05 Fakultät Informatik, Elektrotechnik und Informationstechnik

Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/6

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Publication Type: search.filters.UBSTyp.ZeitschriftenartikelInstitute: search.filters.UBSInstitut.Fraunhofer Institut für Produktionstechnik und Automatisierung (IPA)Author: search.filters.author.Birke, Kai PeterAuthor: search.filters.author.Markthaler, Daniel

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    Analytic free-energy expression for the 2D-Ising model and perspectives for battery modeling
    (2023) Markthaler, Daniel; Birke, Kai Peter
    Although originally developed to describe the magnetic behavior of matter, the Ising model represents one of the most widely used physical models, with applications in almost all scientific areas. Even after 100 years, the model still poses challenges and is the subject of active research. In this work, we address the question of whether it is possible to describe the free energy A of a finite-size 2D-Ising model of arbitrary size, based on a couple of analytically solvable 1D-Ising chains. The presented novel approach is based on rigorous statistical-thermodynamic principles and involves modeling the free energy contribution of an added inter-chain bond DAbond(b, N) as function of inverse temperature b and lattice size N. The identified simple analytic expression for DAbond is fitted to exact results of a series of finite-size quadratic N N-systems and enables straightforward and instantaneous calculation of thermodynamic quantities of interest, such as free energy and heat capacity for systems of an arbitrary size. This approach is not only interesting from a fundamental perspective with respect to the possible transfer to a 3D-Ising model, but also from an application-driven viewpoint in the context of (Li-ion) batteries where it could be applied to describe intercalation mechanisms.
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    Exploring different extrapolation approaches for the critical temperature of the 2D-Ising model based on exactly solvable finite-sized lattices
    (2025) Markthaler, Daniel; Birke, Kai Peter
    The fact that the Ising model in higher dimensions than 1D features a phase transition at the critical temperature Tcdespite its apparent simplicity is one of the main reasons why it has lost none of its fascination and remains a central benchmark in modeling physical systems. Building on our previous work, where an approximative analytic free-energy expression for finite 2D-Ising lattices was introduced, we investigate different extrapolation strategies for estimating Tcof the infinite system from exactly solvable small lattices. Finite square lattices of linear dimension N with free and periodic boundary conditions were analyzed, exploiting their exactly accessible density of states to compute the heat capacity profiles C(T). Different approaches were compared, including scaling models for the peak temperature Tmax(N)and an envelope construction across the set of C(T)-profiles. We find that both approaches converge to the same asymptotic value and compare favorably to the established Binder cumulant method. Remarkably, a model for Tmaxwith a single model parameter following an N/(N+1)-law provides robust convergence, with a physical analogy motivating this proportionality. Our findings highlight that surprisingly few, but highly accurate, finite-size results are sufficient to obtain a precise extrapolation.