Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 33 (2021) 375102 (13pp) https://doi.org/10.1088/1361-648X/ac0f9c The role of dimensionality and geometry in quench-induced nonequilibrium forces M R Nejad1,∗ , H Khalilian2, C M Rohwer3,4,5 and A G Moghaddam6,7 1 The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, United Kingdom 2 School of Nano Science, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran 3 Department of Mathematics & Applied Mathematics, University of Cape Town, 7701 Rondebosch, Cape Town, South Africa 4 Max-Planck-Institut für Intelligente Systeme, Heisenbergstr. 3, 70569 Stuttgart, Germany 5 IV. Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany 6 Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran 7 Research Center for Basic Sciences & Modern Technologies (RBST), Institute for Advanced Studies in Basic Science (IASBS), Zanjan 45137-66731, Iran E-mail: mehrana.raeisiannejad@physics.ox.ac.uk Received 12 March 2021, revised 14 June 2021 Accepted for publication 29 June 2021 Published 15 July 2021 Abstract We present an analytical formalism, supported by numerical simulations, for studying forces that act on curved walls following temperature quenches of the surrounding ideal Brownian fluid. We show that, for curved surfaces, the post-quench forces initially evolve rapidly to an extremal value, whereafter they approach their steady state value algebraically in time. In contrast to the previously-studied case of flat boundaries (lines or planes), the algebraic decay for curved geometries depends on the dimension of the system. Specifically, steady-state values of the force are approached in time as t−d/2 in d-dimensional spherical (curved) geometries. For systems consisting of concentric circles or spheres, the exponent does not change for the force on the outer circle or sphere. However, the force exerted on the inner circles or sphere experiences an overshoot and, as a result, does not evolve to the steady state in a simple algebraic manner. The extremal value of the force also depends on the dimension of the system, and originates from curved boundaries and the fact that particles inside a sphere or circle are locally more confined, and diffuse less freely than particles outside the circle or sphere. Keywords: quench, Brownian, colloids, temperature, active matter, Active Brownian particles (Some figures may appear in colour only in the online journal) 1. Introduction Objects immersed in fluctuating media can experience forces for a variety of reasons. The prototypical example is that of ∗ Author to whom any correspondence should be addressed. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. fluctuation-induced forces (FIFs), also referred to as Casimir and van der Waals forces [1–4], which can arise because the objects modify fluctuation modes of the medium in the presence of long-ranged correlations. Interestingly, these FIFs exhibit many universal properties that are independent of details of the medium and the nature of the fluctuations (e.g., quantum or thermal), and emerge in different contexts rang- ing from atomic and molecular physics condensed matter, and biology to material science, chemistry and biology [5–8]. In 1361-648X/21/375102+13$33.00 1 © 2021 The Author(s). Published by IOP Publishing Ltd Printed in the UK https://doi.org/10.1088/1361-648X/ac0f9c https://orcid.org/0000-0002-4670-9141 mailto:mehrana.raeisiannejad@physics.ox.ac.uk http://crossmark.crossref.org/dialog/?doi=10.1088/1361-648X/ac0f9c&domain=pdf&date_stamp=2021-7-15 https://creativecommons.org/licenses/by/4.0/ J. Phys.: Condens. Matter 33 (2021) 375102 M R Nejad et al classical systems at thermal equilibrium, FIFs generally stem from long-ranged correlations in the vicinity of critical points. In nonequilibrium situations, however, long-ranged correla- tions can emerge much more generally [9, 10]. In particular, the conservation of different global quantities (e.g., the number of particles) can lead to constrained nonequilibrium dynamics which subsequently give rise to long range correlations and FIFs [11–13]. For classical systems, nonequilibrium FIFs have been previously explored in the steady-states of externally driven systems, e.g., in the presence of temperature or den- sity gradients [14–17]. Recently, the role of sudden changes (quenches) of the temperature or of other system parameters in inducing a transient nonequilibrium states and FIFs has been studied by various groups [18–24]. While nonequilibrium dynamics following quenches has attracted much attention in the context of interacting quantum systems [25], various interesting phenomena also emerge in classical quench dynamics. In particular, it has been found that aside from FIFs, post-quench dynamics of the density in classi- cal fluids can give rise to additional forces on immersed objects or surfaces [20, 21, 26]. Such density-induced forces (DIFs) exist (and become longer-ranged) even in non-interacting (ideal) fluids. In contrast, the fluctuation forces discussed above rely on correlations, and disappear in the absence of interactions between fluid particles. It is well-established that geometry and dimensionality play an important role in equilibrium Casimir-type (FIF) systems [27]. However, for forces induced by sudden quenches of tem- perature (or of activity in active fluids), these effects have not been considered thus far. In this paper, we study the transient forces following a quench of temperature in an ideal (Brow- nian) fluid confined by curved surfaces or inclusions. In ref- erence [21] such post-quench DIFs were considered for flat walls immersed in an ideal fluid, and were shown to be inde- pendent of the dimension of the system and the boundaries: the force on the walls approaches its equilibrium value alge- braically in time as t−1/2. Here we aim to understand how this picture is modified by the curvature of the boundaries. We approach the problem with a theory for diffusion in curved geometries, as well as through explicit simulations of Brow- nian dynamics in the given setup. While our study applies to fluids confined by spherical or curved surfaces, it may also be of interest for the dynamics and behavior of fluid droplets in a quenched medium. In addition to temperature quenches in colloidal passive systems or granular matter [28], our findings may be rele- vant for active non-interacting (or weakly-interacting) systems which, to some extent, can be modeled by Brownian motion with an effective temperature [20]. In such a setting, a sudden change in the activity of the active system (e.g., for active Janus particles [29]) confined to curved walls can be described by our formalism in the appropriate coarse-grained regime [30]. While the latter may be of interest especially for living active matter and biological systems, this would require a separate study on the importance of interactions and the role of sol- vent effects. Another possible experimental realisation of this problem could be to apply a fluctuating electric field to a sys- tem of charged Brownian particles in order to mimic the role of the temperature. One could then quench the ‘temperature’ by tuning the strength of the electric field [31–33]. The paper is organized as follows: we first introduce the model system and simulation details in section 2. Using a coarse-grained description, we determine the analytical solu- tions for the post-quench dynamics of the density field of Brownian particles, from which the forces on curved walls can be also obtained. Then, in sections 3 and 4, numerical and analytical results are presented for the pressure and the force exerted on circular and spherical boundaries, respectively. Finally, in section 5, we close the paper with a summarising discussion and conclusion. 2. Model and simulation details We study dynamics of a non-interacting Brownian fluid, inside and outside of a sphere in d dimensions, after a quench in temperature. In particular, we consider the pressure and force exerted on the confining spherical walls in terms of time depen- dence and their scaling behavior. To study the evolution of the system after the quench, we use a coarse-grained analyt- ical approach and show that this model is in agreement with microscopic simulations based on Langevin dynamics. 2.1. Microscopic model for numerical simulations In simulations, we model the walls of the sphere by a repulsive quadratic potential. Dynamics of the system is described by over-damped Langevin equations: dri dt = −μ dVwall dr |ri r̂ + ηi(t), (1) Here, ri is the position of ith particle (i = 1, 2, . . . , N), μ the mobility of the particles, and Vwall represents the con- fining wall potential. When we simulate the interior of the d-dimensional sphere, we use Vwall(r) = λ 2 [Θ(r − Rd)(r − Rd)2]. (2) For simulations of the the exterior region we substituteΘ(Rd − r) by Θ(r − Rd). Interior and exterior regions are simulated separately, so that the corresponding pressures are independent [20, 21]. In equation (2), Rd is the radius of the d-dimensional confining spherical boundaries. From now on and for the sake of clarity, we denote the radius in the case of only one sphere (circle) with r0, whereas in the cases of two spheres (circles), the two radii are denoted by r1 and r2 with a difference Δr (see figure 1). Also, Θ(x) represents Heaviside step function, which is equal to one for x > 0 and vanishes otherwise. The strength of the quadratic potential is given by λ, and is chosen such that the effective range of the wall potential (and corre- spondingly the thickness of the adsorbed particle layer at the surface) is significantly smaller than the confinement length scale. In equation (1), ηi is Gaussian white noise obeying 〈ηiα(t)η jβ(t′)〉 = 2Dδ(t − t′) δi jδαβ , (3) where D is the diffusion constant of the particles, which is related to the mobility by fluctuation–dissipation theorem 2 J. Phys.: Condens. Matter 33 (2021) 375102 M R Nejad et al Figure 1. Panels (a) and (c) show a system with spherical symmetry in 2D; (b) and (d) show the same in 3D. The medium (black dots) is an ideal gas of passive Brownian particles. Panels (a) and (b) correspond to a circle and a sphere with radius r0, respectively. Panels (c) and (d) show a Brownian gas inside and outside of two inclusions with radii r1 and r2, for circular and spherical cases, respectively. according to D = μkBT, and ηiα is the αth component of the noise exerted on particle i. Using equation (1) and (2), we simulated the dynamics of the system of passive Brownian particles inside a spherical geometry in two or three dimensions, following the temper- ature quench. In all simulations, we average over N = 105 particles to obtain the pressure on the walls. In particular, we define a length-scale � = [πkBT/(2λ)]1/2 associated with the wall potential. We also consider the characteristic time- scale τ 0 = �2/D = 1 for diffusion, and choose the system size to be large enough so that it can be compared with the results of the coarse-grained model. In both circular and spher- ical cases, we perform simulations for r1/� = 100, 200 and Δr/� = 100. A forward Euler method is employed to integrate equation (1) in order to study the post-quench dynamics of each particle. In simulations, we instantaneously change the temperature by changing the diffusion coefficient of the par- ticles in equation (1). The overall scalar force exerted on the wall by the interior particles, from which we obtain the pres- sure, is then defined as the sum of the radial components of all single-particle forces as F = N∑ i=1 Fi · r̂i = N∑ i=1 dVwall dr |ri . (4) To obtain the pressure, we simply divide the total force F by the area of corresponding wall. 2.2. Coarse-grained model We also provide a coarse-grained theoretical description based on the Smoluchowski equation, and study the evolution of the density of the non-interacting Brownian particles follow- ing the quench. The particle density is defined as ρ(r, t) =∑N i=1〈δ(r − ri(t))〉. The Langevin equation (1) leads to a spherically symmetric coarse-grained dynamics given by ∂tρ(r, t) = D rd−1 ∂r [ rd−1∂rρ(r, t) ] , (5) in d dimensions. In this coarse-grained picture, walls are assumed to impose no-flux boundary conditions at r = Rd: ∂rρ(r, t)|r=Rd = 0. (6) Before the quench, the system is in a steady state described by the canonical Boltzmann weight ρI ∝ exp[−Vwall(r)/(kBT I)], where kB is Boltzmann’s constant. At time t = 0, we change the temperature to T = TF instantaneously. After this quench, the system evolves to a new equilibrium state which is described by ρF ∝ exp[−Vwall(r)/(kBTF)]. As discussed in reference [21], the quench modifies the boundary layer of particles close to the walls, because the penetration depth of the particles into the wall potential is temperature dependent. We can therefore write the post-quench particle density (for t > 0), as the sum of a homogeneous contribution, representing the region out- side of the wall potential, and a time- and position-dependent excess density, ρ(r, t) = ρ0 +Δρ(r, t). (7) We consider the homogeneous part of the density ρ0 to be the same inside and outside the inclusions. In the coarse-grained description, we consider strongly repulsive walls, so that the initial excess density after the quench is approximated as a δ- function at the boundary, Δρ(r, t = 0+) = αdρ0δ(r − Rd). (8) Here ρ0 and αd scale with length as [ρ0] = 1/�d and [αd] = �, respectively. Indeed,αd corresponds to the change of the width of the boundary layer induced by the quench, and can be com- puted by integrating ρI and ρF over the relevant volume and enforcing conservation of the particle number. We calculate this parameter analytically in appendix A. We note that the density immediately after the quench is exactly the same as before the quench. However, for a sudden quench to a higher temperature, the wall becomes statistically more penetrable for the particles which then possess a higher average energy. Therefore, at t = 0+, the wall accommodates less particles than the equilibrium state corresponding to the new temper- ature and we have desorption of particles on the wall. This gives rise to an effective desorption of some particles on the wall, which can be thought of as the emergence of a very thin layer of empty space at the wall. Conversely, if the tempera- ture is suddenly lowered at t = 0, immediately after the quench additional particles accumulate on the wall (compared to the equilibrium state corresponding to the new temperature). The density contribution of the particles adsorbed or desorbed at the wall due to the nonequilibrium quench is represented by Δρ(r, t = 0+) in equation (8). The excess density therefore has a negative (positive) value when the post-quench temperature TF is higher (lower) than T I. After the quench, the adsorbed (or desorbed, depending on whether the quench is to a higher or lower temperature) layer of particles at the boundary diffuses 3 J. Phys.: Condens. Matter 33 (2021) 375102 M R Nejad et al into the system. This gives rise to dynamics of post-quench forces and pressures acting on the boundaries. It turns out that αd is independent of the system’s dimension; we therefore use the notation α throughout instead. Using separation of variables, we can solve equation (5) and find the evolution of the excess density in time. The ideal gas law then provides the corresponding instantaneous pressure exerted by Brownian particles on the d-dimensional spherical wall after the quench, P(r = Rd, t) = kBTF[ρ0 +Δρ(r = Rd, t)]. (9) Solving equations (5), (6) and (8) for the region outside of the sphere, we can analogously find the evolution of the density of Brownian particles on the exterior of the sphere after the quench (similar no-flux boundary conditions apply at the wall). Subtracting the outside from the inside pressure exerted on the sphere, one obtains the force exerted per area Ad on the sphere following the quench: F(t) Ad = kBTF[Δρin(Rd, t) −Δρout(Rd, t)]. (10) In equation (10), Δρin/out represents the excess density inside/outside the sphere. For time, pressure, and force, we define the dimensionless variables t̄ = Dt R2 d , P̄(t) = P(t)Rd αdρ0kBTF , F̄(t) = F(t)Rd αdρ0kBTFAd . (11) Here Rd takes the values r0, r1, or r2, depending on the consid- ered geometry and the wall for which we calculate the pressure and force. 3. Quench in circular geometries (d = 2) In this section we discuss the post-quench pressure and force acting on a single circular wall or on two concentric circles of different radii. 3.1. Inside the circle Setting d = 2 in equation (5), we have ∂tρ(r, t) = D r ∂r[r∂rρ(r, t)]. (12) We then use separation of variables to write density as Δρ(r, t) = X(r)T (t). Putting this density back to equation (12) we have ∂tT T = D Xr ( ∂rX + r∂2 r X ) = −ζ2. (13) Solving for X and T we find T (t) = e−ζ2t, X(r) = c1J0 ( ζr√ D ) + c2Y0 ( ζr√ D ) , (14) where Jn and Yn are nth order Bessel functions of the first and second kind, respectively [34]. The solution found in equation (14) should be finite inside the circle, so c2 = 0. The no-flux boundary condition at the wall translates into ∂rX(r)|r=r0 = 0, (15) which implies that the parameter ζ takes on discrete val- ues. Defining β = ζr0/ √ D, the allowed values for β are the positive roots of the Bessel function, J1(βn) = 0, n = 1, 2, 3, . . . . (16) We can then write the time-dependent evolution of the density in the form of an infinite series, Δρin(r, t) = ∞∑ n=1 cn e−β2 n t̄J0 ( βn r r0 ) + c0, (17) using dimensionless time t̄ as defined in equation (11) with Rd = r0. The appropriate initial condition for this problem (which involves the excess density at the wall) takes the form Δρin(r, t = 0) = αρ0δ(r − r0), and the constant c0 can be found by integrating over the disk and enforcing particle conservation: ∫ r0 0 r dr [ ∞∑ n=1 cnJ0 ( βn r r0 ) + c0 ] = ∫ r0 0 r drαρ0δ(r − r0), (18) which gives c0 = 2ρ0α/r0. To find the remaining coefficients cn, we use the following orthogonality relation for the Bessel function [34]: ∫ 1 0 x dxJ0(βnx)J0(βmx) = δnm [J0(βn)]2 2 , (19) with βn’s given by equation (16). This yields cm = αρ0r0J0(βm)∫ r0 0 r dr[J0(βmr̄)]2 = 2αρ0 r0J0(βm) . (20) Putting everything together we find the density on the circle as Δρin(r0, t) = ( 1 + ∞∑ n=1 e−βn̄t ) 2αρ0 r0 . (21) 3.2. Outside the circle For the evolution of density outside the circle we similarly use equations (14) and (15), from which we obtain the following result: Δρout(r, t) = ∫ ∞ 0 dγ c(γ)G ( γ, r r0 ) e−γ2̄t, G(γ, x) = Y1(γ)J0(γ x) − J1(γ)Y0(γ x). (22) To find the coefficients c(γ), we insert the above result in the initial condition written in the form∫ ∞ r0 r drΔρout(r, t = 0)G ( γ ′, r r0 ) = αρ0 ∫ ∞ r0 r drδ(r − r0)G ( γ ′, r r0 ) , (23) 4 J. Phys.: Condens. Matter 33 (2021) 375102 M R Nejad et al and evaluate the integrals by invoking the orthogonality con- dition∫ ∞ 1 x dxG(γ, x)G(γ ′, x) = δ(γ − γ ′) [J1(γ)]2 + [Y1(γ)]2 γ . (24) This finally leads to c(γ) = −2αρ0 πr0 [ J2 1(γ) + Y2 1 (γ) ] . (25) Then, the density on the boundary of the circle reads Δρout(r0, t) = ∫ ∞ 0 dγ 4αρ0 e−γ2̄t γπ2r0 [ J2 1(γ) + Y2 1 (γ) ] . (26) In the long time limit, this excess density decays as Δρout(r0, t →∞) ≈ αρ0 2r0 t̄−1. (27) At long times after the quench, the excess pressure exerted on the circle from outside therefore approaches zero as t̄−1. Having computed the density of the particles inside and outside the circle, we can use equation (10) to find the force exerted on the circle after the quench. Figure 2 shows that the force approaches its new steady-state as t̄−1 in time. As can be seen in the inset of the figure, the force exerted on the circle starts from the value F̄2D = 1 at very short times. A similar behavior is also observed for the force exerted on a sphere. Nevertheless, immediately after the quench one would expect the quench-induced force on the wall to be zero. Inded, we show in appendix B (via an appropriate treatment of the summation in equation (21) and the initial condition) that at t = 0+ the quench-induced force vanishes. This means that at very short times there is an ‘abrupt’ transition from zero to the initial finite force seen in the inset of figure 2. This off- set value for the force at very short times is a consequence of the curved geometry of the circle and a clear manifesta- tion of the nonequilibrium character of the quench-induced dynamics. The curvature of the wall implies that diffusion of the excess particles inside the circle occurs in a confined environment, in contrast to the outside region. As a result, and assuming initial excess densities which are entirely local- ized at the boundaries, this finite force is reached immediately (and discontinuously in time) after the quench. However, this observation is a consequence of the coarse-graining assump- tions: if we attribute a finite small width ε to the initial excess density (rather than the delta function), the force becomes finite after a very short time t̄ � (ε/r0)2 (see appendix B for more details). Here we focus on the dynamics of the quench-induced force, and we calculate the value of the total force in subsec- tion 4.4. 3.3. Quench effect on the medium between two circles In this part, we consider a system of non-interacting Brownian particles confined between two circles with radii r1 and r2, as shown in figure 1(c). The details of obtaining analytic solutions for the time evolution of the excess densities between the two Figure 2. Analytical results for the temporal approach of the force acting on the circle, represented in figure 1(a), toward its new steady-state value, shown on logarithmic (main panel) and linear (inset) scales. circular boundaries can be found in appendix C. The results for the post-quench dynamics of the pressure and forces for two concentric circles are represented in figure 3. Both analyt- ical and simulation results, shown in figure 3(a), indicate that the pressures exerted on the small and large circles by the con- fined particles, scale as t̄−1/2 at short times after the quench. This behavior stems from the fact that at very short times the excess particle density is still very localized near the walls, and its evolution is effectively described by a one-dimensional dif- fusion along the direction locally normal to the wall. As will become clear in the next section, the t̄−1/2 behavior of the pres- sure also occurs in a spherical geometry and can be considered as a universal short-term characteristic of the pressure in all dimensions and geometries. We now turn to the force exerted on each circle. As shown in figure 3(b), the force exerted on the large circle approaches its steady-state as t̄−1 at long times. This behavior is the same as for the force on a single circle (figure 2), because the late- time behavior is dominated by the density relaxation in the infinite outside medium. The early time behavior of the force on the smaller circle is also similar to the case of a single cir- cle, and as figure 3(c) shows, the force starts of a finite value F̄1 = 1 after the quench which is similar to the case of a single circular wall. However, the late time behavior differs signifi- cantly: after an initial overshoot, which becomes stronger and takes place at earlier times for larger r1, a steady-state behav- ior is approached. To understand the non-monotonic behavior of the force exerted on the smaller circle, we note that due to the curvature of the walls, at small times after the quench, the particles from inside the circle move toward the walls more than those particles between the circles. Correspondingly, the force exerted on the smaller circle is positive. At later times, the particles which were initially located on the exterior circle (at r = r2) find time to diffuse and reach the smaller circle and, as a result, the force exerted on the smaller circle decreases. This explains the overshoot seen in figure 3(c). Upon varying r1 while keeping Δr fixed, the steady-state force also changes and even undergoes a sign change for large r1 depending on the 5 J. Phys.: Condens. Matter 33 (2021) 375102 M R Nejad et al Figure 3. (a) Time evolution of the pressure exerted on the smaller and larger circle, represented in figure 1(c), measured with respect to their steady-state value. The solid lines and filled symbols (dashed line and hollow symbols) correspond to the smaller (larger) circle. (b) Evolution of the force exerted on the larger circle toward its steady-state value. Panels (c) and (d) show forces exerted on the small and large circles, respectively. Forces are obtained from the analytical solutions. balance between the particles inside/outside the small circle. In contrast to F̄1, the force on the outer circle is always pos- itive (toward outside) and monotonically increases, as shown in figure 3(d). 4. Quench in spherical geometries (d = 3) In this section we investigate the phenomena considered in section 3, but in three spatial dimensions. 4.1. Inside the sphere Dynamics of a diffusive system inside a sphere (d = 3) can be written as ∂tρ(r, t) = D r2 ∂r [ r2∂rρ(r, t) ] . (28) Again we use separation of variables for the excess density, Δρ(r, t) = R(r)T (t). Inserting this into equation (28) we find ∂tT (t) T = D Rr2 ( 2r∂rR+ r2∂2 r R ) = −ζ2. (29) Solving for R and T yields T (t) = e−ζ2t, R(r) = A r cos ( βr r0 ) + B r sin ( βr r0 ) , (30) where β = ζr0/ √ D. For the interior of a sphere, the first term in the solution for R in equation (30) diverges at r = 0, so that one must put A = 0. The no-flux boundary condition on the interior surface of the sphere gives ∂rR(r)|r=r0 = 0. (31) Similar to the 2D case, we find discrete values for the parameter β, which are now the positive roots of βn = tan βn, n = 1, 2, 3, . . . (32) The time-dependent solution for the excess density is then Δρin(r, t) = ∞∑ n=1 cn sin(βnr/r0) r e−β2 n t̄ + c0. (33) In equation (33), the summation is over all positive solutions of β in equation (32). The initial condition Δρin(r, t = 0) = αρ0δ(r − r0) allows us to find the constant c0 in analogy to the 2D case by calculating∫ r0 0 r2 drΔρin(r, t = 0) = ∫ r0 0 r2 drαρ0δ(r − r0), (34) which gives c0 = 3αρ0/r0. The coefficients cn are obtained by applying the orthogonality condition∫ 1 0 dx sin(βmx) sin(βnx) = δnm sin2 βn 2 (35) in the following relation:∫ r0 0 r drΔρin(r, t = 0) sin(βmr/r0) = αρ0 ∫ r0 0 r drδ(r − r0) sin(βmr/r0). (36) One finds cm = 2αρ0 sin βm . (37) 6 J. Phys.: Condens. Matter 33 (2021) 375102 M R Nejad et al Figure 4. (a) Short-term behavior of the pressure exerted on the circle (2D) and sphere (3D) from inside, toward its steady-state value. (b) Approach of the force exerted on the sphere toward its steady state value following the quench. Finally the density on the interior surface of the sphere is obtained as Δρin(r = r0, t) = ( 3 + 2 ∞∑ n=1 e−β2 n t̄ ) αρ0 r0 . (38) The summation is over all positive solutions of β in equation (32). Using equations (21) and (38), we can now calculate the pressure exerted from inside on the circle and the sphere, respectively. The scaling behavior is shown in figure 4(a): at small times after the quench, the pressure decays toward its steady state value as t̄−1/2, similar to the case of a circular wall (d = 2). This behavior originates from the fact that, at short times after the quench, the excess density (and the boundary layer) is still very localized on the wall, so that diffusion of the particles away from the boundary is effectively one-dimensional. Therefore, the short-time scaling of the excess density and pressure measured with respect to their steady state is insensitive to the curvature of the bound- aries. As discussed in the previous section, this is a univer- sal characteristic for the short-term characteristic of quench- induced pressure, and is independent of the dimensionality of the system. 4.2. Outside the sphere Using equations (30) and (31), the time-dependent excess density outside the sphere can be found: Δρout(r, t) = ∫ ∞ 0 dγ c(γ)K ( γ ′, r r0 ) e−γ2̄t, K(γ, x) = γ cos[γ(x − 1)] + sin[γ(x − 1)] x . (39) The unknown coefficient c(γ) is fixed by the initial condi- tion along with equation (39). To this end we need to evaluate the integral relation∫ ∞ r0 r2 drΔρout(r, t = 0)K ( γ ′, r r0 ) = ∫ ∞ r0 r2 drαρ0δ(r − r0)K ( γ ′, r r0 ) . (40) Using the orthogonality relation∫ ∞ 1 dxx2K(γ, x)K(γ ′, x) = π 2 δ(γ − γ ′)(1 + γ2), (41) we find c(γ) = 2αρ0 πr0 γ 1 + γ2 . (42) The density on the exterior of the sphere takes the form Δρout(r = r0, t) = 2αρ0 πr0 ∫ ∞ 0 dγ γ2 1 + γ2 e−γ2̄t = αρ0 r0 [ 1√ πt̄ − et̄ erfc( √ t̄) ] , (43) where erfc(x) is the complementary error function [34]. Using equation (43), we can expand the density of the outside at long times, and obtain the long-time behavior of the pressure which reads lim t̄→∞ P̄out 3D(̄t) = t̄− 3 2 2 √ π . (44) As for the 2D case, the time-dependent density inside and outside of the sphere can now be used to compute the force exerted on the sphere after the quench. This force is shown in figure 4(b) as a function of time. Figures 5(c) and (d) show that the force exerted on the spheres at very short times has a finite offset value F̄ = 2. Similar to the 2D case and as dis- cussed in appendix B, the quench-induced force immediately after the quench is zero, and approaches a finite value after very short times. Moreover, as discussed in the previous section, this sudden increase is an interesting non-equilibrium effect associated with the curvature of the walls. A comparison to figure 2 shows that the late time decay of the force toward its steady-state value follows t−1 in 2D, but t−3/2 in 3D. Accord- ingly, we can generalize these results for d-dimensional spher- 7 J. Phys.: Condens. Matter 33 (2021) 375102 M R Nejad et al Figure 5. Time evolution of the pressure and forces exerted on the two spheres represented in figure 1(d). (a) The solid lines and filled symbols (dashed lines and hollow symbols) correspond to the pressure on the small (large) sphere. (b) Scaling behavior of the force on the large sphere upon approaching its steady-state value. Panels (c) and (d) show the forces exerted on the small and large spheres, respectively. Similar to figure 3, forces are obtained from the analytical results. ical geometries as F̄(̄t) − F̄(̄t →∞) ∝ t̄−d/2 for large t̄. We note that in the case of flat boundaries, the scaling does not depend on the dimension, and the force approaches the steady state as F̄(̄t) − F̄(̄t →∞) ∝ t̄−1/2 for all dimensions. The long- term scaling behavior of the forces acting on the walls stems from the density relaxation in the outer infinite region. In fact, the density of the inner region with finite volume relaxes exponentially to the steady state, whereas the density relax- ation outside the walls is much slower and has a power-law tail ∼ t̄−d/2 governed by the diffusion in a d-dimensional infinite space. 4.3. Quench of medium between two spheres In the last part of this section, we consider a system of non- interacting Brownian particles confined between two spheres with radii r1 and r2, respectively; see figure 1(d). The dis- tance between the surfaces of the two spheres is defined as Δr = r2 − r1. The solution of the post-quench dynamics of the excess density between two spheres are presented with details in appendix D, from which we obtain the pressure and forces on the two spheres as shown in figure 5. As it can be seen in figure 5(a), the pressure exerted on the small and large sphere decay as t̄−1/2 at small times, for different values of r1 and r2. The same scaling was observed for the 2D case and also for the single sphere as shown in figures 3(a) and 4(a), respec- tively. Figure 5(b) shows that the force exerted on the large sphere approaches its steady-state as t̄− 3 2 at long times. Finally, figures 5(c) and (d) represent the time evolution of the forces acting on the two concentric spheres; these are qualitatively very similar to those of the two circles shown in figure 3. In particular, the forces suddenly increase from zero to a finite value (F̄1,2 = 2) at very short times and the force exerted on the small sphere exhibits an overshoot for large enough val- ues of r1. The overshoot becomes more pronounced when r1 is increased for a fixed Δr. 4.4. Net force exerted on spherical boundaries So far we have studied the dynamics of the quench-induced contribution to the pressure and force exerted on the bound- aries. In particular, the results shown in all the figures have been obtained by setting the initial temperature equal to zero (T I = 0) in the simulations and the numerical evaluations of analytical results. As long as we are interested in the quench- induced force, these results remain unchanged even for T I �= 0, because this contribution is proportional to the parameter α ∝ √ TI − √ TF as given by equation (A1). The net force exerted on the walls at non-zero temperatures can be found by adding the pre-quench force to the dynamical force that we found in the previous sections. The pre-quench force is related to the surface tension and arises from the inho- mogeneity of the density due to the soft wall potential, as well as the curvature of the walls [35]. Such a surface tension is indeed present before and after the quench. The contribution of the quench-induced variations in the density and resulting forces has been taken into account in previous sections. There- fore, to find the net force acting on the walls, we only need to add the contribution of the surface tension before the quench. To this end, we calculate the pressure difference between inside and outside, ΔPd(t < 0) = Pin(t < 0) − Pout(t < 0), at 8 J. Phys.: Condens. Matter 33 (2021) 375102 M R Nejad et al the pre-quench temperature T I, for a d-dimensional spherical wall, which reads ΔPd = [ 〈Fin d 〉+ 〈Fout d 〉 ] /Ad = ρ0 rd−1 0 ∫ ∞ 0 rd−1 drλ(r − r0)e− λ(r−r0)2 2kBTI = ρ0kBTI r0 (d − 1) √ 2πkBTI λ , (46) where 〈Fin/out d 〉 = 〈dV in/out wall /dr〉 denotes the average force exerted by the interior/exterior particles on the d-dimensional sphere with an area Ad. It should be noted that the bounds of the integral for finding the averages 〈Fin〉 and 〈Fout〉 are (r0,∞) and (0, r0), respectively, which then sum up to give equation (45). Since the integrand above becomes negative for the range (0, r0), Fout has a negative value as it acts in the direction of −r̂, and therefore Pout ∝ −〈Fout〉. The total force acting on a spherical boundary following the quench can then be found by adding the initial force Fd(t = 0−) = AdΔpd to the dynamical forces introduced in the previous sections. Using the definition of the Laplace pressure for a d-dimensional sphere, Δpd = γ(d − 1)/r0, we can use equation (46) to find the surface tension as γ = ρ0kBTI √ 2πkBTI λ . (47) Note that the surface tension γ in d > 1 dimension has a dimensionality [γt] = [kBT]�−d+1. 5. Conclusions We have studied systems of Brownian particles confined by surfaces in spherical geometries for 2 and 3 spatial dimen- sions, following a quench in temperature. For all cases con- sidered, the analytical results were shown to be in quan- titative agreement with our explicit Brownian dynamics simulations. In particular, we calculated the time-evolution of the density analytically. This allowed us to find the dynamics of pressures and net forces acting on the boundaries. Short-time scaling of pressures on curved boundaries was shown to be insensitive to the dimension of the system, and universally decays as t−1/2 dictated by effective a one-dimensional diffusion. The obser- vations made for forces, which are obtained as the difference of inside and outside pressures on a given boundary, are dif- ferent. Unlike what is observed for flat boundaries [21], the long-time scaling of the forces exerted on curved boundaries was shown to depend on the dimension of the system as t̄−d/2. We further note that in our system, the geometry of the curved walls determines these scalings, while for the case of 2D or 3D flat boundaries the forces behave the same as the 1D case studied in reference [21]. Additionally, we show in appendix E that for very large radii the present results for curved geometries recover those obtained for planar geometries, as expected. Furthermore, we showed that the curvature of the bound- ary differentiates diffusion of particles on the two sides of the boundary after the quench, i.e., particles close to the boundary and inside a sphere have less space to diffuse than particles close to the outside boundary. As a result, the post-quench force exerted on the curved boundaries, increases to a constant value very rapidly; the constant depends on the dimension of the system. We have therefore demonstrated that the curvature of con- fining boundaries has an important (dimension-dependent) effect on non-equilibrium dynamics of an ideal fluid. Our results could, in principle, be used to compute forces acting on boundaries of droplets or bubbles in an ideal fluid. In particu- lar, we expect our model to describe the behavior of colloidal suspensions confined by curved boundaries [36]. In the appro- priate coarse-graining and density regimes, our results are also relevant for dynamics of forces exerted on curved membranes due to a quench in activity of active non- or weakly-interacting systems. Indeed, a gas of active particles can be modeled by Brownian motion with an effective temperature, and a quench of the effective temperature of an active medium has been shown to give rise to qualitatively similar effects to a tem- perature quench in a passive medium [21]. In experiments on passive Brownian particles, sudden changes in the tempera- ture can be achieved by laser. On the other hand, in an active system, employing tunable activity in an experimental set-up similar to that used in reference [29] can lead to an effec- tive temperature quench. However, the role of solvent effects would have to be considered with care. Additionally, the theo- retical framework established here sets the basis for computing curvature corrections for forces on planar surfaces. Another interesting case to be studied would be that of non-concentric circles (spheres), where we expect short-time and late-time behaviors of the force to be similar to those in the concen- tric cases. However, the transient dynamics could be different depending on the distance between the two circles (spheres) centers. It would also be interesting to study the role played by correlations arising from quenches of spherically confined sys- tems in terms of a field-theoretical approach. These questions will be addressed in future work. Acknowledgments We thank P Nowakowski for valuable discussions. Further, we thank S Dietrich for funding a research visit of MRN at the MPI-IS in Stuttgart, during which this work was initi- ated. MRN acknowledges the support of the Clarendon Fund Scholarship. CMR acknowledges funding from the Emerg- ing Researchers Grant of the University of Cape Town. AGM acknowledges financial support and the hospitality of the Leib- niz Institute for Theoretical Solid State Physics (IFW Dresden) during his visit. Data availability statement All data that support the findings of this study are included within the article (and any supplementary files). 9 J. Phys.: Condens. Matter 33 (2021) 375102 M R Nejad et al Appendix A. Boundary layer thickness αd In this appendix, we compute the thickness of the nonequi- librium boundary layer formed immediately after the quench which serves as the initial condition for the quench dynamics. Before the quench (t < 0) the system is at equilibrium, which leads to a constant density everywhere but at the wall, where a position-dependent distribution ρ(r) ∝ exp [ −Vwall(r)/kTI ] is found. In the case of harmonic wall potentials, the density pro- file in the walls is indeed half of a Gaussian function. Assum- ing the coarse-grained regime of steep wall potentials, the Gaussian profile becomes very localized and can be approx- imated by a Dirac delta function. Similarly, at very long times after the quench when another equilibrium is reached, the den- sity becomes constant away from the wall, while in the wall we have ρ(r) ∝ exp [ −Vwall(r)/kTF ] , which can be again approxi- mated by a delta function. Based on superposition and particle conservation, we only need to consider the evolution of the nonequilibrium or quench-induced part of the boundary layer which accounts for the extra adsorption/desorption of particles at the wall immediately after the quench. The thickness of this nonequilibrium boundary layer is, therefore, given by the dif- ference of the pre-quench and the long-term steady values of the density as first discussed in reference [21]. For the circle we have α2 = 1 r0 ∫ ∞ r0 r dr[e−Vwall/(kBTI) − e−Vwall/(kBTF)] ≈ √ πkB 2λ ( √ TI − √ TF), (A1) where in the final step we have used the approximation√ 2kBTI λ , √ 2kBTF λ r0 which means we have considered sys- tem sizes much larger than the characteristic width of the boundary layer. To calculate the coefficient α3, we need to integrate the initial and final density in 3D. We use the same approximation for the system size and find: α3 = 1 r2 0 ∫ ∞ r0 (e−Vwall/(kBTI) − e−Vwall/(kBTF))r2 dr ≈ √ πkB 2λ ( √ TI − √ TF), (A2) which is equal to the coefficient α2. For this reason we have used α instead of αd in the main text. Appendix B. Explicit derivation of the initial offset values of the quench-induced force In sections 3 and 4, we have seen that the forces acting on the walls always have an initial offset of F̄ = 1, 2 in 2D and 3D, respectively. There, we have argued that, in fact, the quench- induced force at t = 0+ should be zero and therefore the appar- ent offset values are the result of the Dirac delta approximation for the initial excess densities. In practice, starting from ini- tial excess densities with a finite but small width, one can see that the force very rapidly (but not immediately) approaches these offset values. Indeed, it is natural to consider the coarse- grained model primarily at time scales beyond the dynamical processes occurring directly at the wall immediately after the quench. Here, we show explicitly that the force exerted on a sphere is indeed equal to zero at t̄ = 0+, but very rapidly reaches a value F̄3D = 2, whereafter it evolves toward its steady state value. We also discuss the reason for this rapid increase, which is absent in a system with flat boundaries. We consider the initial condition for the excess density as Δρ(r, t̄ = 0) = ⎧⎨ ⎩ αρ0 εr0 , | r r0 − 1| < ε 2 0, otherwise. (B1) We take ε to be very small (so that the expression approximates a δ function), and use this initial condition to find the unknown coefficients in the density expansion in equations (33) and (39). The density on the interior of the sphere at t = 0 is then Δρin = αρ0 r0 [ 3 + 2 ε ∞∑ m=1 Λm − βmε cos(βmε) β3 m ] , Λm = sin(βmε)[1 + β2 m(1 − ε)], (B2) where the coefficients βm are the positive solutions of equation (32). The summation in equation (B2) can be approx- imated with an integral based on the Euler–Maclaurin for- mula and the fact that ε 1 [37]. To this end, we first add ε− ε2 + ε3/3, which can be thought of as the m = 0 term of the summation (corresponding to β0 → 0). Then, noticing that the distance between successive roots βm of equation (32) is very close to π for large m, we use the conversion ∑∞ m=0 →∫ dβ/π which yields Δρin = αρ0 r0 [ 3 + 2 ε ∫ ∞ 0 dβ π ω(β) − βε cos(βε) β3 − 2 ε ( ε− ε2 + ε3 3 )] +O(ε), ω(β) = sin(βε)[1 + β2(1 − ε)], (B3) and eventually Δρin = αρ0 r0 1 ε +O(ε). (B4) Similarly and from equation (39), the density exterior to the sphere at t = 0 is found to be Δρout = αρ0 r0 2 ε ∫ ∞ 0 dβ π Ω(β) − βε cos βε β ( 1 + β2 ) ] , Ω(β) = sin βε[1 + β2(1 + ε)], (B5) which, after evaluating the integral, leads to Δρout = αρ0 r0 1 ε . (B6) Therefore the force exerted on the sphere immediately fol- lowing the quench is proportional to Δρin −Δρout ∝ ε which means the force is equal to zero at t̄ = 0 for ε→ 0. 10 J. Phys.: Condens. Matter 33 (2021) 375102 M R Nejad et al As time evolves from zero, the force increases very rapidly to the value F̄ = 2, after which the approach to steady state continues as discussed in the main text. Assuming (ε/r0)2 t̄, we first take the limit of ε→ 0 and then the limit of t̄ → 0 in equations (38) and (39) from which we obtain Δρin(r = r0, t̄ → 0) = αρ0 r0 ( 3 + ∞∑ m=1 2 ) , (B7) Δρout(r = r0, t̄ → 0) = 2αρ0 r0 ∫ ∞ 0 dβ π β2 1 + β2 . (B8) Converting the summation in internal density to an integral in the same way explained above, we can then calculate the force exerted at the sphere for finite and small values of the time as: F̄ = r0 αρ0 [Δρin(r = r0, t̄ → 0) −Δρout(r = r0, t̄ → 0)] = 1 + 2 π (∫ ∞ 0 dβ − ∫ ∞ 0 dβ β2 1 + β2 ) = 2, (B9) which matches very well with the inset of figure 4(b). This rapid change in the force acting on the curved surface of the sphere is absent for flat boundaries. In fact, due to the curvature of the boundary, particles on the inner sur- face of the sphere are more confined compared to the par- ticles on the external surface. As a result, at a finite time, the interior particles diffuse into the wall more rapidly. This causes a rapid increase in the force acting on the boundary. A similar process occurs for the force acting on the circle in 2D. Appendix C. Solution for the region between two circles Using the solution found for the density in equation (14), and imposing no-flux boundary conditions on both circles, ∂rX(r)|r=r1,r2 = 0, we can find discrete values for the parame- ter β ′ from the following equation: J1(β′ nr1) Y1(β′ nr1) = J1(β′ nr2) Y1(β′ nr2) , n = 1, 2, 3, . . . (C1) The time-dependent solution for the density can be written as Δρ(r, t) = ∞∑ n=1 cn f n(r)e−β′ 2 n Dt + c0, fn(r) = J0(β′ nr)Y1(β′ nr2) − J1(β′ nr2)Y0(β′ nr), (C2) where the summation runs over all positive solutions of β ′ in equation (C1). The initial condition, which cor- responds to excess particle layers at both surfaces, has the form of Δρ(r, t = 0) = αρ0 [δ(r − r1) + δ(r − r2)], and the constant c can be found by calculating the fol- lowing integrals related to conservation of the particle number:∫ r2 r1 r dr[cn f n(r) + c0] = α ∫ r2 r1 r dr[δ(r − r1) + δ(r − r2)]. (C3) This gives c0 = 2αρ0 r2 − r1 . (C4) For calculating coefficients cn we need to calculate below integrals:∫ r2 r1 r drαρ0 [δ(r − r1) + δ(r − r2)] f m(r) = ∫ r2 r1 r dr [ ∞∑ n=1 cn f n(r) + c0 ] f m(r), (C5) which gives cn = 2αρ0 r2 fn(r2) − r1 fn(r1) . (C6) In the last step above we have used the orthogonality relation∫ r2 r1 r dr fm(r) fn(r) = δmn 2 [r2 2 f 2 n(r2) − r2 1 f 2 n(r1)], (C7) to find the expression (C6) for the coefficients cn, Appendix D. Solution for the region between two spheres Using the solution found for the density in equation (30), subject to no-flux boundary conditions on the surface of two spheres, ∂rR(r)|r=r1,r2 = 0, we can find discrete values for the parameter β ′ from the following equation: Δrβ′ n β′2 n r1r2 + 1 = tan(Δrβ′ n), n = 1, 2, 3, . . . (D1) The time-dependent solution for the density follows Δρ(r, t) = ∞∑ n=1 cn e−Dβ′ 2 n t r f n(r) + c0, (D2) fn(r) = sin(β′ nr) + an cos(β′ nr), (D3) with an = β′ nr1 − tan(β′ nr1) 1 + β′ nr1 tan(β′ nr1) ≡ β′ nr2 − tan(β′ nr2) 1 + β′ nr2 tan(β′ nr2) . (D4) The initial condition for this problem has the form Δρ(r, t = 0) = α3ρ0 [δ(r − r1) + δ(r − r2)], where c0 is fixed by calculating∫ r2 r1 r2 dr [cn r f n(r) + c0 ] = ∫ r2 r1 r2 drα [δ(r − r1) + δ(r − r2)] . (D5) 11 J. Phys.: Condens. Matter 33 (2021) 375102 M R Nejad et al One finds c0 = 3αρ0(r2 1 + r2 2) r3 2 − r3 1 . (D6) For the coefficients cn, we need to compute∫ r2 r1 r drαρ0 [δ(r − r1) + δ(r − r2)] fm(r) = ∫ r2 r1 r drρ(r, t = 0) fm(r), (D7) which gives cm = 2αρ0 r2 fm(r2) + r1 fm(r1) r2 f 2 m(r2) − r1 f 2 m(r1) . (D8) To find the coefficients cm, we have used the orthogonality condition∫ r2 r1 dr fm(r) fn(r) = δmn 2 [ r2 f 2 m(r2) − r1 f 2 m(r1) ] . (D9) Appendix E. Asymptotic convergence to the case of flat surfaces In this section, we explicitly show that our results asymp- totically approach the flat geometry results of the previ- ous study [21], by taking the limit of large radii (Rd → ∞). We first consider the circular case where we find that the density follows equation (26). Recalling that the dimen- sionless time for a single circle is t̄ = Dt/r2 0, and changing the integration parameter to γ̃ = √ Dtγ/r0, we can re-write equation (26) as Δρout(r0, t) = 4αρ0 π2r0 ∫ ∞ 0 dγ̃ γ̃ e−γ̃2 J2 1 ( γ̃r0√ Dt ) + Y2 1 ( γ̃r0√ Dt ) . (E1) Taking the limit of r0 →∞, we can replace Bessel functions with their asymptotic forms J2 1(z) ∼ √ 2 πz sin(z − π/4), (E2) Y2 1 (z) ∼ − √ 2 πz cos(z − π/4), (E3) which results in Δρout(r0 →∞, t) = 4αρ0 π2r0 πr0 2 √ Dt ∫ ∞ 0 dγ̃ e−γ̃2 = αρ0√ πDt . (E4) In a similar manner, we can find the asymptotic behavior for the spherical case given by equation (43). Using the new integration variable γ̃, the density becomes Δρout(r = r0, t) = 2αρ0 π √ Dt ∫ ∞ 0 dγ̃ e−γ̃2 r2 0γ̃ 2/Dt 1 + r2 0γ̃ 2/Dt , (E5) which, by taking the limit of r0 →∞, again reduces to αρ0/ √ πDt. These results are therefore consistent with the direct calculations for the case of planar surfaces. ORCID iDs M R Nejad https://orcid.org/0000-0002-4670-9141 References [1] Casimir H B G 1948 On the attraction between two perfectly conducting plates Proc. Kon. Ned. Akad. Wet. 51 793 [2] Fisher M E and Gennes P G D 1978 Wall phenomena in a criti- cal binary mixture Comptes Rendus Hebd. Seances Acad. Sci. Ser. B 287 207–9 [3] Kardar M and Golestanian R 1999 The ‘friction’ of vacuum, and other fluctuation-induced forces Rev. Mod. 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Introduction 2. Model and simulation details 2.1. Microscopic model for numerical simulations 2.2. Coarse-grained model 3. Quench in circular geometries () 3.1. Inside the circle 3.2. Outside the circle 3.3. Quench effect on the medium between two circles 4. Quench in spherical geometries () 4.1. Inside the sphere 4.2. Outside the sphere 4.3. Quench of medium between two spheres 4.4. Net force exerted on spherical boundaries 5. Conclusions Acknowledgments Data availability statement 5. Boundary layer thickness Appendix B. Explicit derivation of the initial offset values of the quench-induced force Appendix C. Solution for the region between two circles Appendix D. Solution for the region between two spheres Appendix E. Asymptotic convergence to the case of flat surfaces ORCID iDs References