵ ✂ ✁ 
③
✲
✲✁
✲✂
✵
✂
✁
❘
✄
☎
✻
✽
✭
✆
✝
✞
✟
✠
✡
☎
✻
✽
✭
✆
✝
✞
Figure 3.1: Generic spectrum of the matrix L(z) as function of the parameter z. Real parts
of the eigenvalue are plotted as solid lines and the imaginary part as dashed lines.
The system has four states, all transition rates have been drawn at randoma from a
uniform distribution between 0 and 1, and the observable is the current along the link
1 → 2, specified through ai = 0, d12 = 1 = −d21. The Perron-Frobenius eigenvalue
is shown in blue.
aRate matrix L =
*....,
−2.2814 0.1560 0.6011 0.8324
0.9507 −1.0803 0.7081 0.2123
0.7320 0.0581 −2.2791 0.1818
0.5987 0.8662 0.9699 −1.2266
+////-
.
where we use the normalization ⟨q˜µ(z) |qν (z)⟩ = δµν and ⟨−|qν (z)⟩ = 1.
For large time-intervals t, only the eigenvalue with the largest real part dominates the
expansion (3.14). This eigenvalue has several special properties, which are due to the
Perron-Frobenius theorem [21]. This theorem applies to matrices with entirely non-
negative entries. Since the matrix L(z) contains negative entries only in its diagonal,
it can be readily transformed into such a type of matrix by adding a sufficiently large
multiple of the identity matrix. This transformed matrix
L¯i j (z) = Li j (z) + ηδi j (3.15)
with η > maxi[ri − zai] has a spectrum that is shifted by η to λν (z)+ η and has the same
eigenvectors as L(z). The matrix L¯(z) is primitive, meaning that for a sufficiently large
power L¯(z)n all entries
[L¯(z)n]i j = *.,
n∏
k=2
∑
ik
+/- L¯ii2L¯i2i3 . . . L¯in j (3.16)
46
3.1 Cumulant generating function for time-additive observables
become positive. This property becomes obvious by considering the graph spanned by
the non-zero transition rates ki j . We can assume that this graph is connected, otherwise
one would treat two or more unconnected sets of states as independent systems. We
furthermore recall the micro-reversibility introduced in Sec. 2.2, which excludes the
existence of “dead-ends”. As a consequence, any two states i, j are connected by a
finite path associated with positive transition rates and thereby positive entries of L¯(z),
leading to at least one positive term among the non-negative terms of Eq. (3.16).
For such a primitive matrix L¯(z), the Perron-Frobenius theorem states that the eigen-
value ϱwith the largest absolute value (the so-called spectral radius) is real and positive.
We therefore identify λ(z) = ϱ − η as the eigenvalue with the largest real part of L(z),
for which we drop the index ν. Moreover, this eigenvalue is non-degenerate and the
corresponding right and left eigenvectors are positive for all entries qi (z) = ⟨i |q(z)⟩ and
q˜i (z) = ⟨q˜(z) |i⟩. The eigenvectors to all other eigenvalues each have both positive and
negative entries.
For the special case z = 0, the matrix L(0) = L becomes the generator of the time
evolution of themaster equation (3.1)with eigenvalue λ(0) = 0 and eigenvectors ⟨q˜(0) | =
⟨−| and |q(0)⟩ = |ps⟩. The expansion (3.14) for large t then reads exp(tL) ≈ |ps⟩ ⟨−|
with exponentially decaying corrections and transforms any positive initial distribution
into the stationary one in the long-time limit. For arbitrary z and large t, the matrix
exponential (3.14) becomes dominated by
etL(z) ∼ etλ(z) |q(z)⟩ ⟨q˜(z) | , (3.17)
such that the generating function (3.10) reads
ln g(z, t) ≈ tλ(z) + ln ⟨q˜(z)ps⟩ , (3.18)
with exponentially decaying corrections. The leading order contribution to g(z, t) is
called the “scaled cumulant generating function”
λ(z) = lim
t→∞
1
t
ln g(z, t) = lim
t→∞
1
t
ln
⟨
eXz
⟩
, (3.19)
which is equal to the Perron-Frobenius eigenvalue of L(z). Along with the scaling of
g(z, t), all cumulants of X scale linearly for large t as well and can be rescaled as
Cn ≡ lim
t→∞
1
t
cn(t) = ∂nz λ(z) |z=0. (3.20)
In particular, we have the mean rate of change Js ≡ C1 and the effective diffusion
coefficient
D ≡ lim
t→∞
1
2t
(
⟨X (t)2⟩ − ⟨X (t)⟩2
)
= C2/2. (3.21)
47
3 Fluctuations in stationary Markov processes
When taking the limit t → ∞, contributions to the observable X (t) that do not scale
with time become irrelevant for its fluctuations. In order to make this fact explicit,
consider the time-additive observable X¯ (t) = X (t) + ϕi(t) with a state-function ϕi and
accordingly the distance matrix d¯i j = di j + ϕ j − ϕi. The corresponding tilted matrix
(3.13) can then be written as
L¯(z) = V L(z) V−1 (3.22)
with the diagonal matrix Vi j ≡ exp(ϕiz)δi j , which has the same spectrum as L(z).
Hence, λ(z) and all cumulants Cn must be pairwise equal for X (t) and X¯ (t), thus
forming a class of equivalent observables. For instance, such a relation holds for the
total entropy production (2.33) and the entropy production in the reservoirs (2.31).
However, in sub-leading order in t, the generating functions within such a class of
observables differ due to the differences in the eigenvectors.
The formalism we have discussed so far is also applicable to Markov processes
with several transition pathways with transition rates labeled by kai j , as introduced in
Sec. 2.2. The additive observable X (t) is then specified with a generalized distance
matrix dai j . Along the same lines as above, the scaled cumulant generating function for
the fluctuations of X (t) is derived as the Perron-Frobenius eigenvalue of the tilted matrix
Li j (z) ≡
∑
a
kajie
zdaji + δi j (−ri + aiz) (3.23)
with the exit rates ri ≡ ∑a, j kai j .
As a brief illustration, we revisit the simple motor models from Sec. 2.7 and calculate
the scaled cumulant generating function for the displacement of the mechanical degree
of freedom X (t). The one-state model is described by two rates kc00 ≡ k+ and k c¯00 ≡ k−,
which satisfy the local detailed balance condition (2.66) reading ln
(
k+/k−
)
= A. The
observable of interest is specified by dc00 = −d c¯00 = 1. This setup is equivalent to the
asymmetric random walk X (t) along a uniform one-dimensional track with step size
d = 1. The tilted matrix (3.23) has in this case only one entry which is thus equal to its
eigenvalue
λ(z) = L00(z) = k+ez + k−e−z − k+ − k− = 2
√
k+k−
[
cosh
(
z +
A
2
)
− cosh
(A
2
)]
.
(3.24)
The rate of change and the diffusion coefficient follow as Js = λ′′(0) = k+ − k− and
2D = λ′(0) = k+ − k−, respectively.
The two-state model with rates ka0D ≡ k+, k a¯D0 ≡ k−, kbD0 ≡ w+ and k a¯D0 ≡ k−
corresponds to an asymmetric random walk on a one-dimensional lattice with two
48
3.2 Calculation of low-order cumulants
alternating states with periodicity d = 1 and accordingly alternating rates. Here, the
tilted matrix reads
L(z) =
( −k+ − w− k− + w+ez
k+ + w−e−z −k− − w+
)
(3.25)
and has the Perron-Frobenius eigenvalue
λ(z) = −Σ/2 +
√
(Σ/2)2 + k+w+(ez − 1) + k−w−(e−dz − 1) (3.26)
with Σ ≡ k+ + k− + w+ + w−. The rate of change becomes, in accordance with
Eq. (2.68), Js = λ′(0) = (k+w+ − k−w−)/Σ and the diffusion coefficient is 2D =
λ′′(0) = [k+w+ + k−w− − 2(Js)2]/Σ .
3.2 Calculation of low-order cumulants
An analytic calculation of the scaled cumulant generating function as eigenvalue of the
tilted matrix L(z) is possible for less than three states or particularly sparse transition
matrices. In most other cases, one has to resort to a numerical computation of the
eigenvalues in order to obtain the full generating function. Often one is interested
only in low-order cumulants Cn of the fluctuations of an additive observable X (t), in
particular the rate of change Js = C1 and the diffusion coefficient D = C2/2. However,
a numerical calculation of the derivatives of the generating function is typically prone
to discretization errors and round-off errors. Here, we discuss two methods that allow
for an exact computation of the low-order cumulants, avoiding the direct evaluation of
eigenvalues.
Koza method. The method devised by Koza [48] starts by considering the character-
istic polynomial
χ(y, z) ≡ det(L(z) − y1) (3.27)
of the tiltedmatrix, with the identitymatrix 1. This functionmust be zerowhen evaluated
for the eigenvalue λ(z), i.e.,
χ(λ(z), z) = 0. (3.28)
The characteristic polynomial is expanded in the power series
χ(y, z) =
∞∑
n,m=0
bnmynzm (3.29)
49
3 Fluctuations in stationary Markov processes
with coefficients bnm = n!m!∂ny ∂mz χ(y, z) |y,z=0. The power series of λ(z) is by definition
λ(z) =
∞∑
k=1
Ck zk/k!. (3.30)
Plugging these two series into Eq. (3.28) leads to
0 =
∞∑
n,m=0
bnmzm(C1z + C2z2/2 + . . .)n
= b00 + (b01 + b10C1)z + (b02 + b10C2/2 + b20C21 + b11C1)z
2 + O(z3). (3.31)
Collecting equal powers in z allows for a recursive determination of the cumulants Ck as
a function of the coefficients bnm. In particular, one obtains for the two lowest cumulants
Js = C1 = −b01/b10 (3.32)
and
D = C2/2 = −(b02 + b20C21 + b11C1)/b10. (3.33)
The advantage of this method is that it yields analytic expressions for Js and D with-
out needing the stationary distribution or other eigenvectors. However, with increasing
number of states, these expressions soon become very lengthy. For a numerical evalu-
ation of the cumulants, the Koza method is not suitable, since the determination of the
coefficients bnm requires again error-prone numerical derivatives.
Perturbative method. A second method for the calculation of low-order cumulants
uses perturbation theory for a recursive determination of the coefficients of the power
series for λ(z). This method has been used in applied contexts [57] and has been
discussed in generality in Ref. [49].
We use the expansion
L(z) = L + zL′ + z2L′′/2 + O(z3), (3.34)
where L = L(0) is the Markov transition matrix (3.1) and the derivatives of L(z) at
z = 0 are L′i j = k jid ji + aiδi j and L′′i j = k jid2ji, and so forth for higher derivatives. The
eigenvalue is expanded as in Eq. (3.30), likewise we write for the right eigenvector
|q(z)⟩ = ps⟩ + z q′⟩ + z2 q′′⟩ /2 + O(z3), (3.35)
50
3.3 Large deviation functions
wherewe have used that |q(0)⟩ = |ps⟩. The normalization condition ⟨−|q(z)⟩ is respected
by requiring ⟨−|q(n)⟩ = 0 for all n ≥ 1. Plugging these power series into the eigenvalue
equation L(z) |q(z)⟩ = λ(z) |q(z)⟩ leads to a recursive scheme for the calculation of the
coefficients Cn and |q(n)⟩. The zeroth order gives L |ps⟩ = 0, which is transformed into
a solvable system of equations for |ps⟩ by adding the normalization condition ⟨−|ps⟩. In
linear order, one gets
L′ ps⟩ + L q′⟩ = C1 ps⟩ . (3.36)
Multiplication from the left with ⟨−| makes the term with ⟨−| L = 0 vanish, leading to
Js = C1 =
⟨−L′ps⟩ =∑
i j
d jik jipsj +
∑
i
psi ai, (3.37)
which we have already intuitively used to calculate rates of change in Sec. 2.5. Next, the
vector |q′⟩ is determined through the linear set of equations
L q′⟩ = (C1 − L′) ps⟩ . (3.38)
The matrix L with eigenvalue 0 is not invertible. Nonetheless, Eq. (3.38) can be
transformed into a solvable system of equations by replacing one line with the condition
⟨−|q′⟩ = 0. The second order of the eigenvalue equation gives
1
2
L′′ ps⟩ + L′ q′⟩ + 12L q′′⟩ = 12C2 ps⟩ + C1 q′⟩ . (3.39)
Multiplication with ⟨−| yields the diffusion coefficient
2D = C2 =
⟨−L′′ps⟩+ 2 ⟨−L′q′⟩ =∑
i j
[d2jik jip
s
j + 2d jik jiq
′
j]+ 2
∑
i
q′iai . (3.40)
The term with |ps⟩ may be interpreted as the instantaneous quadratic change of X in
the stationary state, whereas the term with |q′⟩ conveys the contribution from temporal
correlations. Higher order cumulants and vectors |q(n)⟩ can subsequently be derived
from the corresponding powers of z in the eigenvalue equation.
The advantage of the perturbative method is that it can directly be implemented
numerically, requiring only the solutions of linear systems of equations. The resulting
cumulants are exact up to machine precision.
3.3 Large deviation functions
The mathematical theory of large deviations provides a powerful toolbox to describe and
compute the probability of atypical events. In this section, we give a brief introduction
51
3 Fluctuations in stationary Markov processes
into this theory with a special focus on additive observables. More in-depth discussions
of the applications of large deviation theory in physics can be found in the review
articles [26, 50].
In general, large deviation theory is concerned with random variables A, whose
probability distribution p(A, N ) depends on a scaling parameter N . Such a distribution
is said to satisfy a large deviation principle if the limit
I (A) ≡ − lim
t→∞
1
t
ln p(A, N ) (3.41)
exists2. The function I (A) is then called “large deviation function” or “rate function”.
We use the short-hand notation
p(A, N ) ∼ e−t I (A) . (3.42)
The variable A may be a scalar, a vector, or a more complex object such as a field or
a trajectory. Scaling parameters that are typically encountered in physics are the time
(then called t), numbers of particles or the reciprocal of a noise intensity. Since the
distribution must be normalized, the large deviation function must be non-negative for
all values of A. Values of A for which I (A) = 0 are identified as “typical values”. Their
probability scales sub-exponentially with N . For atypical values, I (A) > 0 gives the
rate of the exponential decay of the probability. Values of A whose probability decays
even faster or which are entirely impossible are formally assigned I (A) = ∞.
A classic example for a random variable satisfying a large deviation principle is the
ratio A = M/N of outcomes “head” in a coin-toss experiment that is repeated N times.
If the probability for “head” in an individual experiment is p, the probability for having
M such events is
p(M, N ) =
N!
M!(N − M)!p
M (1 − p)N−M . (3.43)
Using Stirling’s approximation of the factorials, the logarithm of this distribution be-
comes
ln p(M, N ) = (M−N ) ln
(
1 − M
N
)
−M ln M
N
+M ln p+ (N−M) ln(1 − p)+O (ln N ) ,
(3.44)
such that the large deviation function for p(A, N ) can be read off directly as
I (A) = (1 − A) ln(1 − A) + A ln(A) − A ln p − (1 − A) ln(1 − p), (3.45)
2In order to define the large deviation principle also for distributions with singular points (e.g., sums of
δ-functions), the distributions in Eq. (3.41) are understood to be integrated in an interval [A, A + δA]
with small but non-zero δA. The limit δA → 0 is then performed after taking the limit N → ∞. In
addition, this convention ensures that the logarithm has always dimensionless arguments.
52
3.3 Large deviation functions
0 0.2 0.4 0.6 0.8 1
A
0
0.2
0.4
0.6
0.8
1
I(A
)
p=0.5
p=0.3
Figure 3.2: Large deviation function (3.45) (solid) for the coin-toss experiment with p = 0.5
(blue) and p = 0.3 (red). The dashed curves are the quadratic approximations for
the typical values of A.
with 0 ln 0 ≡ 0, and is defined for all possible values 0 ≤ A ≤ 1. As shown in Fig. 3.2,
this function has the minimum I (A) = 0 at A = p = ⟨M⟩ /N , i.e., the average value
is also the typical one. Around this typical value, the large deviation function can be
approximated in quadratic order as I (A) ≈ I′′(p)(A − p)2/2 with I′′(p) = 1/[p(1 − p)].
This leads to the Gaussian approximation
I (A) ≈ 1√
2πp(1 − p)/N exp
[
−N (A − p)
2
2p(1 − p)
]
(3.46)
with variance N/I′′(p) = Np(1−p). This distribution is also the result of the central limit
theorem, treatingM as a sumof independent randomvariables that are 1 (with probability
p) or 0. Thus, the large deviation principle may be regarded as a generalization of the
central limit theorem for extreme events.
We now focus on time-additive observables X (t) in aMarkov process, as introduced in
Sec. 3.1. Scaling such a time-extensive observable by time t leads to the time-intensive
observable J (t) ≡ X (t)/t, which expresses the average change of X along an individual
trajectory of length t. The distribution of this observable satisfies a large deviation
principle of the form
p(X, t) ∼ p(J, t) ∼ e−t I (J), (3.47)
53
3 Fluctuations in stationary Markov processes
where we note that the distributions of X and J, p(X, t) = p(J, t)/t, differ only on a
sub-exponential scale. This large deviation function proves consistent with the scaling
of the cumulant generating function (3.19), as can be seen by writing the exponential
average of X as
etλ(z) ∼
⟨
eXz
⟩
=
∫
dJ p(J, t)et Jz ∼
∫
dJ e−t I (J)+t Jz . (3.48)
The integral on the right hand side can be evaluated using a saddle-point approximation
of the integrand, which reduces in the leading exponential order to taking the maximum
of the exponent, leading to
λ(z) = max
J
[Jz − I (J)]. (3.49)
For convex functions I (J), this relation is equivalent to the Legendre transform of I (J),
which reads
λ(z) = J (z)z − I (J (z)) (3.50)
with J (z) determined by z = I′(J).
If I (J) was non-convex, λ(z) would be the Legendre transform of the convex envelope
of I (J) and would have non-analytic kinks at values of z that correspond to the linear
slopes of the convex envelope. However, for Markovian systems with a finite state space,
λ(z) follows as a non-degenerate root of the analytic characteristic polynomial (3.27)
and must therefore be smooth for all z. Thus, the large deviation function must be
convex. Nonetheless, singular points in generating functions are commonly observed in
Markovian systems with infinite state-spaces and are studied in the context of dynamical
phase transitions [26, 28, 58].
For convex I (J), the Legendre transform (3.49) presents an invertible mapping be-
tween I (J) and the likewise convex λ(z). The former can therefore be gained from the
scaled cumulant generating function by inverting the Legendre transform to
I (J) = max
z
[Jz − λ(z)] = Jz(J) − λ(z(J)) (3.51)
with z(J) determined by J = λ′(z), which is the inverse of the mapping J (z) above.
This relation between scaled cumulant generating function and large deviation function,
known as Gärtner-Ellis Theorem, allows one to calculate the large deviation function of
an additive observable from the Perron-Frobenius eigenvalue of the tilted matrix (3.13)
without need for an explicit knowledge of the probability distribution of the observable.
The typical behavior of an additive observable is described by the mean rate of change
Js = λ′(0), therefore z(Js) = 0 and I (Js) = 0, identifying Js as the unique minimum of
54
3.3 Large deviation functions
-10 -5 0
0
5
(z
)
0 2 4 6
0
5
10
15
I(J
)
Figure 3.3: Plot of the scaled cumulant generating function (3.24) (left) and the large deviation
function (3.54) (right) for the asymmetric random walk with k+ = 1 and the affinity
per stepA = 8. Selected pairs of corresponding points of z and J (z) are marked by
symbols.
the large deviation function. The curvature of the large deviation function at this point
is
I′′(Js) =
1
λ′′(0)
=
1
2D
, (3.52)
such that the Gaussian approximation of the distribution p(X, t) becomes
p(X, t) =
1√
4πDt
exp
[
− (X − J
st)2
4Dt
]
, (3.53)
which matches a normal biased diffusion in one dimension with drift velocity Js and
diffusion coefficient D.
For the example of the asymmetric random walk, the Legendre transform of the
generating function (3.24) can be performed analytically [35]. The mapping between J
and z reads here J = λ′(z) = J0 sinh(z − A/2) with J0 ≡ 2
√
k+k− = Js/ sinh(A/2).
Inserting this relation in Eq. (3.51) leads to the large deviation function
I (J) = J
[
arsinh(J/J0) − arsinh(Js/J0)] −√(J0)2 + J2 +√(J0)2 + (Js)2. (3.54)
Plots of this scaled cumulant generating function and large deviation function are shown
in Fig. 3.3. As a typical feature for time-asymmetric observables, the large deviation
function has a high curvature or “kink” around J = 0 [51, 59]. This value of J
corresponds to the minimum of λ(z) at z = −A/2 with a particularly low curvature.
55
3 Fluctuations in stationary Markov processes
If the distribution of the observable of interest obeys an asymptotic fluctuation relation
of the type (2.51), the corresponding large deviation function satisfies
I (−J) = I (J) + J . (3.55)
In the Legendre transformed picture (3.49), this fluctuation relation expresses as the
symmetry
λ(z) = max
J
[zJ − I (J)] = max
J
[zJ + J − I (−J)] = λ(−1 − z) (3.56)
of the scaled cumulant generating function. As an important example, consider the
entropy production (2.31) by setting di j = ln
(
ki j/k ji
)
and ai = 0. Contributions from
the intrinsic entropy, which is a state function that cannot scale with time, do not have
to be taken into account (see also the discussion leading to Eq. (3.22)). Following the
proof of the Gallavotti-Cohen symmetry provided in Ref. [35], the tilted matrix (3.13)
for this observable can be written as
Li j (z) = k ji
(
k ji
ki j
) z
− riδi j = L ji (−1 − z). (3.57)
Thus, not only the Perron-Frobenius eigenvalue, but also the whole spectrum of L(z)
satisfies the symmetry (3.56) around the center at z = −1/2, for which L(z) becomes
symmetric. The left and right eigenvectors exchange their roles upon inflection of z,
qi (z) ∝ q˜i (−1 − z), (3.58)
which holds due to the transposition of L(z) in Eq. (3.57).
Focusing again on general additive observables X (t), we conclude the introduction
to large deviations with a discussion of sub-exponential contributions to probability
distributions [36,60]. Writing the distribution of J (t) as3 p(J, t) = p0(J, t) exp[−t I (J)],
where p0(J, t) does not scale exponentially with t, the saddle-point approximation of the
generating function (3.48) can be refined to
eg(z,t) =
⟨
eXz
⟩
=
∫
dJ p0(J, t)et[Jz−I (J)] ≈
√
2π
t I′′(J∗)
p0(J∗, t)et[J
∗z−I (J∗)] (3.59)
with the saddle-point J∗ ≡ J (z) as above. Comparison to the scaling (3.18) of the
cumulant generating function reveals
p0(J, t) ≈
√
t I′′(J)
2π
⟨−|q(z)⟩ ⟨q˜(z)ps⟩ (3.60)
3In this context, it is again essential to evaluate distributions at first on a finite interval [J, J + δJ] and to
take the limit δJ → 0 after any long-time limit t → ∞. Otherwise, the absolute value of X (t) = t J (t)
may contain information about the initial or finite state of trajectories, leading to a fine-structure on
the sub-exponential scale [36].
56
3.3 Large deviation functions
for any matching pair of J and z = z(J), with an error that decays exponentially.
Using similar methods, it is possible to infer the probability of state i at an intermediate
time τ conditioned on the observation of a large deviation J in a long time interval t,
which we denote as p(i, τ |J, t). The corresponding conditioned exponential average
reads analogously to (3.59)
⟨
eXzδi,i(τ)
⟩
=
∫
dJ p(i, τ |J, t)p(J, t)eJz ≈
√
2π
t I′′(J∗)
p(i, τ |J∗, t)p0(J∗, t)et[J∗z−I (J∗)].
(3.61)
On the other hand, using the methods for the propagator from Sec. 3.1, we calculate⟨
eXzδi,i(τ)
⟩
=
∫
dX1
∫
dX2
∑
i0,i1
e(X1+X2)zp(i1, X2, t − τ |i)p(i, X1, τ |i0)psi0
= ⟨−|e(t−τ)L(z) |i⟩ ⟨ieτL(z) ps⟩
≈ eλ(z)
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
⟨−|q(z)⟩ ⟨q˜(z) |i⟩ ⟨i |ps⟩ , τ = 0
⟨−|q(z)⟩ ⟨q˜(z) |i⟩ ⟨i |q(z)⟩ ⟨q˜(z) |ps⟩ , 0 < τ/t < 1
⟨−|i⟩ ⟨i |q(z)⟩ ⟨q˜(z) |ps⟩ , τ = t
, (3.62)
where we have used Eq. (3.17) for evaluating the matrix exponentials in the limit of long
time intervals. Comparing this to Eqs. (3.61) and (3.60) yields the following picture
for the ensemble of trajectories with J (t) = J for large t [26]: The initial state of these
trajectories is distributed as
p(i, 0|J, t) = ⟨−|q(z)⟩ ⟨q˜(z) |i⟩ ⟨i |p
s⟩
⟨−|q(z)⟩ ⟨q˜(z) |ps⟩ =
q˜i (z)psi∑
i q˜i (z)psi
, (3.63)
with z = z(J) throughout, and the distribution of final states is
p(i, t |J, t) = ⟨−|i⟩ ⟨i |q(z)⟩ ⟨q˜(z) |p
s⟩
⟨−|q(z)⟩ ⟨q˜(z) |ps⟩ =
qi (z)∑
i qi (z)
. (3.64)
For most times τ in between that are sufficiently far form the boundaries, i.e., τ = O (t)
and t − τ = O (t), the distribution of states is
p(i, τ |J, t) = ⟨−|q(z)⟩ ⟨q˜(z) |i⟩ ⟨i |q(z)⟩ ⟨q˜(z) |p
s⟩
⟨−|q(z)⟩ ⟨q˜(z) |ps⟩ = q˜i (z)qi (z). (3.65)
As we have seen in the derivation, the same distributions of i(τ) show up in the ensemble
of trajectories that is re-weighted by the exponential exp(Xz) but which is otherwise
unconstrained. This correspondence of the ensembles of trajectories in the long time
limit is reminiscent of the equivalence of the micro-canonical and canonical ensemble
in the thermodynamic limit [61].
57
3 Fluctuations in stationary Markov processes
3.4 Level 2.5 large deviation theory
So far, we have focused on the large deviation theory for an individual observable,
which is called the “level 1” of large deviation theory. When dealing with the large
deviations of the joint probabilities of several observables, a useful tool is the following
contraction principle. Consider a joint probability p(J1, J2, t) of two observables J1 and
J2 satisfying a large deviation principle of the type p(J1, J2, t) ∼ exp[−t I (J1, J2)] with a
two-dimensional large deviation function I (J1, J2) (the generalization to more than two
variables will be obvious). The distribution of a third, dependent observable defined as
a function K = K (J1, J2) is then obtained for finite times through the integration
p(K, t) =
∫
dJ1
∫
dJ2 p(J1, J2, t) δ[K − K (J1, J2)], (3.66)
which is typically hard to perform analytically. For large times t, only the values of J1
and J2 that lead to the smallest rate of decay in the integrand contribute significantly
to the overall integral. Therefore, the large deviation function for K can be identified
through a constrained minimization as
I (K ) ≡ lim
t→∞
1
t
p(K, t) = min
J1,J2 |K (J1,J2)=K
I (J1, J2), (3.67)
which should be easier to perform than the full integration (3.66).
This contraction principle allows for a complementary approach to the calculation of
the large deviation function I (J) of a single variable. The basic idea is to write J as a
function of several basic observables whose large deviation function has a closed analyt-
ical form. The function I (J) then follows as the result of a constrained minimization of
this “master”-large deviation function. Several levels of such large deviation functions
are known for Markov processes. Higher levels correspond to higher-dimensional sets
of basic observables, allowing for a larger class of observables that is covered through
contraction. However, performing the contraction from higher levels generally becomes
more tedious.
A large deviation functional taking whole trajectories as arguments is commonly
known as “level 3”. Less powerful is the “level 2” large deviation function, which
takes as arguments the fractions of time spent in each of the states. Most relevant
for the analysis of non-equilibrium steady states is the large deviation function of the
intermediate “level 2.5”, which describes the joint fluctuations of the time spent in each
state Ti (t) and the number of transitions mi j (t) between pairs of states [62, 63]. Scaling
these time-additive observables by time leads to the empirical distribution
pi ≡ Ti (t)/t (3.68)
and the empirical flow
µi j ≡ mi j (t)/t, (3.69)
58
3.4 Level 2.5 large deviation theory
which fluctuate for an ensemble of trajectories of fixed length t around the stationary
values psi and µ
s
i j ≡ psi ki j , respectively. The level 2.5 large deviation function can now
be written as I ({pi}, {µi j }). A scaled general time-additive observable J = X (t)/t as
defined in Eq. (3.4) is a linear combination J = J ({pi}, {µi j }) ≡ ∑i j di j µi j +∑i aipi of
these basic observables, such that its (level 1) large deviation function follows through
the contraction principle
I (J) = min
{pi },{µi j }|J ({pi },{µi j })=J
I ({pi}, {µi j }). (3.70)
For a derivation of the closed form of the level 2.5 large deviation function, we follow
loosely the presentations in Refs. [29, 64, 65]. At first, we note that the empirical flows
are not independent. Since the number of jumps into a state and out of it must be
balanced,∑
j
[m ji (t) − mi j (t)] ∈ {−1, 0, 1} (3.71)
must hold for every trajectory and every state i. Accordingly, the empirical flows are
constrained by the Kirchhoff-type law
lim
t→∞
∑
j
[µ ji − µi j] = 0. (3.72)
For values of {µi j } that do not satisfy this constraint, we assign I ({pi}, {µi j }) = ∞.
The importance of the empirical distribution and flow is founded in their determining
the path weight (2.42) as
p[i(t)] = psi(0) exp
⎡⎢⎢⎢⎢⎢⎣−
∑
i
riTi +
∑
i j
mi j ln ki j
⎤⎥⎥⎥⎥⎥⎦ = psi(0) exp
⎡⎢⎢⎢⎢⎢⎣−t
*.,
∑
i
ripi −
∑
i j
µi j ln ki j+/-
⎤⎥⎥⎥⎥⎥⎦ .
(3.73)
A direct marginalization of this path weight in order to obtain the joint distribution
p({pi}, {µi j }, t) is rather tedious, since it requires to specify an integration measure in
path space. Nonetheless, it is possible to compare these distributions and thus the level
2.5 large deviation function to those of a second, auxiliary process with rates {kˆi j }.
Labeling quantities referring to this process by pˆ, Iˆ, etc., one obtains
I ({pi}, {µi j })− Iˆ ({pi}, {µi j }) = − lim
t→∞
1
t
ln
p({pi}, {µi j }, t)
pˆ({pi}, {µi j }, t) =
∑
i
(ri−rˆi)pi−
∑
i j
µi j ln
ki j
kˆi j
.
(3.74)
59
3 Fluctuations in stationary Markov processes
If the set of rates {kˆi j } can be chosen such that {pi} = {pˆsi } and {µi j } = { µˆsi j } are
the typical distribution and flow for the auxiliary process, respectively, one will have
Iˆ ({pi}, {µi j }) = 0, such that Eq. (3.74) yields the large deviation function for the process
of interest. Luckily, even though it is typically computationally hard to determine the
stationary state and flow for a given set of rates, the reverse is performed straightforwardly
by setting kˆi j = µˆsi j/pˆ
s
i . Plugging this into Eq. (3.74) finally yields the level 2.5 large
deviation function
I ({pi}, {µi j }) =
∑
i
*.,ri −
∑
j
µi j
pi
+/- pi−
∑
i j
µi j ln
piki j
µi j
=
∑
i
ripi+
∑
i j
µi j
(
ln
µi j
piki j
− 1
)
(3.75)
An alternative representation of this result is obtained by writing the empirical flow
as µi j = ( ji j + ti j )/2 with the antisymmetric empirical current
ji j ≡ µi j − µ ji = − j ji (3.76)
and the empirical traffic
ti j ≡ µi j + µ ji = t ji . (3.77)
Eq. (3.75) reads with this substitution
I ({pi}, { ji j }, {ti j }) =
∑
i
ripi +
∑
i< j
[
−ti j + ti j + ji j2 ln
ti j + ji j
2piki j
+
ti j − ji j
2
ln
ti j − ji j
2p j k ji
]
.
(3.78)
Kirchhoff’s law (3.72) constrains the currents as∑
j
ji j = 0 (3.79)
for large times, whereas the traffic is a set of positive but otherwise unconstrained
variables.
In stochastic thermodynamics, we are mostly interested in the fluctuations of time-
antisymmetric observables, which are formed as a linear combination of the currents
{ ji j }. It would therefore be desirable to eliminate the traffic {ti j } and the empirical
distribution {pi} from Eq. (3.78) through contraction. The mutual independence of the
traffic variables allows one to perform the minimization for {ti j } term by term, which
leads to the traffic
t∗i j ≡
√
4piki jp j k ji + j2i j (3.80)
60
3.4 Level 2.5 large deviation theory
that minimizes Eq. (3.78) to
I ({pi}, { ji j }) =
∑
i< j
⎡⎢⎢⎢⎢⎢⎢⎣piki j + p j k ji −
√
4piki jp j k ji + j2i j + ji j ln
ji j +
√
4piki jp j k ji + j2i j
2piki j
⎤⎥⎥⎥⎥⎥⎥⎦ .
(3.81)
Using the identity arsinh x = ln[x +
√
x2 + 1] and defining api j ≡ 2
√
piki jp j k ji and
jpi j ≡ piki j − p j k ji leads to a representation of Eq. (3.81) as
I ({pi}, { ji j }) =
∑
i< j
⎡⎢⎢⎢⎢⎣
√
(api j )
2 + ( jpi j )
2 −
√
(api j )
2 + ( ji j )2 + ji j *,arsinh
ji j
api j
− arsinh
jpi j
api j
+-
⎤⎥⎥⎥⎥⎦ ,
(3.82)
which makes its minimum at pi = psi and ji j = j
s
i j = p
s
i ki j − psj k ji evident.
A similar expression applies to systems with several transition pathways between pairs
of states, as introduced in Sec. 2.2. In that case, transitions are additionally labeled by
the upper index a and the sum in Eq. (3.82) is understood to run over every pair of
forward and backward transition once. For the asymmetric random walk with a single
state 0, the empirical density must be p0 = 1 due to normalization and there is a single
independent current J ≡ ja00. The sum in Eq. (3.82) then has only a single term, which
is equal to the result from Eq. (3.54).
The level 2 large deviation function I ({pi}) for the empirical distribution pi only
follows by contraction of Eq. (3.82) over the empirical current { ji j }. In the general
case of a driven system, this contraction cannot be performed analytically, since the
constraint
∑
j ji j = 0 prohibits an individual minimization of the terms in Eq. (3.82).
Nonetheless, for equilibrium systems with rates satisfying ki j/k ji = peqj /p
eq
i , the current
ji j = 0 presents a unique minimum of I ({pi}, { ji j }) for fixed pi. Indeed, the linear term
in the expansion
I ({pi}, {ε ji j }) = I ({pi}, {0}) − ε2
∑
i
ln
pi
peqi
∑
j
ji j + O
(
ε2
)
(3.83)
vanishes for all viable ji j and the convexity of the rate function ensures that this stationary
point is a unique minimum. The corresponding level 2 large deviation function then
follows from Eq. (3.81) as
I ({pi}) = I ({pi}, {0}) =
∑
i< j
(√
piki j −
√
p j k ji
)2
. (3.84)
A further contraction of Eq. (3.82) to I ({ ji j }) is similarly not possible in a closed
analytic form, because of the constraint
∑
i pi = 1. The main benefit of Eq. (3.82) for
61
3 Fluctuations in stationary Markov processes
non-equilibrium systems is therefore its providing upper bounds on the large deviation
function for a time antisymmetric observable. Such a bound follows by choosing an
arbitrary, typically non-optimal, trial distribution {pti } in the contraction principle
I ({ ji j }) = min{pi } I ({pi}, { ji j }) ≤ I ({p
t
i }, { ji j }). (3.85)
Bounds of this type have been introduced in Ref. [8] and will turn out to be crucial for
the proof of several thermodynamic bounds on current fluctuations.
3.5 Continuous state spaces
So far, we have focused on Markovian processes on discrete state spaces. As motivated
in Chapter 2, this description is appropriate for systems with well separated mesostates,
such as enzymes or quantum dots. In this section, we briefly discuss the generalization of
the present formalism to Markovian processes in continuous state spaces. Such a model
applies to systems with a few continuous mesoscopic degrees of freedom, provided
that their dynamics exhibit a clear time-scale separation from the faster dynamics of
the microscopic degrees of freedom. The latter then reach local equilibria for any
constellation of the mesoscopic degrees of freedom.
A paradigmatic example for such a system is the Brownian motion of a colloidal
particle suspended in a fluid that acts as a heat bath with temperature T . A coarse-
grained, stochastic description of this system involves only the position of the colloid,
whereas its velocity and the degrees of freedom of the fluid particles are assumed to be
locally equilibrated.
Here, we focus on the Brownian dynamics of a single, one-dimensional degree of
freedom x in a homogenous environment. More general cases are discussed in the
literature, e.g., in Refs. [6, 52, 53, 66]. We assume that there is a position-dependent
potential V (x) and that a constant mechanical force f is driving the system. In order to
make the connection to the discrete state spaces, we finely discretize the state space as a
lattice with equidistant sites xi = ih with spacing h. We allow for transitions between
neighboring sites xi and xi+1 with transition rates k+i for the forward transition and k
−
i+1
for the backward transition. The local detailed balance condition (2.21)
k+i
k−i+1
= e[ f h+V (xi )−V (xi+1)]/T (3.86)
for these transitions can be satisfied by setting
k+i = k0e
θi [ f h+V (xi )−V (xi+1)]/T and k−i+1 = k0e
−(1−θi )[ f h+V (xi )−V (xi+1)]/T (3.87)
with a rate of reference k0 and arbitrary real numbers θi. Subsequently expanding in h
leads to the current
ji (t) ≡ pi (t)k+i −pi+1(t)k−i+1 =
k0h
T
[pi (t)(g−∂xV (x))− (pi+1(t)−pi (t))/h]+O
(
h2
)
,
62
3.5 Continuous state spaces
(3.88)
which is independent of the parameters θi (provided that they do not scale with h). We
smoothen the discrete probability pi (t) and current ji (t) to the density fields p(x, t) and
j (x, t), such that p(xi, t) = pi (t)/h and j (xi, t) = ji (t), respectively. The continuum
limit exists if k0 scales inverse quadratically in h, leading to
j (x, t) = µ( f − ∂xV (x))p(x, t) − D∂xp(x, t), (3.89)
where we identify the mobility µ ≡ limh→0 k0h2/T and the diffusion constant D ≡
limh→0 k0h2. The Einstein relation D = µT emerges naturally from the local detailed
balance condition (3.86). The conservation of probability, formerly expressed in the
master equation (2.19), becomes in the continuum limit the Fokker-Planck equation
∂tp(x, t) = −∂x j (x, t) = −∂xµF (x)p(x) + D∂2x p(x, t), (3.90)
with the total force F (x) ≡ f − ∂xV (x). For periodic boundary conditions, these
dynamics lead to a steady state with a stationary probability distribution ps(x) and a
stationary current js(x) ≡ js that is independent of x for the one-dimensional system.
An equivalent description of the stochastic dynamics underlying the Fokker-Planck
equation is given by the Langevin equation. Following standard methods [12,67], it can
be read-off from Eq. (3.90) as
x˙(t) = µF (x(t)) + ζ (t) (3.91)
with a Gaussian white noise ζ (t) with zero mean and correlations ⟨ζ (t)ζ (t′)⟩ = 2Dδ(t −
t′).
A time-additive observable X (t) is defined in the discretized model by specifying the
tensors ai and di j in Eq. (3.4). Preparing for the continuum limit, we write ai ≡ a(xi)h
and di,i+1 ≡ d(xi)h with continuous functions a(x) and d(x). Since the total number
of transitions, or “traffic”, diverges and is thus an ill-defined concept for trajectories
on a continuous state-space, we restrict the distance matrix to be antisymmetric, i.e.,
di,i+1 = −di+1,i. After taking the limit h → 0, the dynamics of X (t) is given by the
Langevin equation
X˙ (t) = a(x(t)) + d(x(t)) x˙(t) = a(x(t)) + d(x(t))µF (x(t)) + d(x(t))ζ (t). (3.92)
Due to the possible dependence of d(x) on x, this Langevin equation contains a multi-
plicative noise term, which is to be interpreted in the Stratonovich sense. This way, the
time-antisymmetric character of the d-dependent term is preserved. The mean rate of
change of X (t) in the stationary state is given by
⟨X˙⟩ =
∫
dx [ps(x)a(x) + js(x)d(x)]. (3.93)
63
3 Fluctuations in stationary Markov processes
For instance, the entropy production in the medium (assuming a constant intrinsic
entropy) is described by ai = 0 and the generalized distance matrix di,i+1 = ln
(
k+i /k
−
i+1
)
,
see Eq. (2.31), which becomes d(x) = [ f − ∂xV (x)]/T in the continuum limit. For the
total entropy production (2.32), the term −∂x ln ps(x) is added, leading with Eq. (3.89)
to d(x) = js(x)/Dps(x) [25]. For both definitions, the mean entropy production rate in
the steady state becomes
σ =
∫
dx js(x)[ f − ∂xV (x)]/T = f js/T, (3.94)
reflecting that the mechanical power delivered by the external force, on average f js,
must ultimately be dissipated into the medium.
Eqs. (3.91) and (3.92) present a system of two coupled stochastic differential equations
with fully correlated noise terms. The corresponding Fokker-Planck equation for the
joint probability of x and X is given by [12]
∂tp(x, X, t) = [−∂xµF (x) + a(x)∂X − d(x)µF (x)∂X + D(∂x + d(x)∂X )2]p(x, X, t)
≡ L(X )p(x, X, t), (3.95)
which is analogous to Eq. (3.7) for discrete state spaces. In the picture of the generating
function (3.12), the operator L(X ) acting on both x and X gets transformed to
L(z) ≡ −∂xµF (x) − a(x)z + d(x)µF (x)z + D(∂x − d(x)z)2 (3.96)
acting on x only. The scaled cumulant generating function λ(z), as defined in Eq. (3.19),
the follows as the largest eigenvalue of L(z), or, equivalently and typically more conve-
niently, of its adjoint
L†(z) ≡ µF (x)∂x − a(x)z + d(x)µF (x)z + D(∂x + d(x)z)2. (3.97)
The framework of level 2.5 large deviations can be used for the continuous state space
as well. The fluctuating empirical density is here defined for an individual trajectory
x(τ) as
p(x) ≡
∫ t
0
dτ δ(x − x(τ)), (3.98)
and the empirical current density is
j (x) ≡
∫ t
0
dτ x˙(τ) δ(x − x(τ)), (3.99)
againwith implied Stratonovich convention. In analogy toKirchhoff’s rule in the discrete
case, the empirical current must be divergence-free in leading order in t, which amounts
64
3.6 Conclusion
to being a scalar quantity in one dimension. The fluctuating observable X (t) can be
expressed through these densities as
X (t) =
∫
dx [p(x)a(x) + j (x)d(x)]. (3.100)
For the steady-state average, where ⟨p(x)⟩ = ps(x) and ⟨ j (x)⟩ = js(x), one recovers
Eq. (3.93). The probability for joint fluctuations of the fields p(x) and j (x) satisfies
a large deviation principle. The corresponding level 2.5 large deviation function is a
functional of these fields and reads [52, 68]
I[p(x), j (x)] =
1
4T
∫
dx p(x)
[
j (x) − jp(x)
p(x)
]2
(3.101)
where
jp(x) ≡ µF (x)p(x) − D∂xp(x) (3.102)
is the Fokker-Planck current (3.89) corresponding to the empirical density p(x).
The same result for the level 2.5 large deviation function can be obtained as a quadratic
approximation of Eq. (3.82) that becomes exact in the continuum limit h → 0, for which
api j ≈ 2p(xi)hk0 ≫ ji j, jpi j holds. Thus, even thoughwe focus in the following on discrete
state spaces, one can expect that all results that do not make explicit reference to the
number of states or “traffic-like” quantities hold also for overdamped Brownianmotion in
a continuous state space as a special, limiting case. The generalization to underdamped
dynamics, however, is somewhat more subtle, requiring a distinction between reversible
and irreversible currents [69].
3.6 Conclusion
In this Chapter we have introduced the mathematical concepts that will be used to
describe the fluctuations of thermodynamic currents and, in general, of time-additive
observables in stationary Markov processes on discrete state-spaces.
We have discussed two different approaches to such a description. First, the scaled
cumulant generating function can be calculated as the solution of an eigenvalue problem
involving the tilted rate matrix. This approach is particularly useful when low order
cumulants of the distribution of a fluctuating observable are to be calculated exactly.
Second, large deviation theory provides a description of the exponential decay of the
probability of untypical fluctuations. While the corresponding large deviation function
of an individual current can generally not be calculated analytically, it is nonetheless
possible to calculate the large deviation function associated with joint fluctuations of all
possible currents and sojourn times in a formalism that has been dubbed level 2.5 large
65
3 Fluctuations in stationary Markov processes
deviation theory. The large deviation function for an individual current can then be gained
from the contraction principle. This approach turns out to be particularly useful for
proving bounds on current fluctuations. Ultimately, both descriptions, though technically
different, are interrelated through Legendre transformations and thus equivalent.
Finally, we have shown that systems involving continuous degrees of freedom are
covered by the present formalism through a suitable continuum limit. This insight
allows us to focus largely on a discrete description throughout this Thesis, without loss
of generality.
66
4 Dissipation-dependent bound on
current fluctuations
In this chapter, we introduce a bound on the fluctuations of current-like observables in
non-equilibrium steady states. This bound holds universally for driven systems that are
modeled in aMarkovian, thermodynamically consistent fashion, as introduced inChap. 2.
Crucially, this bound depends on no other details of the system under consideration than
its average rate of total entropy production.
Technically, we employ large deviation theory, as reviewed in Sec. 3.3, to formulate
this bound as a constraint on the large deviation function capturing the whole spectrum
of fluctuations. Evaluated for the typical fluctuations, this bound implies the ther-
modynamic uncertainty relation, expressing a universal trade-off between the entropy
production, or thermodynamic cost, and the smallness of fluctuations, or precision, of a
thermodynamic current [1].
Here, we mainly follow the presentation of the publication [7], where we have first
described the dissipation-dependent bound as a conjecture based on numerical evidence.
We start with a discussion for systems driven weakly within linear response. In this
regime, the bound is typically strongest and can be derived straightforwardly, allowing
the reader to build intuition. Beyond linear response, the bound is illustrated numerically
for sets of randomly generated Markovian networks.
A full mathematical proof of the dissipation-dependent bound has been accomplished
by T. Gingrich et al. [8]. We will employ their techniques in Chap. 6 to derive an even
stronger, affinity- and topology-dependent bound on current fluctuations.
4.1 General setup for bounds on current fluctuations
We consider a Markovian network consisting of N discrete states {i} and allow for tran-
sitions with rates ki j ≥ 0 from state i to j. All transitions are taken to be reversible,
i.e., ki j > 0 implies k ji > 0. For notational convenience, we exclude the possibility
of several transition pathways between any two states and transitions with i = j. The
time-dependent probability distribution pi (t) of state i at time t evolves according to the
master equation (3.1) with the transition matrixLi j from Eq. (3.2). External forces drive
the system into a non-equilibrium steady state, with a stationary distribution psi . Ther-
modynamic consistency for the system and the reservoirs providing the driving forces
67
4 Dissipation-dependent bound on current fluctuations
✭❛✮
Figure 4.1: (a) Unicyclic network with affinity A and five states. (b) Multicyclic network with
two fundamental cycles, one with three states and affinityA1 and the other with four
states and affinity A2. The red dashed lines indicate a cycle with affinity A1 +A2
and five states. (c) Multicyclic network with three fundamental cycles with three
states each. The affinities of these cycles areA1,A2, andA3. The red dashed lines
indicate a cycle with affinity A1 +A2 and four states.
is implemented through the detailed balance relation (2.17) for the pairs of transitions
along all links of the network.
Following Schnakenberg, we identify a complete set of fundamental cycles {Cβ}
within the network [27], as exemplified in Fig. 4.1. For each cycle, we define a time-
antisymmetric observable X β (t) that counts cycle completions after time t (the so-called
integrated current) and the corresponding average current
Jsβ ≡ ⟨Jβ⟩ ≡ ⟨X β (t)⟩/t . (4.1)
The average is independent of t for initial conditions drawn from the steady state distribu-
tion. The set of cycle currents is complete in the sense that any other time-antisymmetric
observable can be written as a linear combination of the X β (t) up to a state function
that cannot scale with time. Unlike the currents through individual links of the network,
which are constrained through Kirchhoff’s law (3.79), there are no constraints on the
realizations of the X β (t). In order to formally define the cycle currents as time-additive
observables of the type (3.4), we set ai = 0 and define generalized distances d βi j = −d βji
that are constrained to add up to one for every closed loop that completes the cycle once
in forward direction.
The affinityA β is defined as the change of entropy in the reservoirs upon completion
of an individual cycle Cβ. Using the local detailed balance condition (2.17), this affinity
is identified as
A β ≡
∑
(i, j)∈Cβ
ln
ki j
k ji
=
∑
(i, j)∈Cβ
∑
r
(∆Ei jr − Kr ∆Xi jr )/Tr, (4.2)
where the first sum runs over the directed links of the cycle and the second one runs
over the reservoirs. For isothermal systems with equal reservoir temperature Tr = T , the
68
4.1 General setup for bounds on current fluctuations
affinity simplifies to
A β = −
∑
(i, j)∈Cβ
∑
r
Kr ∆Xi jr /T, (4.3)
For instance, recalling unicyclic motor models from Sec. 2.7, one identifies a single
cycle with an affinity given by the difference between chemical work and mechanical
work per cycle. The cycle current then corresponds to the average motor speed in units
of the step size.
The fluctuating entropy production Stot(t), as defined in Eq. (2.32), is given by the
time-additive observable with increments
di j = si j ≡ ln
psi ki j
psj k ji
. (4.4)
The average entropy production (2.38) can be expressed with the cycle affinities and
cycle currents as
σ ≡ ⟨Stot(t)⟩ /t =
∑
β
A β Jsβ . (4.5)
Adopting a vector notation X (t) for the set of all integrated cycle currents X β (t) the
scaled cumulant generating function (3.19) is defined as
λ(z) ≡ lim
t→∞
1
t
ln
⟨
exp[z · X (t)]⟩ , (4.6)
where z is a real vector. As an abbreviation, we will refer to λ(z) simply as the
“generating function”. As shown in Sec. 3.1, λ(z) is the largest eigenvalue of the tilted
Markov generator (3.13), which reads here
Li j (z) ≡ Li j exp
(
z · d ji
)
, (4.7)
where d ji is a vector with components d βji. The probability distribution of the variable
Jβ ≡ X β/t satisfies a large deviation principle (see Sec. 3.3) with a large deviation
function I (J ). Since stationary cycle currents are typically non-zero in driven systems,
we can also define the scaled variable
ξ β ≡ X β/(t Jsβ) = Jβ/Jsβ, (4.8)
with the corresponding large deviation function h(ξ ) ≡ I ({ξ β Jsβ}). The Legendre-
Fenchel transform (3.51) relating the generating function and the large deviation function
then reads
h(ξ ) = max
z
⎡⎢⎢⎢⎢⎢⎣
∑
β
zβ Jsβξ β − λ(z)
⎤⎥⎥⎥⎥⎥⎦ . (4.9)
69
4 Dissipation-dependent bound on current fluctuations
The fluctuation theorem for currents (2.53) imposes on the large deviation function
function the symmetry
−h(ξ ) + h(−ξ ) =
∑
β
ξ β JsβA β . (4.10)
In terms of the generating function this symmetry reads [39]
λ(z) = λ(−A − z). (4.11)
An arbitrary fluctuating current is a linear combination X (t) ≡ ∑β cβX β (t) of the
cycle currents with coefficients cβ. Its generating function (3.19) follows by evaluating
the generating function (4.6) in the direction c, leading to
λc (z) = lim
t→∞
1
t
ln
⟨
exp[zX (t)]
⟩
= λ(zc). (4.12)
In cases where the specific choice of c is not relevant, we will drop the index on the
l.h.s. and distinguish generating functions with scalar and vectorial arguments. As a
special case, the generating function for an individual cycle current follows as
λα (z) ≡ λ(zeα), (4.13)
where eα is the unit vector associated with the current in cycle Cα. Generally, the
generating function for an individual current does not exhibit a symmetry of the form
(4.11) as extensively discussed in [37] (see also [70, 71]). In contrast, the evaluation of
λ(z) along the vectorA yields
λs (z) ≡ λ(zA) = lim
t→∞
1
t
ln
⟨
exp[zStot(t)]
⟩
(4.14)
as the generating function of the entropy change. It is symmetric with respect to
z = −1/2, which expresses the fluctuation theorem that holds for this observable.
The large deviation functions associated with the probability distributions of these
variables read
hα (ξα) = max
z
[zJsαξα − λα (z)] (4.15)
and, introducing the scaled entropy change s ≡ Stot(t)/(σt) in analogy to Eq. (4.8),
hs (s) = max
z
[zσs − λs (z)], (4.16)
respectively.
In the following, we distinguish between unicyclic and multicyclic networks of states,
illustrated in Fig. 4.1. For unicyclic networks, where there is only a single affinity
70
4.1 General setup for bounds on current fluctuations
A ≡ Aα and a single fluctuating current X ≡ Xα, we no longer have to distinguish
between the different types of generating functions and can simply write
λ(z) ≡ λα (z) = λs (z/A). (4.17)
Throughout this Thesis, we will be interested in functions b(z) that bound the gener-
ating function λ(z) from below, i.e.,
b(z) ≤ λ(z) (4.18)
for all z. As special cases, the relation (4.18) can be used to extract bounds for individual
fluctuating currents, e.g., λα (z) ≥ bα (z) ≡ b(zeα), and the entropy change, λs (z) ≥
bs (z) ≡ b(zA). Such bounds immediately imply upper bounds on the corresponding
large deviation functions
hα (ξα) ≤ max
z
[zJsαξα − bα (z)]. (4.19)
and
hs (s) ≤ max
z
[zσs − bs (z)]. (4.20)
For any generating function the coefficients of the Taylor expansion around z = 0
correspond to the cumulants (3.20). The Fano factor that quantifies the dispersion of the
distribution is defined as
F ≡ lim
t→∞
⟨
X (t)2
⟩
− ⟨X (t)⟩2
⟨X (t)⟩ =
2D
Js
=
λ′′(0)
λ′(0)
, (4.21)
where X (t) is an arbitrary current with average Js and diffusion coefficient D. We
denote the Fano factor associated with an individual cycle current by Fα ≡ 2Dα/Jsα and
the one associated with the entropy change in the medium by Fs ≡ 2Ds/σ. Since global
lower bounds b(z) with b(0) = 0 must share a tangent with λ(z) at z = 0 while having
a stronger curvature, every such bound implies with
F ≥ b
′′(0)
b′(0)
(4.22)
a bound on the Fano factor.
Since λ(0) = 0 holds trivially for all networks, we usually require that our bounds
are saturated for z = 0. Hence, if λ(z) is analytic, b(z) must have the same gradient as
λ(z).
71
4 Dissipation-dependent bound on current fluctuations
4.2 Linear response regime
In the limit of small affinitiesA β, the average current Jsα depends linearly on the affinities,
Jsα =
∑
β
LαβA β + O
(
A2
)
(4.23)
with the Onsager matrix
Lαβ ≡ ∂J
s
α
∂A β
A=0 = ∂
2λ
∂zα∂A β (0, 0), (4.24)
where we make the dependence of the generating function on the affinities explicit by
writing λ(z,A). Following Ref. [39], we derive the fluctuation symmetry (4.11) for zα
and A β, which leads to
∂2λ
∂zα∂A β (z,A) =
∂2λ
∂zα∂zβ
(−A − z,A) − ∂
2λ
∂zα∂A β (−A − z,A). (4.25)
Letting z andA to zero yields
Lαβ =
1
2
∂2λ
∂zα∂zβ
(0, 0), (4.26)
which implies the Einstein relation
Dα = Lαα (4.27)
and the Onsager reciprocal relations
Lαβ = L βα . (4.28)
Moreover, since the generating function must be convex (see Sec. 3.3), we see that the
Onsager matrix must be positive definite.
In the region z ≲ O (A), the generating function λ(z) can be expanded as a quadratic
form around its center of symmetry, which is, due to Eq. (4.11), located at z = −A/2.
The requirement λ(0) = 0 and ∇λ(0) = Js fixes this expansion as
λ(z) =
∑
β,γ
(zβ +A β/2) L βγ (zγ +Aγ/2) − σ/4, (4.29)
where the entropy production σ is given in Eq. (4.5), which becomes for linear response
σ =
∑
βγ
A βL βγAγ . (4.30)
72
4.3 Beyond linear response: Unicyclic case
Evaluating this function for z = zeα yields as generating function related to the individual
current
λα (z) = zJsα + z
2Lαα . (4.31)
The positive definiteness of the matrix G βγ ≡ (LααL βγ − LαβLαγ) [1, supplemental
material] (with α fixed) yields∑
β,γ
G βγA βAγ = Lαασ − (Jsα)2 ≥ 0. (4.32)
Hence λα (z) is bounded from below by
λα (z) ≥ zJsα (1 + zJsα/σ). (4.33)
Using the Legendre transform (4.19), this bound can be transformed into a bound for the
large deviation function
hα (ξα) =
Lαα
4(Jsα)2
(ξα − 1)2 ≤ σ4 (ξα − 1)
2. (4.34)
Since the direction eα can be chosen arbitrarily, the bound (4.33) can be stated in a
multidimensional formulation as
λ(z) ≥ z · Js (1 + z · Js/σ). (4.35)
Equality holds along the line z ∝ A, which corresponds to the generating function
λs (z) = λ(Az) associated with entropy change. Within linear response, the large
deviation function for the scaled entropy change s is thus given by
hs (s) =
σ
4
(s − 1)2. (4.36)
Eq. (4.34) shows that the knowledge of the average entropy production is sufficient to
bound the fluctuations of any individual current in the linear response regime. It should
be noted, however, that Eq. (4.34) is still restricted to the fluctuations of the current Jα
that are on the order of Jsα. Surprisingly, as we discuss next, this quadratic bound is also
valid for both driving and fluctuations beyond the linear response regime.
4.3 Beyond linear response: Unicyclic case
The quadratic shape of the generating function for z ≲ O(A) and of the large deviation
function for ξ ≲ O(1) can be regarded as a signature of linear response. It arises only for
nearly vanishing affinities or for freely diffusing particles, where the linearity between
73
4 Dissipation-dependent bound on current fluctuations
✵
✵✳
✶
✲✶✳ ✲✶ ✲✵✳ ✵ ✵✳
✕
❆
❘
❲
✭
③
✱
✁
✱
✂
✮
✄❂☎
☎✪
✵
✶
✷
✸
✹
✺
✲✹ ✲✷ ✵ ✷ ✹ ✻
❤
❆
❘
❲
✭
✘
✱

✱
✁
✮
❂

✂
✄✪
Figure 4.2: The generating function λ(z) and the large deviation function h(ξ) of the asymmetric
random walk for selected affinitiesA (2, 5, 10, 50) and N = 1. Black arrows indicate
the direction of increasingA. The quadratic bound for the generating function (4.42)
and for the large deviation function (4.43) are shown as black dashed curves.
affinity and current persists even for high affinities. Beyond this regime, one universally
observes two characteristic changes in the large deviation function [51,59,72]. First, the
tails for large values of |ξα | grow no longer quadratically but with a scaling somewhere
between linear and quadratic. Second, there is a formation of a “kink” around the value
ξα = 0. For finite numbers of states, the large deviation function is still analytic in this
region, but it exhibits a significantly enhanced curvature. In the Legendre transformed
picture of the generating function λα (z), these two effects show up as a faster than
quadratic growth for large z and a pronounced plateau around the minimum of λα (z).
This behavior of the generating function is best illustrated with an asymmetric random
walk (ARW), as shown in Fig. 4.2. Consider a network consisting of a single cycle with
N vertices and affinity A, as shown in Fig. 4.1a. The hopping rates in forward and
backward directions k+ and k− are uniform with
ln
k+
k−
= A/N . (4.37)
The average current in this model is Js = (k+ − k−)/N and the entropy production is
σ = JsA. As derived in Eq. (3.24) (with a different affinity per step), the generating
function is given by
λ(z) = k+
[
ez/N + e−(z+A)/N − 1 − e−A/N
]
(4.38)
= JsλARW(z,A, N ),
where
λARW(z,A, N ) ≡ cosh[(z +A/2)/N] − cosh[A/(2N )](1/N ) sinh[A/(2N )] . (4.39)
74
4.4 Beyond linear response: multicyclic case
Similarly, the large deviation function corresponding to the generating function is given
by Eq. (3.54), which reads for the present setup
h(ξ) = Js hARW(ξ,A, N ), (4.40)
where
hARW(ξ,A, N ) ≡ N
[
ξ arsinh(ξ/ξ0) − ξA/(2N ) −
√
ξ20 + ξ
2 +
√
ξ20 + 1
]
(4.41)
and ξ0 ≡ 1/ sinh[A/(2N )]. As shown in Fig. 4.2, the generating function (4.38) is
bounded from below by the quadratic function
λ(z,A, N ) ≥ z Js (1 + z/A) = z Js (1 + zJs/σ), (4.42)
and the large deviation function is bounded from above by the quadratic function
h(ξ) ≤ JsA(ξ − 1)2/4 = σ(ξ − 1)2/4. (4.43)
Thus, we see that the quadratic bounds (4.33) and (4.34) hold for the asymmetric
random walk not only in the linear response regime but also for arbitrarily strong
driving. The bound gets weaker when the affinity per stepA/N increases, i.e., for cycles
with large affinity or few states. Likewise, the bound becomes weaker for non-uniform
choices of the transition rates along the cycles, which we will illustrate in Sec. 6.3 in the
context of an even stronger, affinity dependent bound. Taken together, these observations
suggest that the quadratic bounds hold in fact for arbitrary unicyclic networks.
4.4 Beyond linear response: multicyclic case
As has been conjectured in Ref. [7] and proven in Ref. [8], the quadratic, dissipation-
dependent bound
λ(z) ≥ z · Js (1 + z · Js/σ) (4.44)
on the generating function (4.6) holds globally for all vectors z and for all types of
Markovian networks. In terms of the individual current in a cycle Cα, this bound can be
formulated as
λα (z) ≥ zJsα (1 + zJsα/σ), (4.45)
or as a bound on the large deviation function
hα (ξα) ≤ σ(ξα − 1)2/4. (4.46)
75
4 Dissipation-dependent bound on current fluctuations
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-1.5 -1 -0.5 0 0.5
λ
α
(z
)/
σ
zJsα/σ
Figure 4.3: Generating functions λα (z) for an individual current in fully connected networks
with random transition rates. The red and green curves correspond to networks with
N = 4 and N = 6 vertices, respectively. The dissipation-dependent bound (4.45) is
shown as a black curve.
The bound (4.44) becomes the same as (4.35) in the linear response regime. However,
(4.44) is also valid beyond this regime where the currents Js are the actual average
currents in the steady state, as determined from∇λ(0), which are different from the linear
response currents (4.23). Likewise, this bound is no longer restricted to fluctuations on
the order of a small stationary current Js. It can therefore be stated without scaling the
fluctuating current as
I (Jα) ≤ σ(Jα/Jsα − 1)2/4. (4.47)
Analogous bounds of the form
λ(z) ≥ zJs(1 + zJs/σ), λI (J) ≤ σ(J/Js − 1)2/4. (4.48)
hold for all other currents that are linear combinations of the cycle currents with average
Js ≡ c · Js, generating function (4.12) and the corresponding large deviation function
I (J). In particular for the entropy production we have
λs (z) ≥ zσ(1 + z). (4.49)
A numerical illustration for this bound is provided in Fig. 4.3. For this illustration, we
have generated a set of networks with N vertices and random transition rates
ki j = exp
[
5(ϕi j + ϕ ji)/2 + 2θi j
]
, (4.50)
where ϕi j and θi j are independent Gaussian random numbers with zero mean and
variance 1. As the affinity increases, generating functions globally deviate in a positive
76
4.5 The thermodynamic uncertainty relation
direction from the parabolic shape. Only at the trivial points z = 0 and z = −A does
the generating function in Eq. (4.44) acquire with zero the same value for all networks.
For larger networks (N = 6) the left hand side of the plot becomes less populated, since
the probability of the vectors eα andA being nearly parallel becomes smaller in higher
dimensions.
4.5 The thermodynamic uncertainty relation
The local evaluation (4.22) of the parabolic bound (4.48) for an individual current yields
the relation
Fσ/Js ≥ 2 (4.51)
for the Fano factor associated with the current X (t). This result, dubbed the “thermody-
namic uncertainty relation”, has first been conjectured in Ref. [1]. Hence, the quadratic
bound on the large deviation function is a generalization of this relation. Applied to the
current associated with the entropy production in the medium, the bound (4.49) leads to
Fs ≥ 2. (4.52)
In order to appreciate its implications, it is instructive to re-formulate the rela-
tion (4.51). First, writing
Dσ/(Js)2 ≥ 1, (4.53)
the relation is expressed as a dissipation-dependent bound on the diffusion coefficient
associated with X (t). From this relation, measurements of the dispersion and average
of an individual current can provide a lower bound on the average entropy production
σ ≥ 2(Js)2/D. This bound makes it possible to estimate the entropy production
by measuring a single individual current, which is complementary to the methods for
inferring the entropy production described inRefs. [73,74]. The bound (4.53) is saturated
for the biased Brownian motion X (t) of a colloidal particle with mobility µ that is pulled
in one dimension with a constant force f . In this case, we have the entropy production
σ = f /T , the single current Js = µ f , and the diffusion coefficient given by the Einstein
relation as D = µT . If, however, the particle is pulled along a periodically patterned
substrate, there will be finite-time correlations in the current X˙ (t) that lead to deviations
of the effective diffusion coefficient D from µT . These deviations must be such that the
relation (4.53) is respected.
Second, we can define the uncertainty of a process as
ε2 ≡
⟨
(X (t) − ⟨X (t)⟩)2
⟩
⟨X (t)⟩2 , (4.54)
77
4 Dissipation-dependent bound on current fluctuations
which is a dimensionless quantity characterizing the amplitude of fluctuations compared
to the mean. For large time intervals t fluctuations typically average out, such that
ε2 decreases like t−1. On the other hand, the thermodynamic cost associated with
maintaining the underlying stationary process is quantified by the produced entropy
C ≡ σt. Thus, the thermodynamic uncertainty relation can be stated as [1]
ε2C ≥ 2, (4.55)
imposing a minimal energetic cost that must be paid for small uncertainty in the output
of, e.g., an enzymatic reaction. In Chap. 8, we will show that this relation holds in fact
for arbitrary time intervals t. For instance, measuring a certain time interval t with a
biomolecular, “Brownian” clock in a thermal environment with a precision of ε = 0.01
costs at least 20 000 kBT [75].
4.6 Dissipation-dependent bound in driven diffusive one
dimensional systems
The hydrodynamic fluctuation theory for driven diffusive systems in contact with two
reservoirs by Bertini et al. [76–78] has been a major development in nonequilibrium
statistical physics. This theory leads to a (typically hard) variational problem that,
if solved, leads to the exact rate function of the current of particles or heat between
reservoirs. The additivity principle derived in [79] is a more direct method that allows
for the calculation of the generating function related to the current in driven diffusive
systems. For example, this method has been used to calculate this function for the
symmetric simple exclusion process (SSEP) [58, 79], the Kipnis-Marchioro-Pressuti
(KMP) model [80, 81], and the weakly asymmetric simple exclusion process (WASEP)
[82]. These results are valid in the limit of large system size, for which the number of
states diverges, and have been verified numerically [81,82]. Even though the dissipation-
dependent bound appears to be restricted to the case of a finite number of states, we
show that the generating functions obtained from the additivity principle for these three
models lies inside our parabolic bound.
First we consider the WASEP and the SSEP, which is a particular case of the WASEP.
These models are illustrated in Fig. 4.4 and their precise definition can be found in [58].
In the WASEP particles flow from the left reservoir with constant density ϱL to the right
reservoir with density ϱR < ϱL. The current of particles in the system is proportional
to the entropy production, and the affinity that drives the process out of equilibrium is
given by [58, 82]
AWASEP = − ln 1 − ϱL
ϱL
+ ln
1 − ϱR
ϱR
+ (L − 1) ln 1 − ν/L
1 + ν/L
. (4.56)
78
4.6 Dissipation-dependent bound in driven diffusive one dimensional systems
(a)
(b)
Figure 4.4: Schematic illustrations of the WASEP (a) and the KMP (b) models. For the WASEP,
in the bulk the particles jump with rates p ≡ 1/2 + ν/(2L) to the right and q ≡
1/2 − ν/(2L) to the left, where the SSEP corresponds to ν = 0. At the boundaries
particles are exchanged with the reservoirs. The model also has the exclusion
principle, i.e., the maximum number of particles in a site is one. For the KMP
model energy flows from a hot reservoir at temperature TL to a cold reservoir at
temperature TR . In the bulk a randomly chosen pair of sites exchange energy, which
is a continuous variable, in such a way that the total energy is conserved. At the
boundaries energy is exchanged with the reservoirs. The precise rules of these
models can be found in [82] for the WASEP and [81] for the KMP model.
The weak asymmetry of the bulk rates, which scales with 1/L, guarantees that in the
thermodynamic limit L → ∞ the affinity is finite. In Fig. 4.5, we have calculated the
generating function using the additivity principle for the SSEP, as explained in [58], and
for the WASEP, as explained in [82]. In both cases the generating functions are inside
the dissipation-dependent bound.
TheKMPmodel is a driven diffusive system for the transport of energy from a reservoir
at temperature TL to a reservoir at temperature TR < TL, as illustrated in Fig. 4.4. A
key feature of the KMP model is that there is no dissipation in the bulk. The precise
definition of the model can be found in [81]. The heat transfer from the left to the right
reservoir is proportional to the entropy production with the affinity given by [81]
AKMP = (T−1R − T−1L ). (4.57)
The generating function for this model, which is obtained from the additivity principle
as explained in [81], also satisfies the dissipation-dependent bound in Fig. 4.5 within the
finite support −T−1R < z < T−1L of λ(z). As a consequence, the large deviation function
satisfies the corresponding dissipation-dependent bound globally.
These results demonstrate that our dissipation-dependent bound is even more univer-
sal: it seems to be valid for these driven diffusive systems in the thermodynamic limit,
for which the number of states diverges.
Another interesting issue will be to explore whether the bound is still valid in the
L → ∞ limit if the system undergoes a dynamical phase transition as the KMP model
in a ring-like geometry [83].
79
4 Dissipation-dependent bound on current fluctuations
✵
✵✳
✶
✲✶✳ ✲✶ ✲✵✳ ✵ ✵✳
✕
s
✭
③
✮
❂
✛
✁✂❆
♣❛r❛❜♦❧✐❝ ❜♦✉♥❞
❙❙❊P
❲✄
❙❊P
❑▼P
Figure 4.5: Comparison of the dissipation-dependent bound (4.48) with generating functions for
driven diffusive systems. For the SSEP the densities of the left and right reservoirs
were chosen as ϱL = 0.99 and ϱR = 0.01. For theWASEP the parameters are ν = 10,
ϱL = 4/7, and ϱR = 5/18, as in Ref. [82]. For the KMP model the parameters are
TL = 2 and TR = 1, as in Ref. [81].
4.7 Conclusion
In this chapter, we have introduced the dissipation-dependent bound on current fluc-
tuations as the most universal result covered in this Thesis. The total rate of entropy
production, which characterizes the non-equilibrium character of a stationary driven
system, is the only quantity that is required to formulate this bound. Using the lan-
guage of large deviation theory, the bound is stated as a quadratic lower bound (4.44)
on the generating function, or, in the Legendre transformed picture, as a quadratic upper
bound (4.46) on the large deviation function.
The bound is typically strongest for currents that are proportional to the entropy
production. Here, the upper bound on the large deviation function states that spontaneous
fluctuations of the current are at least as likely as predicted by a Gaussian distribution
withmatching average andwhich displays the symmetry of the fluctuation relation (2.48).
In the linear response regime, where the generating function and the large deviation
function are themselves quadratic, the bound is saturated for currents proportional to the
entropy production. With increasing driving forces, the generating function and the large
deviation function develop characteristic deviations from this quadratic shape, which
can be best studied for the example of the asymmetric random walk. These deviations
typically lead to the dissipation-dependent bound becoming weaker for strong driving.
A local evaluation of the dissipation-dependent bound for typical fluctuations yields
the so-called thermodynamic uncertainty relation, which establishes a trade-off between
precision in any fluctuating current and the total rate of entropy production.
80
5 Application: Bounds on the
efficiency of molecular motors
Molecular motors are small machines that can transform free energy liberated in a
chemical reaction into mechanical work. Their thermodynamic efficiency η is defined
as the ratio between the work exerted against an opposing mechanical force f (or
torque for a rotary motor) and the free energy consumed in the chemical reaction
driving the motor. This efficiency is universally constrained through the second law by
one [84–100]. Molecular motors are isothermal machines for which η ≤ 1 replaces the
Carnot expression applicable to heat engines [6].
In this Chapter, which is based on the publication [101], we discuss the consequences
of the thermodynamic uncertainty relation for the efficiency of molecular motors. As a
main result, we show that η is bounded by
η ≤ 1
1 + vT/D f
, (5.1)
where v is the (mean) velocity of the motor, D its diffusion coefficient, and T is the
temperature. The intriguing aspect of this bound arises from the fact that v, D, and
f are experimentally accessible quantities. No knowledge of the underlying chemical
reactions scheme is necessary for applying this bound. Moreover, it holds for a huge
class of motor models, arguably essentially for all models that are thermodynamically
consistent whether based on discrete states or on a continuous potential as often used
in ratchet models [85, 86, 102, 103]. Likewise, it holds for complexes of motors pulling
cooperatively a single cargo particle as, e.g., investigated in Refs. [98, 104–106]. Re-
cently, the tightness of the thermodynamic uncertainty relation has been investigated for
a number of motor models that match experimental data [107].
5.1 Thermodynamically consistent motor model
The molecular motor (complex) has an arbitrary number of internal states {i} that
describe distinct conformations. Moreover, a possible change in conformation can be
related to the binding of a solute molecule Aρ. Here, ρ label the species, like ATP whose
hydrolization to ADP and Pi can drive the motor. The motor steps along a periodic track
81
5 Application: Bounds on the efficiency of molecular motors
Figure 5.1: Schematic network of transitions for a molecular motor walking along a periodic
track against an external force f . For each periodic interval the internal states
i ∈ {1, 2, 3, 4} of the motor can be grouped into a “cell”.
of periodicity d, which means that the set of states {i}, called a “cell”, is attached to a
discrete linear sequence of spatial positions for the center of the motor, see Fig. 5.1 [36].
Transitions from state i to state j can occur within a cell, i.e., without net advancement
of the motor or can be accompanied with a forward or a backward step. In the first case,
we denote the rate by k0i j , in the two latter by k
+
i j and k
−
i j , respectively. In the spirit of
general setup introduced in Sec. 2.2, the upper index distinguishes three sets of transition
rates by the concomitant change in the work reservoir providing the mechanical force f .
The changes in the chemical reservoirs providing the solute molecules are assumed to
be uniquely determined by the initial states and final states i, j of a transition. As usual,
the transitions are microscopically reversible, which means that whenever k0i j , 0, k
0
ji
cannot vanish either. Likewise, k+i j , 0 implies k
−
ji , 0 and vice versa. However, there
may be transitions within a cell, which do not occur with spatial motion, i.e., k+,−i j = 0 is
allowed even if k0i j , 0 and vice versa.
Thermodynamic consistency imposes local detailed balance (2.21) as constraints on
both types of transitions. First, for transitions within one periodicity cell,
k0i j
k0ji
= exp
⎡⎢⎢⎢⎢⎢⎣
(
Fi − Fj +
∑
ρ
bρi j µρ
)
/T
⎤⎥⎥⎥⎥⎥⎦ . (5.2)
Here, Fi, j are the free energies of the two states, which may comprise contributions
from the displacement against the external force for incomplete steps of the motor. If
the transition from i to j requires binding a solute of species ρ (like, e.g., ATP), then
bρi j = 1. Likewise, if this transition leads to a release of such a molecule, b
ρ
i j = −1.
82
5.1 Thermodynamically consistent motor model
In both cases, the chemical potential µρ of this species enters the expression in the
exponent providing a contribution to the total free energy involved in such a transition.
If a transition additionally involves a step in the forward direction, against the applied
force, then the ratio between such a step and the corresponding backward step becomes
k+i j
k−ji
= exp
⎡⎢⎢⎢⎢⎢⎣
(
Fi − Fj +
∑
ρ
bρi j µρ − f d
)
/T
⎤⎥⎥⎥⎥⎥⎦ . (5.3)
For a fixed applied external force f and externally maintained chemical potentials
{µρ}, the motor reaches a steady state with velocity
v =
∑
i j
psi (k
+
i j − k−i j )d (5.4)
with psi the steady state probability to find the motor in the internal state i. This velocity
v = ⟨X (t)⟩ /t = ⟨n+(t) − n−(t)⟩ d/t (5.5)
is given by the steady state average of the stochastic displacement X (t), where n+,−(t)
are the number of forward and backward steps along the track in time t. In order to
apply the thermodynamic uncertainty relation, besides the diffusion coefficient D for the
displacement of the motor, as defined in Eq. (3.21), we need the rate of thermodynamic
entropy production σ. Summing in Eq. (2.38) over all types of transition, one obtains
σ =
∑
i j
psi
(
k0i j ln
k0i j
k0ji
+ k+i j ln
k+i j
k−ji
+ k−i j ln
k−i j
k+ji
)
. (5.6)
Using the detailed balance conditions (5.2,5.3), the steady state condition (2.20), reading
here ∑
j
psi (k
0
i j + k
+
i j + k
−
i j ) =
∑
j
psj (k
0
ji + k
−
ji + k
+
ji), (5.7)
and the symmetry bρi j = −bρji leads to the expression
σ =
∑
i j
psi (k
0
i j + k
+
i j + k
−
i j )
∑
ρ
bρi j µρ/T − v f /T . (5.8)
The chemical work put into the motor arises from reactions of the type∑
ρ
r ρν A
ρ ⇌
∑
ρ
sρν A
ρ, (5.9)
83
5 Application: Bounds on the efficiency of molecular motors
where r ρν = 1 if species ρ is an educt of the forward reaction of type ν and s
ρ
ν = 1 if
species ρ is a product of this reaction. In the simplest case of just ATP hydrolysis, as for
F1-ATPase [84, 93, 95, 99] or kinesin [91, 108, 109], we have only one net reaction
ATP ⇌ ADP + Pi. (5.10)
In general, the rate of consumption of species ρ is given by
Rρ =
∑
ν
γν (r
ρ
ν − sρν ) (5.11)
where γν is the effective net rate of the reaction of type ν with γν < 0 if the reaction goes
backward on average. Likewise, Rρ < 0 if, on average, species ρ is rather produced than
consumed. Since we assume that all reactions are catalyzed by the motor acting as an
enzyme, thus requiring binding and release as introduced above, this net rate can also
be written as
Rρ =
∑
i j
psi (k
0
i j + k
+
i j + k
−
i j )b
ρ
i j . (5.12)
The rate of consumption of chemical (free) energy, i.e., the rate with which chemical
work is put into the motor, becomes
W˙ chem =
∑
ρ
Rρµρ. (5.13)
Inserting these expressions into the entropy production rate (5.8), we get
σ = (W˙ chem − f v)/T = f v(1/η − 1)/T . (5.14)
with the thermodynamic efficiency [85]
η ≡ f v/W˙ chem = f v/( f v + Tσ). (5.15)
5.2 The bound
Inserting the uncertainty relation (4.53) for the displacement current Js = v in (5.15),
we obtain our main result stated in the introduction, copied here for convenience as
η ≤ 1
1 + vT/D f
. (5.16)
Remarkably, this universal bound has been derived without any assumptions about
the specific molecular mechanism driving the motor. It holds for a single motor as well
84
5.2 The bound
1 10 100
1.5
1.0
0.5
0.0
1000 0 1 2 3 4 5 6 7
1.5
1.0
0.5
0.0
Figure 5.2: Efficiency bound (5.16) for experimental data for kinesin from Ref. [111]. The
randomness parameter r = 2D/vd was measured for various ATP-concentrations at
constant force f = 3.59 pN (left panel) and for various forces f at constant ATP-
concentration cATP = 2 mM (right panel). The shaded regions are labeled with the
maximal efficiency η that applies to data points falling into the respective region.
Reprinted from Ref. [65], Copyright (2018), with permission from Elsevier.
as for complexes of several motors pulling a cargo to which an external force is applied.
Hence, it is sufficient to measure the velocity and the diffusion constant of the motor
to bound its thermodynamic efficiency. This bound can also be written in terms of the
often used randomness parameter r ≡ 2D/vd [88,108,110–112]. With this quantity, the
universal bound on efficiency (5.16) reads
η ≤ r
r + 2T/ f d
. (5.17)
In Fig. 5.2, we apply this bound to experimental data for kinesin obtained by Visscher
et al. [111]. For strong forces and low fuel concentrations, the bound on the efficiency is
rather close to unity, whereas for high fuel concentrations and weak forces the maximal
efficiency is about 60% or even less.
If only one type of chemical reaction, e.g, ATP hydrolysis is involved, the rate of
chemical work becomes W˙ chem = RATP∆µ where ∆µ ≡ µATP − µADP − µPi . The rate of
ATP-consumption can then be bounded as
RATP ≥
(
f +
vT
D
)
v
∆µ
. (5.18)
This relation makes it possible to infer the minimal rate of ATP consumption by mea-
suring the first and second moments of the displacement of the motor.
85
5 Application: Bounds on the efficiency of molecular motors
5.3 Variants
So far, we have focused on the thermodynamic efficiency η where the motor runs against
an external load. Formotors pulling cargo through a viscous environment with no further
external load, often the Stokes efficiency
ηS ≡ γ (0)v2/W˙ chem (5.19)
is used [103]. It compares the chemical work spent to the (mechanical) power required
by a fictitious force to pull the cargo with a bare friction coefficient γ (0) at the same
velocity through the viscous medium. Using σ = W˙ chem/T in the uncertainty relation
(4.53), one easily gets
ηS ≤ γ (0)D/T = D/D(0), (5.20)
where the second equality follows from the Einstein relation between the bare friction
coefficient and the bare diffusion coefficientD(0) = T/γ (0) of the cargo. Hence the Stokes
efficiency is universally constrained by the ratio between the full diffusion coefficient
of the motor cargo complex and the bare diffusion coefficient of the cargo. Again, this
bound holds independently of all molecular details.
So far, we have assumed that the motor can be described by a set of internal states
replicated along a spatially periodic track using a discrete master equation dynamics.
For motors modeled in a continuous potential possibly changing due to internal (chemi-
cal) transitions [85] and for motors pulling big probe particles described by a Langevin
equation [96, 113], these universal results remain valid. Formally, one has to discretize
the spatial coordinate and introduce (biased) transition rates between neighboring spa-
tial positions, as detailed in Sec. 3.5. After such a discretization, which can become
arbitrarily fine, one is back at the model investigated above.
For a motor that performs chemical work W˙ chemout > 0 by synthesizing molecules with
high chemical potential driven by an applied external force f dr > 0 [114] similar results
hold with velocity v > 0. The efficiency for this situation reads
η ≡ W˙ chemout / f drv = ( f drv − Tσ)/ f drv ≥ 0. (5.21)
The inequality (4.53) leads to
η ≤ 1 − vT/ f drD. (5.22)
Again, the efficiency of this molecular machine can be bounded without measuring the
rate at which it produces molecules.
86
5.4 Numerical case study
(a)
A
T
P      AD
P
+
P
i
(b)
0 0.2 0.4 0.6 0.8 1
f d /∆µ
0
0.2
0.4
0.6
0.8
1
Th
er
m
od
yn
am
ic 
ef
fic
ie
nc
y
η
bound ( ∆µ=4 k
B
T)
bound ( ∆µ=10 k
B
T)
bound ( ∆µ=16 k
B
T)
bound ( ∆µ=22 k
B
T)
(c)
5 10 15 20 25 30 35 40
∆µ [ k
B
T]
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
St
ok
es
 e
ffi
cie
nc
y
ηS
bound
Figure 5.3: (a) Illustration of the model for an ATP driven motor (blue) elastically coupled to a
probe particle (red), to which the external force f is applied. (b) Comparison of the
thermodynamic efficiency η as a function of the scaled force f d/∆µ (black line) and
the bound (5.16) at various values of ∆µ (colored curves). (c) Comparison of the
Stokes efficiency as a function of ∆µ (black) and the bound (5.20) (red). For large
∆µ, where the bound surpasses 1, it is continued as a dashed line. Model parameters
were chosen as detailed in Ref. [96], different values of ∆µwere generated by varying
the concentration of ATP while keeping the concentrations of ADP and Pi fixed.
5.4 Numerical case study
We illustrate the bound for a simple model of the rotary motor F1-ATPase coupled
elastically to a colloidal probe particle [96, 115]. As shown in Fig. 5.3a, after mapping
the rotary motion to a linear one, the motor performs discrete steps of length d, with
each forward step hydrolizing an ATP molecule and each backward step synthesizing an
ATP molecule. The stepping rates depend on the chemical potential difference ∆µ of
the ATP reaction and the elongation of the linker to the probe particle. The continuous
dynamics of the probe particle with friction coefficient γ (0) is modeled as overdamped
Brownian motion subject to the external force f and the potential force of the linker [96].
The average velocity v and the diffusion constant D for this model can be calculated
numerically from the Master equation that is obtained by finely discretizing the possible
87
5 Application: Bounds on the efficiency of molecular motors
elongations of the linker and truncating at large elongations. Due to the tight coupling
between the chemical reaction and the motion of the motor, the rate of the chemical work
is W˙ chem = ∆µv/d [96], so that the thermodynamic efficiency (5.15) becomes trivially
η = f d/∆µ (5.23)
while the Stokes efficiency (5.19) is
ηS = γ
(0)vd/∆µ. (5.24)
In Fig. 5.3b, we compare the thermodynamic efficiency (5.23) to the bound (5.16) for
forces ranging from 0 to ∆µ/d and selected values of ∆µ. The bound is trivially saturated
for f = 0 and for the case f d = ∆µ, where the chemical and themechanical force balance
each other, leading effectively to equilibrium conditions with vanishing velocity. Since
the uncertainty relation becomes exact within linear response for unicyclic systems [1,8],
the bound is also saturated for forces close to ∆µ/d. Similarly, for ∆µ → 0, the bound
approaches η for all 0 < f < ∆µ/d.
In Fig. 5.3c, the Stokes efficiency (5.24) is compared to the bound (5.20) for a range of
∆µ and constant f = 0. While this bound is also saturated in the linear response regime
for ∆µ→ 0, it becomes rather loose for large values of ∆µ. It even surpasses the bound
ηS < 1, proven in Ref. [103], when D becomes larger than the bare diffusion coefficient
D(0) [116, 117]. Obviously, the bound (5.20) is not useful in this range of parameters.
5.5 Conclusion
In this Chapter, we have shown that the thermodynamic uncertainty relation implies a
universal bound on the efficiency ofmolecularmotors that depends only on the fluctuating
displacement of the motor and an external load force. The sole knowledge of both the
average and the dispersion of the experimentally accessible displacement yields a bound
that is independent of the underlying mechano-chemical reaction scheme. This result
applies to any nano or micro machine operating in an environment of fixed temperature.
88
6 Affinity- and topology-dependent
bound on current fluctuations
The main strength of the quadratic bound on current fluctuations covered in Chapter 6
lies in its dependence on solely the rate of entropy production – other details of the system
under consideration do not have to be known in order to apply this result. However, as a
consequence of this universality, the bound can be fairly loose when applied to specific
systems, especially when they are driven far away from linear response.
One may expect that with more knowledge on the details of a system, stronger bounds
on current fluctuations can be devised. Indeed, as we will derive in this chapter, a refined
variant of the bound exists, which requires additional information on the driving affinities
and the topology of the Markovian network underlying the system. This information
will typically be much better accessible than the full set of transition rates that would be
required to calculate the spectrum of fluctuations exactly.
We have conjectured this affinity- and topology-dependent bound along with the
dissipation dependent bound in Ref. [7], where it is referred to as “hyperbolic cosine”
bound according to its functional form as a bound on the generating function. We
could publish a full proof of the bound in Ref. [118], where we build on the techniques
developed for the dissipation-dependent bound in Ref. [8] and construct a suitable cycle-
decomposition.
For unknown affinities and network topology, the dissipation-dependent bound follows
naturally as the weakest instance of the affinity- and topology-dependent bound. Thus,
the proof presented in this chapter serves also as a proof of the former. As applied to
typical fluctuations, the affinity- and topology-dependent bound implies a refinement of
the thermodynamic uncertainty relation, which has first been conjectured in Ref. [119].
6.1 Setup
Just as the dissipation-dependent bound, the affinity- and topology-dependent bound is
stated for a stationary Markov process on a finite network of N states, using the same
notation as introduced in Sec. 4.1. We recall the definition of the stationary currents
along links of the network,
jsi j ≡ psi ki j − psj k ji = − jsji, (6.1)
89
6 Affinity- and topology-dependent bound on current fluctuations
which are the expectation values of the fluctuating currents ji j , defined in Eq. (3.76) as
the net number of transitions in a stochastic trajectory along the link (i, j) divided by the
observation time t. An individual transition i → j contributes [6]
si j = ln
psi ki j
psj k ji
(6.2)
to the entropy production in a NESS, leading to the average total entropy production
σ =
∑
i< j
jsi j si j . (6.3)
We denote by
∑
i< j the sum over those links (i, j) in the network graph for which
transitions are possible, i.e., for which ki j > 0 and k ji > 0.
The decomposition of networks into cycles is an important concept providing a link
between (bio-)physical properties of a system and its description as a stochastic process
[22, 27, 120, 121]. After the intuitive introduction given in Sec. 4.1, we here formalize
the concept of a cycle decomposition. A cycle Cα of length nα ≥ 3 (smaller cycles
are not relevant for our discussion) is defined as a directed, self-avoiding, closed path
[ℓ(1) → ℓ(2) → · · · → ℓ(nα) → ℓ(1)] along links of the network with kℓ(n)ℓ(n+1) > 0.
We define the directed adjacency matrix χα = ( χαi j ) of such a cycle as
χαi j ≡
nα∑
n=1
δℓ(n),iδℓ(n+1), j − δℓ(n), jδℓ(n+1),i, (6.4)
which is +1 for links (i, j) where Cα passes in forward direction, −1 for the backward
direction and zero otherwise. A set of cycles {Cα} is called complete if any set of currents
{ ji j } along the links of the network consistent with Kirchhoff’s law (3.79)∑
j
ji j = 0 for all i (6.5)
can be decomposed into a set of cycle currents {Jα} such that
ji j =
∑
α
Jα χαi j . (6.6)
In particular, the stationary currents jsi j can be decomposed into stationary cycle currents
Jsα. The affinity Aα of a cycle Cα is, as in Eq. (6.7), given by
Aα =
∑
i< j
χαi j ln
ki j
k ji
=
∑
i< j
χαi j si j =
∑
(i, j)∈Cα
si j, (6.7)
which can be used to write the average entropy production (6.3) as in Eq. (4.5) as
σ =
∑
α
JsαAα . (6.8)
90
6.2 Main result
6.2 Main result
With the above definitions at hand, we can now state the central result from our publica-
tion [118]. The generating function, defined in Eq. (4.6) as
λ(z) = lim
t→∞
1
t
ln
⟨
ez·Jt
⟩
(6.9)
for the fluctuations of the vector J = (Jα) of cycle currents is bounded from below by
λ(z) ≥ σ cosh[(z · J
s/σ + 1/2)A∗/n∗] − cosh[A∗/(2n∗)]
(A∗/n∗) sinh [A∗/(2n∗)] , (6.10)
where A∗/n∗ is defined as the smallest positive value for the affinity per cycle length
over all cycles in the network. In a scalar version, this bound implies for the generating
function (4.12) of any generic current J, which can be written as an expansion in the
cycle currents, the bound
λ(z) ≥ σ cosh[(zJ
s/σ + 1/2)A∗/n∗] − cosh[A∗/(2n∗)]
(A∗/n∗) sinh [A∗/(2n∗)] , (6.11)
where Js is the stationary value of J. In particular, for the entropy current
∑
α JαAα we
have Js = σ, which further simplifies (6.11). Based on strong numerical evidence we
have conjectured the relation (6.10) in Ref. [7]. The proof from Ref. [118] is given in
Sec. 6.5 below.
As shown in Appendix 6.C, the right hand side of Eq. (6.10) decreases monotonically
for decreasing A∗ and all other quantities fixed. Taking the limit A∗ → 0 leads to the
quadratic, dissipation-dependent bound (4.44), which Eq. (6.10) is therefore a refinement
of.
For the typical fluctuations, a local evaluation of the from (4.22) of the bound (6.11)
leads to [119]
F ≥ A
∗Js
n∗σ
coth
(A∗
2n∗
)
(6.12)
as a bound on the Fano factor. This bound can be used to estimate the number of
intermediate states in enzymatic schemes from measurements of the Fano factor in
single molecule experiments, as discussed in Ref. [119].
Overdamped Brownian dynamics in a continuous state space is covered by our discrete
formalism through the continuum limit described in Sec. 3.5. However, in this limit the
affinity per cycle length vanishes with the discretization length h, such that the bound
(6.10) degenerates to the dissipation-dependent bound. The latter can also be derived
directly for overdamped dynamics from the level 2.5 large deviation function (3.101), as
described in Refs. [64, 122].
91
6 Affinity- and topology-dependent bound on current fluctuations
-10
-5
0
5
10
15
-100 -80 -60 -40 -20 0
λ
(z
)/
Js
z
SD = 2
1
0.5
0.2
0.1
0.05
0.01
Figure 6.1: Generating function λ(z) scaled by the steady state current Js in unicyclic networks
with N = 10 states and fixed affinityA = 100. The color-scale encodes the standard
deviation (SD) used for sampling the transition rates (see Ref. [7, Appendix B]).
Blue corresponds to a nearly uniform distribution of transition rates and yellow to a
broad distribution of transition rates. The lower and upper bounds in Eq. (6.15) are
shown as red dashed lines.
6.3 Numerical illustration
Physical intuition for the result (6.11) can be gained from numerical case studies. To
begin with, consider an arbitrary unicyclic model with N states and periodic boundary
conditions. The transition rates are denoted by
ki,i+1 = k+i and ki,i−1 = k
−
i , (6.13)
where i = 1, 2, . . . , N . A fixed affinity A implies the constraint∏N
i=1 k
+
i∏N
i=1 k
−
i
= eA (6.14)
on the transition rates. Different choices of the transition rates that fulfill this restriction
can lead to different generating functions, as shown in Fig. 6.1 for a randomly generated
set of unicyclic networks. In particular, if the transition rates are uniform, i.e., k+i = k
+
and k−i = k
−, as considered in Sec. 4.3, the generating function divided by the average
cycle current λ(z)/Js becomes λARW(z,A, N ), which is given in Eq. (4.39). The
opposite extreme choice for the transition rates is the case where the network behaves
effectively like there was only one link between states (N = 1) and all the affinity is
concentrated in this single link. In this case, λ(z)/Js becomes λARW(z,A, 1), which
fulfills λARW(z,A, 1) ≥ λARW(z,A, N ).
92
6.3 Numerical illustration
-1.5
-1
-0.5
0
0.5
-8 -7 -6 -5 -4 -3 -2 -1 0 1
λ
(z
)/
Js
z
A = 7 A = 5 A = 3
Figure 6.2: Generating function λ(z) scaled by the steady state current Js in randomly generated
unicyclic networks with N = 10 for three families of affinities A. The black curves
refer to the lower and upper bound from Eq. (6.15).
These considerations suggest for unicyclic networks the bounds
λARW(z,A, 1) ≥ λ(z)/Js ≥ λARW(z,A, N ). (6.15)
The lower bound corresponds to Eq. (6.11), which becomes particularly simple in the
unicyclic case, where A∗ and n∗ are uniquely determined by the single cycle affinity A
and length N , and where there is only a single cycle current that relates to the entropy
production rate as σ = AJs. The upper bound in Eq. (6.15) is specific for unicyclic
networks and is proven in Appendix 6.A.
In Fig. 6.2, we show the bounds in Eq. (6.15) for different values of the affinity.
For smaller A the bounds are closer to each other. In the linear response regime, up
to quadratic order in z, they become the same and equal to the dissipation-dependent
bound in Eq. (4.42).
The bounds in Eq. (6.15) lead to the bounds
coth
(A
2
)
≥ F ≥ 1
N
coth
( A
2N
)
(6.16)
on the Fano factor defined in Eq. (4.22). The lower bound is an affinity dependent
bound on the Fano factor that has been obtained in [1]. In the formal limit A → ∞ it
becomes F ≥ 1/N . This bound for formally divergent affinity is a key result in statistical
kinetics [112] as it allows for an estimate on the number of states N from measurements
of the Fano factor. The upper bound on F in Eq. (6.16) is a new result. It is a refinement
of the known result F ≤ 1, which is also valid in the limit A → ∞ [112].
Next, we illustrate the bound (6.11) for multicyclic networks. In order to explain how
to identify A∗/n∗ we consider the network of states in Fig. 6.3a. We arbitrarily choose
93
6 Affinity- and topology-dependent bound on current fluctuations
(a)
-0.2
0
0.2
-1 0
λ
s(
z)
/σ
z
(b)
-0.2
0
0.2
-1 0
λ
s(
z)
/σ
z
(c)
-0.2
0
0.2
-1 0
λ
s(
z)
/σ
z
(d)
-0.2
0
0.2
-1 0
λ
α
(z
)/
σ
zJsα/σ
(e)
Figure 6.3: Generating functions for entropy change (b-d) and randomly selected individual
currents Jα (e) for the network shown in panel (a). The affinities are A = (1, 2, 3)
in (b), A = (11, 12, 13) in (c) and (e), and A = (−11, 12, 13) in (d). The black
curves correspond to the dissipation-dependent bound and the red dashed curves
correspond to the affinity- and topology-dependent bound. The random transition
rates were generated as explained in Ref. [7, Appendix B]. In panel (b)A∗/n∗ = 1/3,
in panels (c) and (e)A∗/n∗ = 11/3, and in panel (d)A∗/n∗ = 1/4. For small values
of A∗/n∗, as in panels (b) and (d), the parabolic and hyperbolic cosine bound are
closer to each other.
the cycles (1, 4, 2, 1) with affinity A1, (2, 4, 3, 2) with affinity A2, and (1, 3, 4, 1) with
affinity A3 as the three fundamental cycles. Any other cycle in the network is just a
composition of these fundamental cycles; for example, the cycle (1, 4, 3, 2, 1), which is
marked with a red dotted line in Fig. 6.3a, with affinity A1 +A2 is the sum of the first
and second fundamental cycles. If the affinities areA = (1, 2, 3) then the cycle with the
smallest affinity per number of states is the fundamental cycle (1, 4, 2, 1). Therefore, in
this case A∗ = 1 and n∗ = 3. If the affinities areA = (−11, 12, 13), where the negative
affinity is with respect to counter-clockwise direction in Fig. 6.3a, the cycle with minimal
affinity per number of states is (1, 4, 3, 2, 1). In this case A∗ = 1 and n∗ = 4.
The bound in Eq. (6.11) for the current associated with the entropy production can be
interpreted as follows. Given a network of states with fixed affinities, the transition rates
that lead to the smallest possible generating function λs (z)/σ are those for which the
cycle with smallestA/n dominates the network. This cycle dominates the network if the
94
6.4 Uniform cycle decomposition
transition rates within the cycle are large, transition rates to leave the cycle are small, and
transition rates to return to the cycle are large. With this choice for the transition rates,
the multicyclic network is effectively a unicyclic network with affinity A∗ and number
of states n∗, for which the lower bound in Eq. (6.15) holds. Any other choice of rates
will just add cycles with a smaller affinity per number of states, which cannot decrease
fluctuations.
In Fig. 6.3 we illustrate the bound (6.11) along with the dissipation-dependent bound
for sample realizations of a four-state network with various combinations of cycle affini-
ties. If the cycle relevant for the bound has a rather small affinity per number of states,
which is often the case in a large network of states, the bound (6.11) is only slightly
stronger than the dissipation-dependent bound (4.45), as visible in Fig. 6.3b and 6.3d.
An often tighter bound for this situation is derived in Sec. 7.1. Our numerics indicates
that an affinity-dependent upper bound on the generating function in the multicyclic case
does not exist. For fixed affinities, the generating function can become arbitrarily close
to the trivial bound λs (z) < 0 for −1 < z < 0, visible in Fig. 6.3b-d.
6.4 Uniform cycle decomposition
For multicyclic networks, the choice of a minimal complete set of cycles is not unique.
We make use of this ambiguity to expand the stationary current jsi j in a specific set of
cycles {Cβ}. This set is constructed in a way that every cycle Cβ contributing to the
stationary current, i.e., for which Jsβ , 0, satisfies the following properties, which will
turn out to be essential for our proof of the affinity dependent bound (6.10):
(i) The links of every cycle are aligned with the stationary current, i.e.,
sgn χ βi j = sgn j
s
i j = sgn si j (6.17)
for all i, j and β. We call such cycles uniformwith respect to jsi j . The equality with
sgn si j follows directly from the definitions (6.1) and (6.2) and shows via Eq. (6.7)
that uniform cycles have positive affinity.
(ii) All stationary cycle currents are strictly positive, Jsβ > 0 for all β.
It can be easily checked that an arbitrary cycle decomposition, for example the de-
composition into fundamental cycles by Schnakenberg [27], does not necessarily meet
these conditions. A construction of a cycle decomposition that satisfies at least condi-
tion (ii) was introduced by J. MacQueen [123], its physical relevance and proof is also
discussed in Ref. [124]. This decomposition, however, refers only to networks with irre-
versible transitions. The network of our present setup with reversible transitions could
be mapped on such a network by replacing every link by two irreversible links with
95
6 Affinity- and topology-dependent bound on current fluctuations
1
1
3
4
2 2
2
3
3
2 2
1
2
2
2 2
Figure 6.4: Example for a decomposition of a 5-state network into a complete set of three
independent uniform cycles. Links are labeled by the strength of the currents j (β)i j ,
whose direction is marked by arrows. For each of the three iteration steps, a link
with minimal current is labeled in red and the uniform cycle Cβ including this link
is shown as a yellow polygon with red border.
antiparallel direction, yet this procedure does not necessarily lead to a decomposition
satisfying condition (i).
Here, we present a variation of the algorithmofMacQueen for networkswith genuinely
reversible transitions, which generates a set of cycles satisfying both conditions (i) and
(ii), as exemplified in Fig. 6.4.
• Initialization. Set j (1)i j ≡ jsi j .
• Iteration over β ≥ 1. Locate a link (i∗, j∗) with minimal positive current j (β)i∗ j∗ and
set
Jsβ ≡ min
i, j | j (β)i j >0
j (β)i j = j
(β)
i∗ j∗ . (6.18)
Construct Cβ as a closed self avoiding path starting with the link (i∗, j∗) and
passing only along links in the direction for which j (β)i j > 0. This path is not
necessarily unique, one of several such paths can be chosen freely as Cβ. Using
the directed adjacency matrix χ βi j corresponding to Cβ, as defined in Eq. (6.4), set
j (β+1)i j = j
(β)
i j − Jsβ χ βi j . (6.19)
• Terminate when j (β+1)i j = 0 for all i, j.
By construction, the currents j (β)i j satisfy Kirchhoff’s law (6.5) for every β. As a
consequence, it is always possible to construct an appropriate uniform cycle Cβ following
the direction of j (β)i j : For every node where there is a way in, there must also be a way
96
6.5 Proof
a) b) c)
Figure 6.5: Illustration of a uniform cycle Cβ starting with the link (i∗, j∗). The direction of j (β)i j
is indicated by the arrows, the (attempted) cycle is shown as solid blue lines. Neither
dead ends (a) nor “dead-cycles” (b) are consistent with Kirchhoff’s law (6.5).
out, which rules out dead ends (see Fig. 6.5a). Similarly, it is not possible to run into
a cycle without exit that does not include the starting node i∗ (see Fig. 6.5b). On a
finite set of states, every self avoiding path must reach the starting point i∗ again after at
most N steps. Typically, there is more than one cycle starting with (i∗, j∗) meeting these
conditions. Any of these cycles can be selected as Cβ. Since we use in every iteration
the minimal current as new cycle current, the individual currents j (β+1)i j are either zero or
have the same sign as jsi j . Thus, every cycle that is uniform with respect to the currents
j (β)i j is also uniform with respect to j
s
i j , as required by condition (i). The matrices χ
β
i j
are linearly independent, since for every β the link (i∗, j∗) is no longer contained in all
the subsequent cycles. By construction, the algorithm leads to the decomposition
jsi j =
∑
β
Jsβ χ
β
i j . (6.20)
Typically, the algorithm terminates when β reaches the number of fundamental cycles.
It cannot terminate later since all cycles Cβ are linearly independent. It may happen that
the algorithm terminates earlier, so that the set of cycles {Cβ} is not complete, i.e., it
cannot be used to represent arbitrary currents different from the stationary current. In
this case the set {Cβ} can be completed by adding further linear independent non-uniform
cycles. The stationary current of these cycles is always zero.
6.5 Proof
Equipped with the decomposition in uniform cycles, we can now prove the affinity
dependent bound (6.10) on the large deviation function. As a starting point we use,
building on Ref. [29] and analogously to Refs. [8, 64], the universally valid bound for
the large deviation function of the distribution of the vector j = ( ji j ) of all fluctuating
currents along links,
I ( j) ≤
∑
i< j
⎡⎢⎢⎢⎢⎣
√
(asi j )
2 + ( jsi j )
2 −
√
(asi j )
2 + ( ji j )2 + ji j *,arsinh
ji j
asi j
− arsinh
jsi j
asi j
+-
⎤⎥⎥⎥⎥⎦ , (6.21)
97
6 Affinity- and topology-dependent bound on current fluctuations
with asi j ≡ 2
√
psi p
s
j ki j k ji. This relation holds for all values of j that are consistent
with Kirchhoff’s law (6.5), otherwise I ( j) = ∞. Eq. (6.21) follows from the exact
expression (3.82) for the level 2.5 large deviation function I ( j, p) for the joint distribution
of the fluctuating current j and the fluctuating density p = (pi). Using the contraction
principle (3.85) with p = ps as a trial function yields immediately Eq. (6.21) as bound.
This particular choice of the trial function ensures that the bound is saturated for the
most likely, stationary current and that one can ultimately identify physically meaningful
steady-state averages such as the entropy production σ in the bound.
At first, we assume for simplicity that the stationary current jsi j is non-zero along all
links with ki j > 0. Then, we can rewrite the bound (6.21) as
I ( j) ≤
∑
i< j
jsi jψ( ji j/ j
s
i j, si j ), (6.22)
where the function ψ(ζ, s) is defined as
ψ(ζ, s) ≡ ζ arsinh(ζ/b) − ζ s/2 −
√
b2 + ζ2 +
√
b2 + 1 (6.23)
with
b ≡ [sinh(s/2)]−1. (6.24)
The functionψ(ζ, s) is the large deviation function (4.41) for the current in an asymmetric
random walk with uniform transition rates, with affinity per step s, and with stationary
current Js = 1 [35]. For this highly symmetric network, there is only one possible
fluctuating current J and psi = p
∗
i (J) = 1/N holds independently of J, so that Eq. (6.22)
becomes an equality. For positive s and fixed ζ , ψ(ζ, s) is concave in s, moreover,
ψ(ζ, s)/s decreases monotonically with increasing s, as proven in Appendix 6.B and
Appendix 6.C, respectively. These essential properties are demonstrated in Fig. 6.6. The
Legendre transform of ψ(ζ, s) is
λ(y, s) ≡ max
ζ
[yζ − ψ(ζ, s)] = [cosh(y + s/2) − cosh(s/2)]/ sinh(s/2). (6.25)
Using the decomposition in the uniform cycles {Cβ} from Eq. (6.20), for which all si j
along a cycle become positive, we can write Eq. (6.22) as
I ( j) ≤
∑
β
Jsβ
∑
(i, j)∈Cβ
ψ( ji j/ jsi j, si j ). (6.26)
Next, we set ji j = ξ jsi j for all links (i, j) of the network. This choice simplifies the
following steps and ensures that the fluctuating currents ji j are consistentwithKirchhoff’s
98
6.5 Proof
0
0.5
1
1.5
2
−2 −1 0 1 2 3 4
ψ
(ζ
,s
)/
s
ζ
s → 0
s → ∞
0
5
10
15
20
0 5 10 15
ψ
(ζ
,s
)
s
ζ = −1.6
ζ = −1
ζ = 0
ζ = 1
ζ = 5
Figure 6.6: Left: The function ψ(ζ, s)/s as a function of ζ for values of s ranging from 1
(violet) to 101.2 (red) with logarithmic spacing. For fixed ζ and s > 0, ψ(ζ, s)/s
decreases monotonically in s, as required in Eq. (6.28). The limiting curves for
lims→∞ ψ(ζ, s)/s = (|ζ | − ζ )/2 and for lims→0 ψ(ζ, s)/s = (ζ − 1)2/4 are shown in
black. Right: The function ψ(ζ, s) as a function of s for values of ζ ranging from
−2 (violet) to 6 (red). Selected values of ζ are shown as solid lines, in particular
ψ(−1, s) = s and ψ(1, s) = 0. Obviously, ψ(ζ, s) is concave for s ≥ 0.
law (6.5). Having ensured via the uniform cycle decomposition that all si j are positive,
we can apply Jensen’s inequality to the second sum in Eq. (6.26), in which the summation
index runs over the nβ links of the cycle Cβ. Since the Jsβ are positive, this procedure
leads to
I (ξ js) ≤
∑
β
Jsβnβ
∑
(i, j)∈Cβ
1
nβ
ψ(ξ, si j ) ≤
∑
β
JsβA β (nβ/A β) ψ(ξ,A β/nβ), (6.27)
where we have used Eq. (6.7). For a coarser bound, we identify the uniform cycle with
minimal A β/nβ ≡ A∗/n∗, which is positive as explained below Eq. (6.17). Using the
monotonic decrease of ψ(ζ, s)/s, the positivity of JsβA β, and Eq. (6.8), we obtain
I (ξ js) ≤ σ (n∗/A∗) ψ(ξ,A∗/n∗). (6.28)
Finally, we adopt a more elegant formulation of the large deviation function taking the
cycle currents Jβ as argument instead of ( ji j ) = j, i.e., we write
I(J ) ≡ I *.,
∑
β
Jβχ β
+/- . (6.29)
We thus avoid currents that are inconsistent with Kirchhoff’s law, for which the large
deviation function would become infinite. The generating function corresponding to the
99
6 Affinity- and topology-dependent bound on current fluctuations
large deviation function (6.29) turns out to be bounded by
λ(z) = max
J
⎡⎢⎢⎢⎢⎢⎣
∑
β
zβ Jβ − I( j)
⎤⎥⎥⎥⎥⎥⎦
≥ max
J=ξJs
[
z · Jsξ − I (ξ js)]
≥ max
ξ
[
z · Jsξ − σ (n∗/A∗) ψ(ξ,A∗/n∗)]
=
σ
A∗/n∗ λ(z · J
s(A∗/n∗)/σ,A∗/n∗)
= σ
cosh[(z · Js/σ + 1/2)A∗/n∗] − cosh[A∗/(2n∗)]
(A∗/n∗) sinh [A∗/(2n∗)]
≡ B(z · Js,A∗/n∗, σ), (6.30)
where we have used Eq. (6.28) for the third line and Eq. (6.25) for the fifth line. Although
the uniform cycle decomposition was essential in the steps leading to this result, its final
statement is no longer dependent on the specific choice of independent cycles, as can
be seen from the following argument. In a different complete set of cycles the cycle
currents are represented as J˜α =
∑
β Cαβ Jβ with some transformation matrix Cαβ. The
corresponding generating function then reads
λ˜( z˜) = lim
t→∞
1
t
ln
⟨
exp
⎡⎢⎢⎢⎢⎣
∑
α
z˜α J˜αt
⎤⎥⎥⎥⎥⎦
⟩
= lim
t→∞
1
t
ln
⟨
exp
⎡⎢⎢⎢⎢⎢⎣
∑
α,β
z˜αCαβ Jβt
⎤⎥⎥⎥⎥⎥⎦
⟩
= λ
(
CT z˜
)
(6.31)
and satisfies the bound
λ˜( z˜) ≥ B(CT z˜ · Js,A∗/n∗, σ) = B( z˜ · J˜s,A∗/n∗, σ), (6.32)
which finally proves Eq. (6.10) for arbitrary cycle decompositions. It should be noted
that, a priori, A∗/n∗ still refers to the minimal A β/nβ in a set of uniform cycles.
However, in a possible application, where the network topology and the cycle affinities
are known but individual transition rates are not known, the more general result (6.10),
where A∗/n∗ refers to the minimum over all cycles, might lead to a weaker bound that
is more useful. Since uniform cycles always have affinityA β > 0, the minimization can
be restricted to cycles with positive affinity. Thus, in the physically important case of a
network containing cycles with zero affinity, Eq. (6.10) provides a bound that is stronger
than the dissipation-dependent bound that is obtained for A∗/n∗ → 0. This fact is in
accordance with the insight from Ref. [125], stating that coarse graining such cycles
with zero affinity has little effect on the fluctuations of the entropy production.
100
6.6 Vanishing currents and equilibrium
6.6 Vanishing currents and equilibrium
So far, we have considered only the case of non-vanishing stationary currents jsi j along
all links. If this is not the case for some links (although ki j > 0), we have to split the
bound for the large deviation function in Eq. (6.21) in two parts as
I ( j) ≤
∑
i< j | jsi j,0
jsi jψ( ji j/ j
s
i j, si j )+
∑
i< j | jsi j=0
⎡⎢⎢⎢⎢⎣ ji j arsinh
ji j
asi j
−
√
(asi j )
2 + j2i j + a
s
i j
⎤⎥⎥⎥⎥⎦ . (6.33)
Evaluating this bound along currents j = ξ js, the second sum vanishes while the
first sum leads along the same lines as before to the bound (6.30). However, in the
equilibrium case, where all stationary currents vanish, this procedure leads merely to
the trivial statement λ(z) ≥ 0. While typical fluctuations in equilibrium systems can be
well described within linear response theory, rare fluctuations exhibit many features akin
to systems far from equilibrium [126]. A non-trivial bound for these rare fluctuations
in equilibrium systems, similar to the one derived in Ref. [7] for unicyclic networks, is
obtained by letting the affinityAα of a single fundamental cycle go to zero while keeping
the affinities of the other fundamental cycles fixed at zero. Then, none of the cycles can
have an affinity smaller than Aα, so that A∗ = Aα. The length of the relevant cycle n∗
can be bounded by the total number of states N ≥ n∗. Due to the Einstein relation (4.27),
the stationary current Jsα is given for small Aα by
Jsα = DαAα + O(A2α), (6.34)
where Dα is the diffusion coefficient associated with the current Jα. With the entropy
production (6.8) reducing to σ = JsαAα = DαA2α, we thus obtain as bound on the
generating function for the equilibrium fluctuations of Jα
λα (z) ≥ limAα→0 B(zDαAα,Aα/N, DαA
2
α) = 2N2Dα (cosh(z/N ) − 1). (6.35)
This bound is saturated for z ≪ 1, where λα (z) depends quadratically on z according to
the Gaussian distribution of typical fluctuations within linear response. The probability
of extreme fluctuations beyond linear response deviates the more from this Gaussian
shape the smaller the number of states in the network is.
6.7 Conclusion
In this Chapter, we have presented the bound (6.10) as a major refinement of the
dissipation-dependent bound (4.44) on the spectrum of current fluctuations in a Markov
network. This refined bound depends additionally on the minimumA∗/n∗ of the affinity
per cycle length in any cycle of the network, which is accessible when the network
101
6 Affinity- and topology-dependent bound on current fluctuations
topology and cycle affinities are known. Just as the dissipation-dependent bound, the
refined bound is universally valid for arbitrary currents in arbitrary networks and entails
a bound on the Fano factor in enzyme kinetics, previously conjectured in Ref. [119].
The bound (6.10) on the generating function of current fluctuations has the shape of a
hyperbolic cosine, which is also the shape of the generating function of the current in an
asymmetric random walk with uniform rates. Indeed, for this simple system the bound
is saturated. Making the rates non-uniform and adding further cycles to the network
typically drives the generating function away from this bound, as we have illustrated
numerically.
As a preliminary for the proof of the bound, we have shown that for every Markov
network it is possible to construct a set of independent cycles such that all affinities
and stationary cycle currents are positive and that all stationary currents along links are
aligned with the direction of the cycles. We call this a decomposition in uniform cycles.
Based on such a decomposition of the stationary currents, the bound (6.10) can be proven
within the framework of level 2.5 large deviations. As a corollary, we have also proven
a universal bound on equilibrium fluctuations, given in Eq. (6.35), that depends only on
the diffusion coefficient and on the number of states.
102
Appendices to Chapter 6
6.A Proof of the upper bound on the generating function
for unicyclic networks
For unicyclic networks with N states, the tilted Markov generator (3.13) has the tridiag-
onal shape
L(z) =
*............,
−r1 k˜21 0 0 . . . k˜N1
k˜12 −r2 k˜32 0 . . . 0
0 k˜23 −r3 k˜43 . . . 0
0 0 k˜34 −r4 . . . ...
...
...
. . .
. . .
. . . k˜N,N−1
k˜1N 0 0 . . . k˜N−1,N −rN
+////////////-
(6.36)
with k˜i j ≡ ki jezdi j . The displacements di j must satisfy di j = −d ji. If the observable of
interest is the number of turnovers, the displacements must add up to the cycle affinity
d12 + d23 + · · · + dN1 = 1. It should be kept in mind that any statistical quantity in
the long time limit (in particular the generating function and the rate function) must be
independent of the specific choice of the individual di j .
The characteristic polynomial associated with the matrix (6.36) reads
χ(z, x) ≡ det(L(z) − x1N )
=
∑
π
(−1)π
N∏
i=1
(Liπ(i) (z) − xδiπ(i)), (6.37)
where the sum runs over all permutations π of the indices i = 1, . . . , N and 1N is the
N × N identity matrix. We identify 0 ≡ N and N + 1 ≡ 1 for the indices of matrix
entries. There are two types of terms in (6.37) that contain a specific rate k˜i+1,i: the
contribution from the next row i + 1 can either be k˜i,i+1 or k˜i+2,i+1. For the former type
the z-dependence cancels out due to di+1,i = −di,i+1 and we end up with the constant
factor k˜i+1,i k˜i,i+1 = ki+1,iki,i+1. Terms of the latter type must also contain k˜i,i−1 as the
only possible contribution from the previous column i − 1. Iteratively, we see that there
can be only one term of this type, namely the one that contains all forward transitions
k˜12 k˜23 . . . k˜N1 = k12k23 . . . kN1ez ≡ Γ+ez . (6.38)
103
6 Affinity- and topology-dependent bound on current fluctuations
①
P✭①✮
②
✶
①
✶
Figure 6.A.1: The polynomial P(x) for a generic unicyclic network with N = 6 states. The
tangent at x = 0 and the values y = y1 and x = x1 are shown as dashed lines.
An analogous argument can be set up for the lower off-diagonal of the matrix with the
z-dependent term
k˜21 k˜32 . . . k˜1N = k21k32 . . . k1Ne−z
≡ Γ−e−z = Γ+e−(z+A) . (6.39)
All other terms in the determinant (6.37) do not depend on z and we can write
χ(z, x) = (−1)N+1[Γ+ez + Γ−e−z − (Γ+ + Γ−) − P(x)] (6.40)
with some polynomial P(x) that is independent of z and the specific choice of the di j .
The alternating prefactor is due to the fact that the permutations associated with the
terms (6.38) and (6.39) are either odd or even, depending on the number of states N .
The generating function is thus given by
λ(z) = P−1
(
Γ+ez + Γ−e−z − Γ+ − Γ−)
= P−1
(
2
√
Γ+Γ−[cosh(z +A/2) − cosh(A/2)]
)
, (6.41)
where the function P−1(y) returns the root of the polynomial P(x)− y that has the largest
real part. Due to the Perron-Frobenius theorem, this root must be real for all arguments
occurring in (6.41), i.e., for all y ≥ y1 ≡ 2
√
Γ+Γ−[1− cosh(A/2)]. The root associated
with the minimal argument y1 is x1 ≡ P−1(y1) = minz λ(z) = λ(−A/2). Obviously,
the polynomial P(x) (see Fig. 6.A.1) has the properties P(0) = χ(0, 0) = 0 and
lim
x→∞ P(x) = (−1)
N lim
x→∞ χ(z, x)
= lim
x→∞(−1)
N det(−x1N ) = +∞. (6.42)
104
6.B Proof of the concavity of ψ(ζ, s) in s
Since the matrix L(−A/2) can be brought to a symmetric form by choosing di j =
ln
(
ki j/k ji
)
/A, the corresponding characteristic polynomial P(x) − y1 has only real
roots xi with x1 denoting the largest one. The second derivative of P(x) is
P′′(x) =
d2
dx2
⎡⎢⎢⎢⎢⎣y1 +
N∏
i=1
(x − xi)
⎤⎥⎥⎥⎥⎦ =
N∑
i=1
N∑
j=1
j,i
N∏
ℓ=1
i,ℓ, j
(x − xℓ). (6.43)
For x > x1 this expression is positive so that P(x) is convex. As a consequence, the
inverted function P−1(y) is concave for the relevant arguments y > y1. Hence it satisfies
P−1(y) ≤ (P−1)′(0) y (6.44)
with equality for y = 0. This relation leads to the upper bound
λ(z) ≤ 2√Γ+Γ−(P−1)′(0) [cosh(z +A/2) − cosh(A/2)] (6.45)
holding with equality for z = 0. From Eq. (6.41), we see that the prefactor in this bound
is equal to the stationary current
Js = λ′(z) = 2
√
Γ+Γ−(P−1)′(0) sinh(A/2), (6.46)
which leads to the upper bound in Eq. (6.15).
6.B Proof of the concavity of ψ(ζ, s) in s
The derivative of ψ(ζ, s) in Eq. (6.23) can be written as
∂sψ(ζ, s) =
1
2
√
b2 + ζ2
√
b2 + 1 − b
2
2
− ζ
2
. (6.47)
As a function of b = [sinh(s/2)]−1 (which decreases monotonically in s), this expression
increases monotonically for b > 0, since
∂b∂sψ(ζ, s) =
b
2
*.,
√
b2 + ζ2
b2 + 1
+
√
b2 + 1
b2 + ζ2
− 2+/- ≥ 0 (6.48)
(note that x + 1/x ≥ 2 for x > 0). Therefore, ∂sψ decreases monotonically in s and
∂2s ψ(ζ, s) < 0 (6.49)
holds for all ζ ∈ R and s > 0.
105
6 Affinity- and topology-dependent bound on current fluctuations
6.C Proof of the monotonic decrease of ψ(ζ, s)/s in s
The monotonic decrease of ψ(ζ, s)/s in s for s > 0 is equivalent to the monotonic
increase of its Legendre transform
µ(z, s) ≡ max
ζ
[ζ z−ψ(ζ, s)/s] = λ(z f , s)/s = cosh[(z + 1/2)s] − cosh(s/2)
s sinh(s/2)
≡ A(z, s)
B(s)
.
(6.50)
The derivative ∂sµ(z, s) is non-negative if
0 ≤ C(z) ≡∂sA(z, s) B(s) − A(s) ∂ f B(s)
= [(z + 1/2) sinh((z + 1/2)s) − (1/2) sinh(s/2)] s sinh(s/2)
− [cosh((z + 1/2)s) − cosh(s/2)] [sinh(s/2) + (s/2) cosh(s/2)] .
(6.51)
Equating ∂zC(z) to zero leads to
2(z + 1/2) tanh(s/2) = tanh[(z + 1/2)s], (6.52)
which has the three solutions z = −1,−1/2, 0. The stationary points z = 0 and z = −1
are minima since
∂2zC(z = 0) =
s2
2
(sinh s − s) > 0 (6.53)
and C(z) is symmetric with respect to −1/2. Thus, the center of symmetry at z = −1/2
is a maximum of C(z) and the global minimum of C(z) is given by C(0) = C(−1) = 0.
106
7 Activity-dependent and asymptotic
bound
In this Chapter, we explore further universal bounds on fluctuations of time-additive
observables that depend on few characteristic features of the system and its steady
state. These bounds are complementary to the dissipation-dependent bound, as they can
become strong in cases where that bound is rather weak, such as for strong driving far
away from linear response and for extreme fluctuations far away from typical fluctuations.
The first main result in this Chapter is a bound that depends on the overall activity
of the system, i.e., the total number of transitions per unit time. As a bound on the
generating function, this bound has the shape of an exponential function. Evaluated
for the typical fluctuations, it implies a relation akin to the thermodynamic uncertainty
relation, where the activity replaces the entropy production rate.
The second main result is a bound which becomes strong for extreme fluctuations as
described by large values of the parameter in the generating function. It captures the
asymptotic behavior of the generating function and depends essentially on the topology
of the Markov network.
7.1 Activity-dependent bound
In Ref. [7], we have derived an activity-dependent, exponential bound on the generating
function for currents in a stationary state. The proof therein builds on inequalities for
the Perron-Frobenius eigenvalue of the tilted Markov generator [127]. A simpler proof
using the formalism of level 2.5 large deviations has been devised by Garrahan [128]
and will be followed here.
We consider a Markovian network with a finite set of states and use the same notation
as introduced in Sec. 4.1. The observable of interest, X (t), is specified according to
Eq. (3.4) with a distance matrix di j and ai = 0. Unlike for the previous bounds, we do
not require di j to be antisymmetric. Hence, the result to follow holds not only for current
observables but also for counting observables [128].
The scaled fluctuating observable is given by J = X (t)/t =
∑
i j µi jdi j with the
empirical flow defined in Eq. (3.69). Its expectation value in the stationary state is
Js = ⟨X (t)⟩ /t =
∑
i j
µsi jdi j =
∑
i j
psi ki jdi j . (7.1)
107
7 Activity-dependent and asymptotic bound
The large deviation function for J can be obtained through contraction from the level
2.5 large deviation function (3.75) as
I (J) = min∑
i j µi jdi j=J
I ({pi}, {µi j }). (7.2)
The ansatz pi = psi and µi = ξµ
s
i with ξ ≥ 0 satisfies Kirchhoff’s law and leads to a
scaled current J = ξ Js. The incomplete contraction (3.85) using this ansatz leads to the
bound
h(ξ) ≡ I (J = ξ Js) ≤ I ({psi }, {ξµsi j }) =
∑
i
ripsi +
∑
i j
psi ki jξ (ln ξ−1) = R[1−ξ−ξ ln ξ]
(7.3)
with the overall stationary activity
R ≡
∑
i
ripsi =
∑
i j
psi ki j (7.4)
that gives the average number of transitions in the whole network per unit time. The
Legendre transformation of this bound yields as bound on the generating function
λ(z) ≥ R(ezJs/R − 1). (7.5)
In particular, this bound holds for any current observable, such that the generating
function (4.6) for the set of cycle currents can be written in a vectorial notation as
λ(z) ≥ R(ez·Js/R − 1). (7.6)
Due to the Gallavotti-Cohen symmetry (4.11) the activity-dependent bound can also be
written as
λ(z) ≥ R(e(−A−z)·Js/R − 1). (7.7)
This bound is sharper than (7.6) for z · Js < −A · Js/2 = σ/2. Combining Eqs. (7.6)
and (7.7) we obtain the activity-dependent bound
λ(z) ≥ R
[
e(|σ/2+z·J
s |−σ/2)/R − 1
]
. (7.8)
For an individual current Jα, the activity-dependent bound reads
λα (z) ≥ R
[
e( |σ/2+zJ
s
α |−σ/2)R − 1
]
. (7.9)
The choice z = zA in Eq. (7.8) leads to
λs (z) ≥ R
[
e(|σ/2+zσ |−σ/2)/R − 1
]
(7.10)
108
7.1 Activity-dependent bound
−10
0
10
20
−20 −15 −10 −5 0 5
0
10
20
30
40
50
−1 0 1 2 3
z
λ(z)
bound
ξ
h(ξ)
bound
Figure 7.1: Generating function (left) and rate function (right) for a five-state unicyclic network
with rates ln k+i = (3, 3, 2, 3, 2) and ln k
−
i = (−1, 0,−1, 0, 0), affinityA = 15, current
Js ≃ 2.25, and activity R ≃ 12.9. The functions are shown as solid lines and
the activity-dependent bound (7.8) as dashed lines. Analytic continuations of the
piecewise defined functions are shown as dotted curves.
for the entropy change.
In terms of the large deviation function for an individual current Jα, the application
of the fluctuation refines Eq. (7.3) to
hα (ξ) ≤
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
R[1 + ξ − ξ ln |ξ |] − σξ, ξ ≤ −e−σ/(2R),
R[1 − ξ + ξ ln |ξ |], ξ ≥ e−σ/(2R),
R[1 − e−σ/(2R)] − σξ/2, otherwise,
(7.11)
where the intermediate, linear part stems from the construction of the convex envelope.
An illustration of the activity-dependent bound (7.8) and (7.11), respectively, is pro-
vided in Fig. 7.1. This bound is typically tighter than the dissipation-dependent bound
for far from equilibrium conditions, i.e., for large affinity. For example, for a random
walk on a unicyclic network with N sites, a uniform forward stepping rate k and a van-
ishing backward stepping rate, which implies divergent affinity, the bound in Eq. (7.8)
is saturated. Specifically, in this case the generating function is
λ(z) = k
(
ez/N − 1
)
, (7.12)
the activity is R = k and the cycle current Js = k/N . For vanishing current at
equilibrium, the activity-dependent bound reduces to the trivial statement λ(z) ≥ 0.
109
7 Activity-dependent and asymptotic bound
Our numerics indicates that the affinity- and topology-dependent bound is always
tighter than the activity-dependent bound in unicyclic networks. For multicyclic net-
works the activity-dependent bound can be tighter. Furthermore, contrary to the affinity-
and topology-dependent bound, the activity-dependent bound does not require knowl-
edge of the topology of the network of states, only the average activity and, when using
the variant (7.8), the average entropy production are required.
Using (4.22) in the activity-dependent bound (7.8) for an individual current leads to
F ≥ Js/R. (7.13)
This relation provides a lower bound on the dispersion of an individual current, charac-
terized by the Fano factor F, in terms of its average Js and the activity R.
7.2 Asymptotic bound for unicyclic networks
The asymptotic bounds discussed in the following are exact results that become tighter
than all previous bounds for large values of |z |. First we consider a unicyclic network
with N states and affinity A. In this case, we can prove the following bound on the
generating function:
λ(z) ≥ JλARW(z,A, N ) + rARW − 1N
N∑
i=1
ri, (7.14)
where λARW(z,A, N ) is the generating function of the asymmetric random walk defined
in Eq. (4.39), rARW ≡ k+ + k−, and
k± ≡ *,
N∏
i=1
k±i +-
1/N
. (7.15)
Our numerics indicate that with increasing |z | the difference between this bound and
the actual generating function tends to zero. This fact is quite remarkable given the
exponential growth of both functions. Unlike all other bounds presented so far, the
bound (6.13) is not saturated at z = 0. Only for the case of uniform rates, i.e., k±i = k
±,
the generating function (4.38) saturates the bound (7.14) globally.
In order to prove this asymptotic bound, we consider the path weight (3.73) of a
trajectory i(τ) in the unicyclic network, which is characterized by sojourn times Ti
and the number of jumps m±i out of the state i in forward and backwards direction,
respectively. The path weight then reads
p[i(τ)] = psi(0) exp
⎡⎢⎢⎢⎢⎣−
∑
i
(riTi + m+i ln k+i + m−i ln k−i )
⎤⎥⎥⎥⎥⎦, (7.16)
110
7.2 Asymptotic bound for unicyclic networks
where we use the same notation for the rates as in Eq. (6.13). The weight of the same
paths in a process with modified transition rates kˆ±i and corresponding exit rates rˆi and
stationary distribution pˆsi becomes
pˆ[i(τ)] = pˆsi(0) exp
⎡⎢⎢⎢⎢⎣−
∑
i
(rˆiTi + m+i ln kˆ+i + m−i ln kˆ−i )
⎤⎥⎥⎥⎥⎦ . (7.17)
For these modified transition rates we choose an asymmetric random walk with the
uniform rates k±i = k
± from Eq. (7.15), which leads to rˆ ≡ k+ + k−. Ensemble averages
using the path weight pˆ[i(τ)] are denoted as ⟨. . .⟩ARW. Since we are ultimately interested
in the long-time limit, we are free to choose a fixed initial state i(0) = 1 for all trajectories,
without loss of generality. The generating function for the integrated cycle current X (t)
can be rewritten as
λ(z) = lim
t→∞
1
t
ln
⟨
ezX[i(τ)]
⟩
= lim
t→∞
1
t
ln
∑
i(τ)
ezX[i(τ)]
p[i(τ)]
pˆ[i(τ)]
pˆ[i(τ)]
= lim
t→∞
1
t
ln
⟨
ezX[i(τ)]
N∏
i=1
(
k+i
k+
)m+i ( k−i
k−
)m−i
e−(ri−rˆi )Ti
⟩
ARW
, (7.18)
where the sum in the first line denotes a path integral over all stochastic trajectories.
The path dependent variables m±i count the jumps out of state i in forward or backward
direction and Ti is the total sojourn time in state i. These variables are identically
distributed in theARW-ensemble. The constant contribution from the initial probabilities
ps1 and pˆ
s
1 vanishes in the long-time limit.
The probability pˆ(X ) is the probability that the fluctuating current is X in the ARW-
ensemble. It is the sum of the weight of all trajectories for which the current is X . Using
this pˆ(X ), Eq. (7.18) can be written as
λ(z) =
1
t
ln
∑
X
pˆ(X )ezX
⟨
exp
[ N∑
i=1
m+i ln
(
k+i /k
+
)
+
N∑
i=1
m−i ln
(
k−i /k
−) − N∑
i=1
(ri − rˆ)Ti
] X
⟩
ARW
≥ 1
t
ln
∑
X
pˆ(X )ezX exp
[ N∑
i=1
⟨
m+i |X
⟩
ARW
ln
(
k+i /k
+
)
+
N∑
i=1
⟨
m−i |X
⟩
ARW
ln
(
k−i /k
−) − N∑
i=1
(ri − rˆ)t/N
]
, (7.19)
where the conditioned average in the first line represents a functional integration over
all trajectories with fluctuating current equal to X and we used Jensen’s inequality from
111
7 Activity-dependent and asymptotic bound
③
✷

✶
✲✁✂
✲✄✂
✲☎✂
✲✆✂
✂
✆✂
☎✂
✄✂
✁✂
✲✁✂ ✲✄✂ ✲
☎
✂ ✲
✆
✂ ✂
✆
✂
☎
✂ ✄✂ ✁✂
✂
✂✵☎
✂✵✁
✂✵
✝
✂✵✞
✆
Figure 7.1: Asymptotic bound for the house-shaped network with two fundamental cycles shown
in Fig. 4.1b. The color code represents the ratio (7.24) between the generating
function and the bound. Black dashed lines indicate the borders between sectors
with constant relevant cycles Cˆ(z). For each sector, the relevant cycle Cˆ(z) is shown
in white. The affinity of the three-cycle is A1 = 8 and the affinity of the four-cycle
is A2 = 6.
the first to the second line. Due to (7.15) the terms with the logarithms vanish, leading
to the final result in Eq. (7.14).
7.3 Asymptotic bound for multicyclic networks
In order to obtain an asymptotic bound that is also valid for multicyclic networks we
define an arbitrary closed path C, which is a sequence of jumps that finishes at the state
it started, as
C ≡ [i(1) → i(2) → · · · → i(nC) → i(1)], (7.20)
where nC is the length of the closed path. With this path we associate a geometric mean
of the transition rates
γC ≡ (ki(1),i(2)ki(2),i(3) . . . ki(nC ),i(1))1/nC (7.21)
and integer winding numbers m βC that count how often the elementary cycle β is com-
pleted within the path C. Applying a theorem valid for arbitrary non-negative matri-
ces [21, Lemma 3.5.3] to the matrix Li j (z) + δi j maxℓ rℓ we obtain
λ(z) + max
ℓ
rℓ ≥ f (z, C) ≡ γC exp *.,
1
nC
∑
β
m βC zβ
+/- (7.22)
112
7.3 Asymptotic bound for multicyclic networks
for any closed path C. The best bound on λ(z) in Eq. (7.22) is obtained by choosing
an optimal path Cˆ(z), which in principle depends on z, that maximizes the r.h.s of
Eq. (7.22) in the large z regime.
First we consider this optimal path for the unicyclic network. In this case, the optimal
path is a single cycle in the forward direction with mC = 1 if z > 0. If we consider a
path C with two cycles, i.e., mC = 2, the bound remains the same as the number of states
nC also doubles. If the closed path is not a direct cycle but contains, for example, one
backward jump, then γC can become larger. However, such a backward jump also makes
nC larger and hence, the exponent in Eq. (7.22) smaller. Since we are interested in the
large z regime, this second effect should be dominant. Hence, for z > 0 the bound (7.22)
leads to
λ(z) ≥ k+ exp(z/N ) −max
ℓ
rℓ . (7.23)
Even though this bound is different from (7.14), they both predict the same exponential
growth, with the same prefactor, for large z > 0. The same reasoning is valid for z < 0,
with the optimal path being a single cycle in the negative direction.
For multicyclic networks we consider the house-shaped network with five states shown
in Fig. 4.1b. This network consists of a cycle with three states and affinity A1 and a
cycle with four states and affinity A2. We choose these cycles to be the fundamental
cycles. This network also has a third cycle, which is the cycle with five states and affinity
A1 +A2. Given a vector z = (z1, z2), the optimal path is the cycle that maximizes the
r.h.s of Eq. (7.22). For large enough |z |, this optimal path depends only on the direction
of the vector. Clearly a path that includes other cycles will lead to a weaker bound.
A contour plot of the ratio f (z, Cˆ(z))/[λ(z) + maxℓ] for this house-shaped network
is shown in Fig. 7.1. Remarkably, the r.h.s. of (7.22) captures the leading order of the
asymptotics for large |z |, i.e., for large |z |
f (z, Cˆ(z))
λ(z) + maxℓ rℓ
→ 1. (7.24)
Only in the lines separating regions dominated by different cycles in Fig. 7.1 does this
ratio tend to slightly lower values. Along this line the dominant cycle is degenerate. As
we show below, relation (7.24) is valid for any multicyclic network. Hence, we conclude
that the asymptotic bound predicts the exponential growth of the generating function,
apart from exceptional regions in z where the optimal cycle is degenerate.
In order to prove the relation (7.24), we write the characteristic polynomial (3.27) of
the tilted Markov generator (4.7) as
0 = χ(z, λ(z)) =
N∏
i=1
[−ri − λ(z)] +
∑
C
(−1)CγnCC emC ·z
∏
j<C
[−r j − λ(z)], (7.25)
113
7 Activity-dependent and asymptotic bound
where the sum runs over all combinations of disjoint cycles in the underlying network and
(−1)C denotes the sign of the corresponding permutations in the determinant. Dividing
Eq. (7.25) by λ(z)N leads to
0 =
N∏
i=1
[−ri/λ(z) − 1] +
∑
C
(−1)C f (z, C)nCλ(z)−nC
∏
j<C
[−r j/λ(z) − 1]. (7.26)
We now analyze the limit |z | → ∞ with the direction z/|z | kept fixed. Making use of
the (already proven) lower bound (7.22) with the optimal path Cˆ ≡ Cˆ(z), we see that
ri/λ(z) and the terms with (mC · z)/nC < mCˆ · z/nCˆ vanish in this limit. Provided that
the optimal cycle is unique, we are left with
0 = (−1)N + lim
|z |→∞
(−1)1+nCˆ f (z, Cˆ)nCˆλ(z)−nCˆ (−1)N−nCˆ, (7.27)
which leads to
lim
|z |→∞
f (z, Cˆ)
λ(z)
= 1. (7.28)
In Eq. (7.24), the constant maxℓ rℓ is added to the denominator without harm, in order
to make the ratio positive everywhere. The essential ingredient in this proof is the
uniqueness of the optimal cycle Cˆ. Only in peculiar regions the vector z leads to more
than one cycle with the same value of mC · z/nC . For example, these regions show up
in Fig. 7.1 as the lines along which the ratio (7.24) differs from 1.
In Ref. [129], we have used this asymptotic approximation of the generating function
in the context of a current that corresponds to an apparent entropy production in a
system with a hidden degree of freedom. There, the bound is rather tight even for typical
fluctuations and helps to explain the approximate validity of a fluctuation relation.
7.4 Conclusion
In this Chapter, we have presented further bounds on current fluctuations, which are
complementary to the dissipation-dependent and affinity- and topology-dependent bound
discussed in Chapters 4 and 6, respectively. All of these bounds are summarized in
Fig. 7.1 for the generating function of a sample unicyclic network.
The activity-dependent bound introduces the activity, or total rate of transitions,
as a further characteristic property of the steady state that can be used to bound the
spectrum of current fluctuations. When expressed in terms of the generating function,
this bound has the shape of an exponential function. Interestingly, it holds not only for
current fluctuations but also for fluctuations of time-symmetric counting observables.
For current-like observables and given entropy production, the fluctuation relation can
114
7.4 Conclusion
−50
0
50
100
150
−20 −15 −10 −5 0 5
z
λ(z)
dissipation
aff.-top.
activity
asymptotic
Figure 7.1: Summary of the dissipation-dependent, the affinity- and topology-dependent, the
activity-dependent, and the asymptotic bound for a unicyclic network with four
states. Transition rates are ln k+i = (3, 4, 5, 4) and ln k
−
i = (0,−1, 1, 1), leading to the
affinity A = 15 and the current Js ≃ 10.307.
be used to symmetrize this bound. For systems with few states that are driven far
away from linear response, the activity-dependent bound is typically stronger than the
dissipation-dependent one.
The other type of bound discussed in this chapter is what we call the asymptotic bound.
It is typically weak when used as bound on the typical fluctuations, but becomes strong
for extreme fluctuations, where it outperforms all other bounds, capturing the asymptotic
behavior of the generating function exactly. The asymptotic bound depends essentially
on the topology of the underlying network, in particular, it requires the identification of
individual cycles that dominate ranges in the spectrum of current fluctuations.
115

8 Finite-time generalizations
All the bounds on current fluctuations we have considered so far have been focused on
fluctuations that occur in the limit of long time intervals. In fact, taking this long-time
limit has enabled us to formulate the results in terms of diffusion coefficients, Fano
factors, scaled cumulant generating functions, or large deviation functions. Estimating
large deviation functions experimentally is possible on the basis of large sets of data
and if the probability of untypical fluctuations decays slowly enough to make the long-
time limit accessible [130]. In contrast, the theory of stochastic thermodynamics has
proven most fruitful for experimental applications in cases where it provides relations
that hold on finite time scales. Most prominently, the Jarzynski relation [23] and
the Crooks fluctuation theorem [131] allow one to infer free energy differences from
the measurement of the fluctuating work during finite-time protocols, see, e.g., [132].
Similarly, the concept of stochastic entropy [25] allows for a generalization of the
detailed fluctuation theorem for the entropy production in a NESS (2.48) to finite and
thus experimentally accessible time scales [133].
In this Chapter, we show that most of the bounds, and in particular the thermodynamic
uncertainty relation, can be generalized to fluctuations on finite time scales as well.
We illustrate this finite-time version with experimental data for fluctuations of work
performed on a colloidal particle in a dichotomously switching trap [15, 16, 134]. This
illustration serves as a proof of principle for applying the uncertainty relation in the future
to more complex experimental systems with more than one input or output current such
as Brownian heat engines [130, 135] and molecular motors, see, e.g., [111]. Moreover,
we illustrate the finite-time generalization of the dissipation-dependent bound for the
full spectrum of fluctuations.
We have conjectured the finite-time generalization of the dissipation-dependent bound
based on extensive numerics in Ref. [136], which this Chapter is largely based on. There,
we also give a proof for short time-intervals as well as for weak driving. A full proof
was later found by Horowitz and Gingrich [137] using large deviation methods devised
by Maes et al. [138]. Here, we review this proof briefly, showing that finite-time
generalizations hold analogously for the affinity-and topology-dependent bound and the
activity-dependent bound. Building on these ideas, we show that a generalization of the
thermodynamic uncertainty relation holds also for the transient relaxation into a steady
state.
117
8 Finite-time generalizations
8.1 Main result
For a thermodynamic system modeled as a Markovian network and driven into a NESS
by time independent forces, we consider the fluctuations of an arbitrary time-integrated
current X (t) with X (0) = 0. While the average of such a current increases linearly in
time t as
⟨X (t)⟩ = Jst, (8.1)
where ⟨· · ·⟩ denotes the steady-state average, other characteristics of the distribution of
X (t) typically exhibit a more complex dependence on the observation time t. For the
variance Var[X (t)] ≡ ⟨X (t)2⟩ − ⟨X (t)⟩2, we demonstrate that
Var[X (t)]σ/(Js)2t ≥ 2 (8.2)
holds for arbitrary times t > 0. Thus, the fluctuations of X (t) at finite times can be
related to the rate of total entropy production σ associated with the driving. In the
limit of large observation times the variance of X (t) settles to a linear increase with
the effective diffusion coefficient D = limt→∞ Var[X (t)]/2t (3.21). On this infinite time
scale, the uncertainty relation reads Dσ/(Js)2 ≥ 1, as is discussed in Sec. 4.5.
8.2 Experimental illustration
As an experimental illustration of the relation (8.2), we analyze data for a colloidal
particle in a dichotomously switching optical trap [16]. The center of the trap is switched
along a one-dimensional coordinate λ(τ) between the positions +λ0 and −λ0 at points
in time that are generated by a Poisson process with rate γ [see Fig. 8.1(a)]. The force
f (τ) which is exerted on the bead along this dimension is measured directly from the
deflection of the light. We consider two different definitions of work [139,140]
w1(τ) ≡
∫ τ
0
dλ(τ′) f (τ′) ≈
τ/δτ∑
n=1
(λn − λn−1) fn + fn−12 (8.3a)
and
w2(τ) ≡ −
∫ τ
0
d f (τ′) λ(τ′) ≈ −
τ/δτ∑
n=1
( fn − fn−1) λn + λn−12 . (8.3b)
The discrete integration schemes with fn ≡ f (n δτ) and λn ≡ λ(n δτ) define the
integrals for discontinuous λ(τ) and f (τ) via the limit δτ → 0 and are used to compute
the work for experimental data captured with a finite time resolution of δτ ≃ 1 ms. We
118
8.2 Experimental illustration
10-2 10-1 100 101
t [s]
100
101
102
2k
B
T
W
1
(t)
W
2
(t)
V
a
r[
W
(t
W
(t
)
[k
B
T
]
(c)
(a)
(b)
-100
0
100
[n
m
]
-1
0
1
f
[p
N
]
0 1 2 3 4 5
[s]
0
100
200
300
w
[k
B
T
]
w
1
( )
w
2
( )
Figure 8.1: Experimental data for the bound (8.2) for a colloidal particle in a stochastically
switching trap, as sketched in (a). Panel (b) shows the time dependent position λ(τ)
of the trap, the force f (τ) exerted on the colloid and the work w1,2(τ) according
to the two definitions (8.3) for a short part of the trajectory. In (c), the quantity
Var[w(t)]/ ⟨w(t)⟩ is shown as a function of the length t of the time interval and
compared to the lower bound 2kBT . Data refer to the amplitude λ0 ≃ 170 nm and
a trap with inverse relaxation time τ−1rel ≃ 4.6 s−1 throughout. For the blue lines the
switching rate is γ ≃ 2.88 s−1. In (c), we show additional data for γ ≃ 8.73 s−1 (red)
and γ ≃ 12.3 s−1 (yellow).
119
8 Finite-time generalizations
-50 0 50 100 150 200
W
1
0
0.02
0.04
0.06
p
(W
1
,t
)
-50 0 50 100 150 200
W
2
0
0.02
0.04
0.06
p
(W
2
,t
)
-50 0 50 100 150 200
W
1
0
0.02
0.04
0.06
p
(W
1
,t
)
-50 0 50 100 150 200
W
2
0
0.02
0.04
0.06
p
(W
2
,t
)
-50 0 50 100 150 200
W
1
0
0.02
0.04
0.06
p
(W
1
,t
)
-50 0 50 100 150 200
W
2
0
0.02
0.04
0.06
p
(W
2
,t
)
Figure 8.2: Full distributions of thework underlying the data formean and variance in Fig. 8.1(c).
Left column: γ ≃ 2.88 s−1 [blue in Fig. 8.1(c)], middle column γ ≃ 8.73 s−1 [red in
Fig. 8.1(c)], and right column γ ≃ 12.3 s−1 [yellow Fig. in 8.1(c)]. Time t increases
from 0.1 s (black) to 1 s (light brown) in steps of 0.1 s.
interpret w1(τ) as the work performed by moving the trap against the force f . The
second definition, w2(τ), is equivalent to w1(τ) up to a finite boundary term of the form
λ f . Figure 8.1b shows sample data for λ(τ) and f (τ) together with w1,2(τ).
Due to the stochastic switching of the trap, the system reaches a NESS for long
observation times T . Hence, the steady state averages and cumulants for the work
W1,2(t) ≡ w1,2(τ)−w1,2(τ− t) performed on finite time intervals t ≪ T can be obtained
from the time average over τ ∈ [t,T ].
In Fig. 8.2, we show the full distributions of work performed on the colloidal particle.
Unlike for deterministic switching [141], these distributions can be highly non-Gaussian
at finite times. The time scale chosen in these plots covers the transition from work
fluctuations in a typically resting trap for short times to work fluctuations that are
directly affected by switching the trap. Since the work W1 increases in a step-like
fashion [see Fig. 8.1(b)], its distribution exhibits a sharp peak corresponding to time
intervals where the trap does not switch. With increasing length of the time interval
the height of this peak decreases and a second bulge in the distribution starts growing.
This part of the distribution is much broader since the work performed while switching
the trap is stochastic. For the work W2, fluctuations occur also while the trap is at rest,
leading to a broader peak at short times. With increasing switching rate γ of the trap,
the effects of the resting trap become less pronounced, leading to an overall smoother
work distribution.
For long time intervals t, both definitions of the work measure the area enclosed by the
120
8.3 Bound on the generating function
trajectory in the (λ, f ) space up to a finite contribution that does not scale with t. Thus,
in the long-time limit, cumulants ofW1(t) become equal to the respective cumulants of
W2(t) to leading order in time. In particular, as the mean is independent of t, we have
⟨W1(t)⟩ /t = ⟨W2(t)⟩ /t = σT , where T is the temperature of the surrounding heat bath.
Since the work that is performed on the system must ultimately be dissipated, we can
indeed identify these averages with the rate of entropy production σ. Thus, specifying
W1,2(t) as integrated current in (8.2), we obtain the bound
Var[W1,2(t)]⟨
W1,2(t)
⟩ ≥ 2kBT (8.4)
on the fluctuations of W1,2, where we insert Boltzmann’s constant kB for compatibility
with the units used in the experiment. As Fig. 8.1c shows, this bound is satisfied for
arbitrary times t, various values of the switching rate γ, and for both definitions W1(t)
andW2(t). In the limit of large t, for which the uncertainty relation has previously been
shown to hold, the expression on the left-hand side of Eq. (8.4) becomes equal for both
definitions. In contrast, for finite time intervals the fluctuations ofW1(t) andW2(t) differ
by a whole order of magnitude. Thus, the finite-time generalization of the uncertainty
relation allows one to infer stronger lower bounds on the entropy production by choosing
the most suitable among various currents that become equivalent in the long-time limit.
Most remarkably, the difference to the bound can be smaller for finite times than it is in
the long-time limit, as the minimum of the blue dashed curve in Fig. 8.1, corresponding
to a slow switching rate γ, shows. The finite-time bound evaluated at t ≃ 0.03 s yields
5.4 kBT and is thus about a factor of 2 better than the long-time value 12.0 kBT .
The relation between the variance and mean of work fluctuations has previously been
discussed for transient non-equilibrium processes [142,143]. For those, it is possible to
obtain a ratio of these quantities that is smaller than the bound set by Eq. (8.4), which
applies to steady states.
8.3 Bound on the generating function
In the following, we discuss the finite-time bound (8.2) in a broader theoretical frame-
work. As before in Sec. 4.1, we represent the system as a set of states {i} and Markovian
transition rates ki j ≥ 0 from state i to state j and denote the corresponding stationary
distribution as psi . A time-integrated current X (t) is defined by specifying its change
di j = −d ji upon a transition from i to j. The steady-state average of this current is
calculated as
Js = ⟨X˙ (t)⟩ =
∑
i j
psi ki jdi j . (8.5)
121
8 Finite-time generalizations
In particular, the choices
dmi j ≡ ln
ki j
k ji
and dsi j ≡ si j = ln
psi ki j
psj k ji
(8.6)
define the entropy production in the reservoirs or “medium” S(t) (up to contributions
from the intrinsic entropy) and the total entropy production Stot(t), respectively, see
Sec. 2.4 and Eq. (4.4). The steady state averages (8.5) of these two currents are equal,
defining the entropy production rate σ ≡ ⟨s˙m⟩ = ⟨s˙tot⟩ . The fluctuations of any current
X (t) can conveniently be analyzed in terms of the generating function (3.10), stated here
again for convenience as
g(z, t) =
⟨
ezX (t)
⟩
=
⟨
−etL(z) ps⟩ (8.7)
with the tilted transition matrix (3.13) and the vector ⟨−| containing 1 in every entry.
This function allows one to infer the mean of the current as
⟨X (t)⟩ = ∂z ln g(z, t) |z=0 (8.8)
and its variance as
Var[X (t)] = ∂2z ln g(z, t)
z=0 . (8.9)
The generating function satisfies the bound
(1/t) ln g(z, t) ≥ Jsz(1 + zJs/σ) (8.10)
for arbitrary times t, which is themost general theoretical result of Ref. [136]. In the limit
t → ∞, the left-hand side of this expression converges to the scaled cumulant generating
function λ(z), which we have used so far for the discussion of current fluctuations on
infinite time scales. Eq. (8.10) thus generalizes the dissipation-dependent bound (4.48)
to the regime of fluctuations on finite time scales. Crucially, the difference between
(1/t) ln g(z, t) and the quadratic bound can be smaller for finite times t than it is in the
long-time limit. Such a behavior of the generating function is necessary for a minimum
of the ratioVar[X (t)]/ ⟨X (t)⟩ at finite time t as in our experimental illustration in Fig. 8.1
for the workW2(t) at low switching rate.
The bound (8.10) is globally saturated for a Gaussian distribution of the current 1, as
observed for a biased diffusion in a flat potential. This process can be approximated by a
discrete asymmetric random walk on a ring where the number of states is let to infinity
while the affinity per step is let to zero. Otherwise, the bound is only trivially saturated
1Even though the distribution of work W2 often looks Gaussian in our experimental case study (see
Fig. 8.2), the generating function would in these cases not be quadratic due to non-Gaussian tails of
the distribution. Accordingly, the uncertainty relation is not saturated.
122
8.4 Short-time and linear response limit
for z = 0 and, as a consequence of the fluctuation theorem (2.48), for the generating
function of Stot(t) at z = −1. For other currents that become equal to Stot(t) on large
time scales, such as the medium entropy production S(t), the bound is approached at
z = −1 only in the long-time limit.
Of experimental relevance is mainly the variance (8.9) of the current X (t). Since
(1/t) ln g(z, t) touches the bound at z = 0 for all t, the finite-time version (8.2) of the
thermodynamic uncertainty relation follows from the relation (8.10).
8.4 Short-time and linear response limit
While the full proof of the quadratic bound (8.10) requires sophisticated large deviation
techniques, a weaker bound, which becomes equivalent to (8.10) for small t, can be
derived frommerely the fluctuation relation (2.49). This relation expresses the symmetry
p(−Stot,−X, t)/p(Stot, X, t) = exp(−Stot), (8.11)
of the joint probability distribution of the total entropy production and the current of
interest at arbitrary time t. Using this relation, the generating function (8.7) can be
written (dropping the index ‘tot’) as
g(z, t) =
∫
dS
∫
dX p(S, X, t) ezX =
1
2
⟨
ezX + e−zX−S
⟩
=
⟨
e−S/2 cosh(zX + S/2)
⟩
. (8.12)
Bounding the hyperbolic cosine by a parabola that touches it at z = 0 and z = −S/X , we
obtain
g(z, t) ≥1 +
⟨
(1 − e−S)zX (1 + zX/S)
⟩
/2
= 1 + zJst + z2σt
∫ ∞
0
dS
∫ ∞
−∞
dXψ(S, X ) (X/S)2
≥ 1 + t Jsz(1 + zJs/σ). (8.13)
In the last step we have used Jensen’s inequality for the averages with the distribution
ψ(S, X ) ≡ p(S, X, t)S(1 − e−S)/σt for S ≥ 0, which is non-negative, normalized, and
gives ∫ ∞
0
dS
∫ ∞
−∞
dXψ(S, X ) (X/S) = Js/σ. (8.14)
While the bound (8.13) is rigorous for arbitrary times t, it is useful mainly for short
times as a first order expansion of the otherwise stronger bound on g(z, t) that follows
123
8 Finite-time generalizations
-1.5 -1 -0.5 0 0.5
zJ /
-0.2
0
0.2
0.4
ln
[g
(z
,t
=
1)]
/
Figure 8.1: Generating function of the average current at time t = 1 in a unicyclic network
with perturbations of strength ε ∈ {0.05, 0.1, 0.5, 5} (from blue to orange). The
unperturbed network has five states and rates k+ = e1 and k− = 1. The bound (8.10)
is shown as a red curve.
from Eq. (8.10). Indeed, for the variance of the current, the bound (8.13) implies for
arbitrary t
⟨X (t)2⟩ ≥ 2t(Js)2/σ. (8.15)
Equation (8.2) differs from this relation only by the term ⟨X (t)⟩2 = (Js)2t2 and is thus
proven for small times t in linear order.
In the linear response regime for small driving affinityA, the current scales as Js ≃ A
and the entropy production rate as σ ≃ A2. Hence the bound (8.15) implies Eq. (8.2)
in the linear response limit for any fixed time, as follows from the scaling Js ∼ A and
σ ∼ A2 for small driving affinities A.
8.5 Illustration for unicyclic networks
As a simple example, for which the generating function can be calculated explicitly, we
consider the asymmetric random walk on a ring with N states and uniform forward and
backward transition rates k+ and k−. For the current averaged along all links, the tilted
transition matrix (3.13) reads
Li j (z) = k+ez/Nδi, j+1 + k−e−z/Nδi+1, j − (k+ + k−)δi, j, (8.16)
where we identify the states N +1 ≡ 1. The average current is Js = (k+− k−)/N and the
entropy production is σ = (k+ − k−) ln(k+/k−) . The stationary distribution psi = 1/N
124
8.6 Proof
is an eigenvector of L(z) for every z, hence the generating function (8.7) becomes
g(z, t) = exp
[
t
(
k+ez/N + k−e−z/N − k+ − k−
)]
. (8.17)
It can be easily checked that this generating function satisfies the bound (8.10) at all
times t. The bound is saturated for small z in the linear response limit of vanishing
affinity ln
(
k+/k−
)
per step.
These unicyclic asymmetric random walks are “optimal” in the sense that they mini-
mize the generating function at any given z and t. Changing the rates non-uniformly and
adding further cycles only increases the distance from the bound. In order to illustrate
this observation, we show in Fig. 8.1 the effects of perturbations of the rate matrix of the
type
ki j = k+eεθ
+
i δi, j+1 + k−eεθ
−
i δi+1, j + εϕi j, (8.18)
where the θi are independently drawn from a standard normal distribution and the ϕi j
are zero for |i − j | ≤ 1 and exponentially distributed otherwise. While the terms
with θ±i make the unicyclic rates non-uniform, the terms ϕi j add further cycles to the
network. We calculate the generating function numerically for t = 1, which qualifies as
an intermediate time scale for transition rates of order 1. The bound (8.10) is satisfied
in all cases.
In Fig. 8.2 we illustrate the bound (8.10) for fully connected networks with N ∈
{3, 4, 5, 7, 10} states and random transition rates with ln ki j distributed uniformly between
−12 and 5. A sample of such a network with N = 5 states is illustrated in the inset
of Fig. 8.2. For these networks we have calculated the stationary distribution and the
generating function g(z, t = 1) via Eq. (8.7) with Jsz/σ ranging from −2 to 1. This
procedure has been repeated for the currents of total entropy production (di j = dsi j in
Eq. (8.6)) and medium entropy production (di j = dmi j ), as shown in Fig. 8.2. In order to
show the tightness of the bound, each of the random networks has then been used as a
starting point for a constrained local minimization procedure that varies the rates ki j to
minimize g(z, t = 1) while keeping σ and Jsz/σ fixed.
8.6 Proof
A first proof of the finite-time uncertainty relation (8.2) was achieved by Pigolotti et al.
for the special case X (t) = Stot(t) using martingale methods [144].
The full proof for arbitrary currents is due to Horowitz and Gingrich [137]. Here, we
will follow their argumentation, which allows us to derive finite-time variants of other
bounds as well.
The central idea in this proof is to augment the description of the setup from a single
system to a set ofN independent copies of the system. This setup can be imagined as a set
125
8 Finite-time generalizations
-1 -0.5 0
z
-0.3
-0.2
-0.1
0
0.1
0.2
-1 -0.5 0
z
random rates
minimization
parabolic bound
ln
[g
(z
,t
=
1
)]
/
Figure 8.2: Numerical illustration of the bound on the generating function for a fully connected
network with five states and random transition rates, as shown in the inset of the left
panel and indicated by the different thicknesses of the arrows. We show generating
functions g(z, t = 1) for Stot (left) and S (right) calculated numerically for uniformly
distributed ln ki j ∈ [−5, 5] and scaled by the entropy production rate σ (blue).
For each set of rates a local minimization of g(z = −0.5, t = 1) was performed,
the corresponding generating functions are shown in red. In all cases, the bound
σz(z + 1) (shown in black) is satisfied.
ofN non-interacting “particles” populating the state-space {i}, each of which undergoes
transitions with possibly time dependent transition rates ki j (τ). This augmentation
allows one to recover concepts from large deviation theory for finite time intervals, with
N replacing t as a large parameter [68, 145].
The set of trajectories {in(τ)}, with 0 ≤ τ ≤ t and 1 ≤ n ≤ N labeling the individual
particles, gives rise to several time-dependent empirical, i.e., fluctuating, fields. First,
we have the empirical density
pi (τ) ≡ 1N
∑
n
δi,in (τ) . (8.19)
Writing the number of directed transitions of particle n from i to j up to the time τ as
mni j (τ), we define the empirical flow as
µi j (τ) ≡ 1N
∑
n
m˙ni j (τ), (8.20)
where the dot denotes the derivative for τ. Moreover, we define the empirical traffic
ti j (τ) ≡ µi j (τ)+ µ ji (τ) and the empirical current ji j (τ) ≡ µi j (τ)− µ ji (τ). Conservation
126
8.6 Proof
of the number of particles is expressed in the continuity equation
∂τpi (τ) = −
∑
j
ji j (τ), (8.21)
which must be satisfied for any consistent pair of empirical density and current.
In the ensemble average, which is the average over realizations of the total system, one
obtains an average density p¯i (τ) ≡ ⟨pi (τ)⟩ with a prescribed initial distribution p¯i (0)
fromwhich the initial states of the particle trajectories are sampled. This average density
is governed by the master equation
∂τ p¯i (τ) = −ri (τ)pi (τ) +
∑
j
k ji (τ)p j (τ) (8.22)
with the time-dependent exit-rate ri (τ) ≡ ∑ j ki j (τ). Correspondingly, one identifies the
average flow µ¯i j (τ) ≡ ⟨µi j (τ)⟩ = p¯i (τ)ki j (τ) and likewise the average traffic t¯i j (τ) and
current j¯i j (τ).
We now postulate a large deviation principle for the probability p[p(τ), µ(τ),N ] of
the joint fluctuations of the density p(τ) = {pi (τ)} and the flow µ(τ) = {µi j (τ)} of the
form
I[p(τ), µ(τ)] = − lim
N→∞
1
N ln p[p(τ), µ(τ),N ], (8.23)
which is a functional of p(τ) and j (τ). The path weight for the joint realization of the
N trajectories follows as the product of the individual path weights as
p[{in(τ)}] = exp *.,
∑
n
ln p¯in (0) (0) +
∑
n
∫ t
0
dτ
⎡⎢⎢⎢⎢⎢⎣−rin (τ) +
∑
i j
m˙i j (τ) ln ki j (τ)
⎤⎥⎥⎥⎥⎥⎦
+/-
= exp *.,N
∑
i
pi (0) ln p¯i (0) +N
∫ t
0
dτ
⎡⎢⎢⎢⎢⎢⎣−
∑
i
pi (τ)ri (τ) +
∑
i j
µi j (τ) ln ki j (τ)
⎤⎥⎥⎥⎥⎥⎦
+/- .
(8.24)
Analogously to Eq. (3.74), we now consider an auxiliary process with rates kˆi j , initial
distribution pˆi (0) and large deviation function Iˆ[. . .]. Since the path weight (8.24) is a
functional of the empirical density and flow, we can calculate the ratio of the probabilities
for the realizations of p(τ) and j (τ) under the two different dynamics as
I[p(τ), µ(τ)] − Iˆ[p(τ), µ(τ)] = − lim
N→∞
1
N ln
p[p(τ), µ(τ),N ]
pˆ[p(τ), µ(τ),N ]
=
∑
i
pi (0) ln
pˆi (0)
p¯i (0)
+
∫ t
0
dτ
⎡⎢⎢⎢⎢⎢⎣
∑
i
pi (τ)[ri (τ) − rˆi (τ)] −
∑
i j
µi j (τ) ln
ki j (τ)
kˆi j (τ)
⎤⎥⎥⎥⎥⎥⎦ .
(8.25)
127
8 Finite-time generalizations
Next, we choose for the auxiliary process the rates kˆi j (τ) = µi j (τ)/pi (τ) and the initial
distribution pˆi (0) = pi (0), such that p(τ) and j (τ) become typical under the auxiliary
dynamics and hence Iˆ[p(τ), µ(τ)] = 0. This finally yields the large deviation functional
I[p(τ), µ(τ)] = D(p(0) | | p¯(0))+
∫ t
0
dτ
⎡⎢⎢⎢⎢⎢⎣
∑
i
pi (τ)ri (τ) +
∑
i j
µi j (τ)
(
ln
µi j (τ)
pi (τ)ki j (τ)
− 1
)⎤⎥⎥⎥⎥⎥⎦
(8.26)
with the Kullback-Leibler divergence
D(p(0) | | p¯(0)) ≡
∑
i
pi (0) ln
pi (0)
p¯i (0)
. (8.27)
Notably, the integrand in Eq. (8.26) has the same form as the large deviation func-
tion (3.75). In complete analogy to Eq. (3.78), we can thus write the large deviation
functional using the empirical traffic and current as I[p(τ), j (τ), t (τ)]. Since the traffic
is not constrained by the continuity equation (8.21), we can do a contraction for the traffic
by minimizing I[p(τ), j (τ), t (τ)] over ti j (τ) for every link and every time individually.
As a result, we obtain along the same lines as for Eq. (3.82) the large deviation functional
I[p(τ), j (τ)] = D(p(0) | | p¯(0)) +
∫ t
0
dτ L(p(τ), j (τ), τ), (8.28)
where the function L(p, j, τ) is given by the right-hand side of Eq. (3.82) with time-
dependent transition rates ki j (τ).
As usual, we define a fluctuating observable X (t) = X[i(τ)] through a distance matrix
di j . In theN -particle ensemble, every particle nwill lead to a different realization X n(t)
of this observable. For the total system, we define the fluctuating observable
X(t) ≡ 1N
∑
n
X n(t) (8.29)
as the average over all particles, which can be written as a functional of the empirical
flow
X(t) = X[µ(τ)] =
∫ t
0
dτ
∑
i j
µi j (τ). (8.30)
Since the particles are independent, its ensemble average
⟨X(t)⟩ = ⟨X n(t)⟩ ≡ X¯ (t) = ∫ t
0
dτ
∑
i j
µ¯i j (τ)di j (8.31)
128
8.7 Uncertainty relation for transient non-equilibrium states
is equal to the ensemble average for any individual particle. With increasing particle
number N , the distribution of X(t) concentrates around this average, satisfying a large
deviation principle of the form
I (X, t) ≡ − lim
N→∞
1
N ln p(X, t,N ). (8.32)
The Legendre transform of this large deviation function yields the scaled cumulant
generating function
Λ(z, t) ≡ lim
N→∞
1
N ln
⟨
ezNX(t)
⟩
= min
X
[zX − I (X, t)]. (8.33)
Crucially, again due to the independence of the particles, this function is related to the
generating function (8.7) of an individual particle through
Λ(z, t) = lim
N→∞
1
N ln
⟨
ez
∑
n Xn (t)
⟩
= ln
⟨
ezX (t)
⟩
= ln g(z, t). (8.34)
Thus, any bound we find for I (X, t) and Λ(z, t), respectively, will also provide a bound
on the generating function of an individual particle at finite time.
The large deviation function I (X, t) can be obtained from the contraction principle
I (X, t) = min
p(τ),µ(τ) |X[µ(τ)]=X
I[p(τ), µ(τ)] = min
p(τ), j (τ) |X[ j (τ)]=X
I[p(τ), j (τ)] (8.35)
where the second equality holds only for current-like observables with di j = −d ji. As
before in Eq. (3.85), any non-optimal ansatz for the emirical fields leads to an upper
bound on the large deviation function. The simplest class of such ansatzes are time-
independent ones, forwhich the integrals in Eqs. (8.26) and (8.28) become trivial, leading
to the familiar expressions (3.75) and (3.82), respectively, multiplied by t. Moreover, for
initial states sampled from the stationary distribution psi , the choice p(τ) = p
s leads to
a vanishing Kullback-Leibler distance for the initial distribution.
Using as before the ansatz j (τ) = ξ js then shows that the dissipation-dependent
bound (4.48) and the affinity- and topology-dependent bound (6.11) hold not only as
bounds on the scaled cumulant generating function λ(z) in the long time limit, but also
as bounds on (ln g(z, t))/t for any t, as in Eq. (8.10). Likewise, the activity-dependent
bound (7.5) holds on finite time scales as well.
8.7 Uncertainty relation for transient non-equilibrium
states
So far, the finite-time thermodynamic uncertainty relation (8.2) has been proven for
ensembles of finite trajectories whose initial states are sampled from the stationary
129
8 Finite-time generalizations
distribution psi . However, in the context of a typical finite-time experiment, one may
also consider ensembles of trajectories with a fixed initial state or with an arbitrary other
initial distribution. Such an distribution may arise from a sudden change (or “quench”)
of the transition rates. We assume that during the relaxation process the transition rates
remain fixed. Using the methods reviewed in the previous Section, we show that a
generalization of the uncertainty relation to such transient states.
The probability distribution p¯i (τ) relaxes from the initial distribution p¯i (0) to the
stationary distribution psi , which is in case of (global) detailed balance the equilibrium
distribution peqi . For a current-like observable with a time-independent distance matrix
di j = −d ji the ensemble average X¯ (τ) = ⟨X (τ)⟩ evolves as
˙¯X (τ) =
∑
i< j
j¯i j (τ)di j (8.36)
with initial condition X¯ (0) ≡ 0. Likewise, the average total entropy (2.33) evolves as
S˙tot(τ) =
∑
i< j
σ¯i j (τ) ≡
∑
i< j
j¯i j (τ) ln
p¯i (τ)ki j
p¯ j (τ)k ji
. (8.37)
We now consider the ensemble of N independent realizations of the process with
initial conditions distributed according to p¯i (0). The empirical densities pi (τ) and
currents ji j (τ) in this ensemble satisfy a large deviation principle with the large deviation
functional I[p(τ), j (τ)], given by Eq. (8.28). Bounding the function L(p, j) from above
by a quadratic function leads to
I[p(τ), j (τ)] ≤
∫ t
0
ds
∑
i< j
( ji j (τ) − jpi j (τ))2
4 jpi j (τ)2
σ
p
i j (τ) + D(p(0) | | p¯(0)), (8.38)
where jpi j (τ) ≡ pi (τ)ki j − p j (τ)k ji and σpi j (τ) ≡ jpi j (τ) ln[pi (τ)ki j/p j (τ)k ji]. The fluc-
tuations of the average X[ j (τ)](t) of the observable X (t) in the N -ensemble at the
observation time t obey a large deviation function that is obtained via the contrac-
tion (8.35).
We now consider as empirical density and current a time-scaled version of p¯(τ) and
j¯ (τ), respectively, i.e., we set p(τ) = p¯(ζτ) and j (τ) = ζ j¯ (ζτ). The corresponding
value of the observable X is X[ j (τ)] = X¯ (ζ t) and, since p(0) = p¯(0), the Kullback-
Leibler divergence D(p(0) | | p¯(0)) vanishes. Thus we obtain
I (X = X¯ (ζ t), t) ≤ I[ p¯(ζτ), ζ j¯ (ζτ)]
≤ 1
4
(ζ − 1)2ζ
∫ t
0
dτ
∑
i< j
σ¯i j (ζτ)
=
1
4
(ζ − 1)2ζ ∆Stot(ζ t). (8.39)
130
8.8 Conclusion
Since both sides of the inequality are zero for ζ = 1, we can derive both sides twice for
ζ and set ζ = 1, leading to
I′′(X¯ (t), t) (t ˙¯X (t))2 ≤ 1
2
∆Stot(t). (8.40)
Finally, with Var X (t) = limN→∞NVarX(t) = 1/I′′(X¯ (t), t), we arrive at
1
t
Var X (t) ≥ 2
˙¯X (t)2
∆Stot(t)/t
. (8.41)
For the relaxation into a NESS, we recover the usual form of the uncertainty relation in
the limit t → ∞ [1,8]. If the initial condition is already p¯(0) = ps, we have ˙¯X (t) = ˙¯X (t)/t
and thus recover the finite-time uncertainty relation in the form stated in Refs. [136,137].
For the relaxation into an equilibrium state, where ˙¯X (t) tends exponentially to zero, the
relation (8.41) is useful on finite time scales.
It should be noted that the naive generalization of the finite-time uncertainty relation
of the form Var X ≥ X¯2/∆Stot ≥ 2 does not apply to transient processes. This can easily
be checked for the relaxation from state 1 of a two-state system with two equal rates and
with d12 = 1 = −d21, where we have for large t the values Var X = 1/4, X¯ = 1/2 and
∆Stot = ln 2.
8.8 Conclusion
In this Chapter, we have shown that the thermodynamic uncertainty relation between
the fluctuations of any current and the rate of entropy production in a NESS holds
on arbitrary time scales. This result follows from a quadratic bound on the cumulant
generating function associated with such a current, which generalizes the dissipation
dependent bound from Sec. 4 to finite time scales. This bound can be proven using large
deviation theory for an ensemble of independent copies of the system. This method
allows us to generalize the affinity- and topology-dependent bound and the activity-
dependent bound to finite time scales as well. Moreover, we derive a variant of the
finite-time uncertainty relation that applies to transient non-equilibrium processes.
For an experimental illustration in the case where the entropy production is measur-
able, we have analyzed this finite-time uncertainty relation for the work that is performed
on a colloidal particle in a stochastically switching trap. A similar setup involving a
sliding harmonic potential has recently been analyzed theoretically [146]. As a next
experimental step, it will be interesting to apply this relation to systems driven by chem-
ical reactions like molecular motors, in order to bound the then a priori unknown rate
of entropy production from below. Our generalization of the thermodynamic uncer-
tainty relation should then become a valuable tool for inferring hidden thermodynamic
properties of driven systems from experimental trajectories of finite length.
131

9 Concluding perspective
Since its publication in early 2015 [1], the thermodynamic uncertainty relation has
spurred remarkable and ongoing research efforts concerning related problems. The
strand of research we have documented in this Thesis puts this relation into the general
context of large deviation theory. The seed of this strand of research has been the formu-
lation of a universal, dissipation-dependent bound on the large deviation function for the
fluctuations of any current in a Markovian system, as described in Chap. 4. A major con-
tribution in this endeavor have been the publications by Gingrich et al. [8, 64], invoking
level 2.5 large deviation theory as a versatile tool for proving bounds on current fluctu-
ations. These techniques have enabled us to derive the refined affinity- and topology-
dependent bound in Chap. 6 and yield a simplified proof of the activity-dependent bound
in Chap. 7. Complementary works have helped to elucidate the nature of these bounds
from a formal perspective. These works provide further refinements [122], consider the
canonical structure of currents and activity in Markov networks [145, 147], and give
physical interpretations of the processes that produce large deviations in the fluctuations
of currents [148, 149].
The universality that is inherent to the thermodynamic uncertainty relation allows for
its application in various contexts, leading in many cases to valuable new insights. For
instance, the relation has been applied to overdamped Brownian transport processes in
one dimension [150] and in a general multi-dimensional setup allowing for position-
dependent diffusion coefficients [151]. Other contexts in which the uncertainty relation
has been applied include enzyme kinetics [119], self-propelled particles [152], magnetic
systems [153], and self-assembly [154]. In Chap. 5 we describe how the application
of the uncertainty relation to a very general model for molecular motors leads to a
universal bound on the thermodynamic efficiency of molecular motors. Crucially, this
bound depends only on experimentally accessible quantities, namely the velocity and
diffusivity of the motor and the external load force. A comprehensive analysis of the
uncertainty relation for molecular motors has been provided in Ref. [107].
For illustrations of the bounds discussed in this Thesis, we have chosen systems
that operate in an isothermal environment, for which the entropy production equals
the dissipation into the environment. Nevertheless, the rate of entropy production, as
defined in Secs. 2.4 and 2.5, is also a well-defined quantity in a setup with several heat
reservoirs at different temperatures. It then involves contributions from the amount of
heat that is transferred to or from each of the reservoirs. In Ref. [155] we have applied
the uncertainty relation to a general model for steady state heat engines. The efficiency
133
9 Concluding perspective
of such engines is generally bounded by the famous Carnot efficiency. If one measures
for such an engine not only the average of the mechanical power output but also its
fluctuations, the bound on the entropy production inferred via the uncertainty relation
allows for a refinement of the bound on the efficiency. These results show that there
is a general trade-off between the closeness of the efficiency to Carnot efficiency, the
average power and the power fluctuations for any steady state heat engine. Usually, when
the efficiency of a heat engine approaches Carnot efficiency, the power fluctuations stay
finite which leads to vanishing power. On the other hand, if the parameters of the engine
are adjusted such that the power remains finite, the fluctuations of power must diverge.
Similar trade-offs have been reported for systems containing large numbers of coupled
heat engines [156,157].
We have focused in this Thesis on a characterization of current fluctuations in terms of
the probability distribution of the integrated current in a time interval of fixed length. In
the original formulation, this length had to be large compared to the typical correlation
time of the system, which is lifted by the finite-time generalization discussed in Chap. 8.
Other characteristics of current fluctuations that are not linked to such a fixed time
interval are conceivable and have indeed been studied. One such a characteristic is
the first passage time associated with a fixed value of the integrated current. The
fluctuations of such first passage times have been shown to be constraint by bounds
that are similar to the activity-dependent bound [128] and to the dissipation-dependent
bound or thermodynamic uncertainty relation [158]. Similarly, the fluctuations of cycle
completion times in cyclic networks containing an irreversible step can be characterized
by low-order cumulants that are bounded by the number of states [159]. The current
associated with the total entropy production exhibits several characteristic statistical
properties that are founded in its exponential being a martingale process [160]. Most
interestingly for our context, this approach can be used to prove the thermodynamic
uncertainty relation for Langevin dynamics completely without using large deviation
theory [144]. It will be interesting to explore whether similar concepts can also be
applied to other currents and to discrete state-spaces. Another characteristic property
that applies to any thermodynamic current is the “arcsine law”, describing the fluctuations
of the time during which an integrated current is above or below its average [161].
Another thread of research is related to pushing the limits of applicability of the
thermodynamic uncertainty relation and related bounds. These results hold already quite
universally for systems that can be described using continuous-time Markov processes
with time-independent dynamics. The state space can be discrete or continuous, the
latter requires an overdamped type of Langevin dynamics. Nevertheless, one can ask
whether the uncertainty relation still holds if any of these conditions are lifted, and, in
case it does not hold, whether one can derive suitable modifications.
For instance, it is straightforward to show that for a discrete-time Markov process the
thermodynamic uncertainty relation can be violated [162]. A discrete-time process can
134
have a higher precision than a continuous one with the same entropy production, which
can be retraced to the influence of the external clock providing an exact timing of the
jumps. Proesmans and Van den Broeck have shown that for discrete-time processes a
variant of the thermodynamic uncertainty relation holds, in which the entropy production
rate is replaced by an expression that becomes exponentially small when the entropy
production per time-step is much larger than one [163]. For a fine discretization of time,
where the entropy production per time-step is small, the original uncertainty relation
is recovered. The general connection between the continuous-time and discrete-time
variants of this relation has recently been discussed in Ref. [164].
Another variant of the setup is that of a continuous-time Markov process with deter-
ministically time-dependent transition rates. This corresponds physically to an external
manipulation of the free energies of the mesostates or of the driving forces. Recently, the
trade-off between dissipated work and work fluctuations has been considered for finite-
time protocols [165]. Even more research has been devoted to time-periodic protocols
that drive the system into a time-periodic stationary state. In Ref. [75], it has been shown
that in this case the synchronization with the external clock can lead to a increased pre-
cision that violates the uncertainty relation. Optimizing the design of simple, externally
driven Brownian clocks, it can even be shown that arbitrarily high precision at finite
entropic cost is possible. Furthermore, for stochastic pumps that mimic a NESS through
time-periodic manipulation of states, the dissipation-dependent bound has been shown
to hold only in the limit of weak perturbations [166]. Yet, the uncertainty relation can be
restored in general if one takes into account the entropic cost associated with the external
control in a setting where the latter is reversible [167]. So far, the only strong result
in the context of time-periodic driving and involving the entropy production proper is
provided by the discrete-time bound from Ref. [163], which can be shown to hold also
for time-symmetric driving in continuous time. It will be most interesting to see whether
similar bounds can be derived for the much larger class of time-asymmetric protocols.
A further open challenge is the question whether the thermodynamic uncertainty
relation and the related dissipation-dependent bound hold for underdamped Langevin
dynamics. The naive application of the level 2.5 large deviations formalism leads
in this case to a formally divergent entropy production, since it does not distinguish
between reversible and irreversible currents. In Ref. [69] we have derived a level 2
large deviation formalism that applies to underdamped dynamics, which yields several
remarkable bounds on the large deviation function for the current in a one-dimensional
system. However, these bounds are different from the dissipation-dependent bound from
Chap. 4. So far, numerics has shown no violation of the uncertainty relation for a
large number of random systems, yet the high numerical cost for calculating diffusion
coefficients prohibits to make a clear case.
A new perspective on this and related problems is provided by a recent publication
by Brandner et al. [168]. They study a multi-terminal model where non-interacting
135
9 Concluding perspective
particles are exchanged along deterministic paths through a target. Stochasticity arises
in this model only through the Maxwell-Boltzmann distributions for the initial momenta
of the particles. For this setup, the thermodynamic uncertainty relation can be proven in
the absence of magnetic fields. In the presence of magnetic fields, the original relation
can be broken, but is restored in a somewhat weaker fashion, replacing the factor 2
by a numerically determined constant ≃ 0.9. It will be interesting to see whether this
modification is also relevant in more general setups, e.g., for underdamped Langevin
dynamics in the presence of magnetic fields. Moreover, Ref. [168] demonstrates that
quantum coherences can decrease current fluctuations such that the uncertainty relation
is violated, which should likewise be investigated in more general setups.
On a broader scope, one can ask how in general the rate of dissipation relates to the
performance of a non-equilibrium system, that is either artificially designed or evolution-
arily adapted to perform a specific task [169, 170]. In this Thesis, we have focused as a
measure of the performance on the precision, as defined via the fluctuations compared to
an average output. In other contexts, other observablesmay be useful for an identification
of precision. For instance, kinetic proofreading is a non-equilibrium process that is able
to reduce the frequency of errors in the transcription of DNA [171,172]. Here, onewould
identify the error rate as a measure of precision, rather than the fluctuations of some
current. In a related context, recent studies have evaluated the thermodynamic cost for
maintaining a probability of a state that is different from its Boltzmann weight [173,174].
Further insight may be provided by trade-off relations for information-theoretic concepts
of precision [175–178]. Future research will show whether these considerations of ther-
modynamic cost and performance are only apparently related, or whether an overarching
theory can be found.
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150
Danksagungen
An erster Stelle gilt mein besonderer Dank Herrn Prof. Dr. Udo Seifert, der während
meiner sechs Jahre am II. Institut für Theoretische Physik mich stets mit großem per-
sönlichem Einsatz gefördert hat und unseren gemeinsamen Projekten entscheidend zum
Erfolg verholfen hat. Durch die zahlreichen und tiefgründigen Gespräche wurde mir
physikalische Intuition vermittelt und mir wurden wichtige Ratschläge für die Welt der
Wissenschaft mit auf den Weg gegeben.
Herrn Prof. Dr. Siegfried Dietrich und Herrn Prof. Dr. Andreas Engel danke ich
herzlich für die freundliche Übernahme des Mitberichts. Herrn Prof. Dr. Engel möchte
ich auch für dieGastfreundschaft und die ausführlichenDiskussionen beimeinemBesuch
seiner Arbeitsgruppe in Oldenburg danken.
Herrn Dr. Andre Barato danke ich für die erfolgreiche Zusammenarbeit an vielen der
in dieser Arbeit dokumentierten Ergebnisse und die vielen hilfreichen Gespräche in der
Anfangsphase meiner Promotion und darüber hinaus.
Unter denKollegen am II. Institut für Theoretische Physikmöchte ich besonders Herrn
Lukas Fischer und Herrn Matthias Uhl danken für ihr großes Engagement bei unseren
gemeinsamen Projekten. Allen aktuellen und ehemaligen Kollegen am Institut danke ich
für die freundschaftliche Atmosphäre und die vielen fachlichen und/oder unterhaltsa-
men Diskussionen. Dafür möchte ich insbesondere den langjährigen Kollegen Dr. Kay
Brandner, Dr. Sebastian Goldt, Dr. David Hartich und Dr. Daniel Schmidt danken. Herrn
Dr. Eckhard Dieterich danke ich auch für die wertvollen Tipps zu Barcelona und bei der
Aufbereitung experimenteller Daten. Besonderer Dank gilt Frau Dr. Eva Zimmermann,
die mir in meiner ersten Zeit am Institut stets als Mentorin zur Seite gestanden hat; auch
in der Zeit danach war es mir immer eine große Freude den Kontakt zu halten.
I wish to thank Prof. Dr. Fèlix Ritort for the successful collaboration, the support,
and the hospitality during my stay in Barcelona in 2016. It was a great experience for
me to get to know the experimental techniques. I would also like to thank the whole
team in Barcelona, in particular Álvaro Martínez, for their help in the lab and for an
unforgettable summer of 2016. Moltes gràcies!
In Stuttgart hatte ich die spannende Aufgabe bei der Betreuung von Bachelor- und
Masterarbeiten mitzuwirken. Für die gute Zusammenarbeit in den jeweiligen Projekten
danke ich Kevin Kleinbeck, Lukas Fischer, Matthias Uhl, Luca Blessing, Timur Koyuk
und Christoph Lohrmann.
Frau Steinhauser hat mir auf ihre persönliche und professionelle Art stets in allen
organisatorischen Belangen zur Seite gestanden, wofür ich mich ganz herzlich bedanken
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möchte. Auch alle Admins am Institut haben immer beste Arbeit geleistet, danke dass
ihr für jedes noch so wundersame Problem immer eine Lösung parat hattet.
Ganz besonders möchte ich mich bei meiner Familie, vor allem meinen Eltern, für die
liebevolle und großzügige Unterstützung in vielerlei Hinsicht bedanken.
152
Ehrenwörtliche Erklärung
Ich erkläre, dass ich diese Arbeit selbstständig verfasst und keine anderen als die ange-
gebenen Quellen und Hilfsmittel verwendet habe.
Stuttgart, den 26. März 2018 Patrick Pietzonka
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