Contact topology and non-equilibrium thermodynamics

Entov, Michael; Polterovich, Leonid

doi:10.1088/1361-6544/acd1ce

Cited by 5 publications

(8 citation statements)

References 29 publications

(65 reference statements)

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“…There are several approaches to express nonequilibrium thermodynamic processes by means of contact geometry, [7,15,20]. In section 3.2 the class of contact Hamiltonians of the form…”

Section: Contact Geometrymentioning

confidence: 99%

“…, n) is appropriate when q is treated as a set of static thermodynamic variables (see section 3.2). The contact Hamiltonian (22) has been focused in [15] and its related one in [7]. In [15], integral curves of X H induced from (22) are interpreted as relaxation processes, because the long-time limit of a point of an integral curve of X H is on L ψ , where L ψ is interpreted as equilibrium thermodynamic phase space.…”

Section: Contact Geometrymentioning

confidence: 99%

“…Their applications include not only to physical sciences, but also to mathematical engineering [36,39]. In addition, by applying these equations to toy models, explicit mathematical expressions of thermodynamic quantities have been obtained, where such toy models include the Glauber equation [7,14], Brownian motion [25], and gas systems [48].…”

Section: Introductionmentioning

confidence: 99%

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From the Fokker–Planck equation to a contact Hamiltonian system

Goto

2024

J. Phys. A: Math. Theor.

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The Fokker-Planck equation is one of the fundamental equations in nonequilibrium statistical mechanics, and this equation is known to be derived from the Wasserstein gradient ﬂow equation with a free energy. This gradient ﬂow equation describes relaxation processes and is formulated on a Riemannian manifold. Meanwhile contact Hamiltonian systems are also known to describe relaxation processes. Hence a relation between these two equations is expected to be clariﬁed, which gives a solid foundation in geometric statistical mechanics. In this paper a class of contact Hamiltonian systems is derived from a class of the Fokker-Planck equations on a Riemannian manifold. In the course of the derivation, the Fokker-Planck equation is shown to be written as a diﬀusion equation with a weighted Laplacian without any approximation, which enables to employ a theory of eigenvalue problems.

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“…There are several approaches to express nonequilibrium thermodynamic processes by means of contact geometry, [7,15,20]. In section 3.2 the class of contact Hamiltonians of the form…”

Section: Contact Geometrymentioning

confidence: 99%

Section: Contact Geometrymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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From the Fokker–Planck equation to a contact Hamiltonian system

Goto

2024

J. Phys. A: Math. Theor.

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“…The task to decide where the truth is is all the more difficult, as the mathematical methods of approach differ. Thermodynamic entropy is a deterministic notion, mainly studied by means of the GH equation, whose modelization is based on contact geometry ( [28][29][30][31][32][33][34][35][36][37][38] and references therein) and/or on different non-holonomic associated invariants. The BGS entropy study rests on probability and statistical tools; there exist, however, some geometric objects associated to it, e.g., the Fisher metrics, the statistical manifolds, etc.…”

Section: Historymentioning

confidence: 99%

Holonomic and Non-Holonomic Geometric Models Associated to the Gibbs–Helmholtz Equation

Pripoae,

Hirica,

Pripoae

et al. 2023

Mathematics

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By replacing the internal energy with the free energy, as coordinates in a “space of observables”, we slightly modify (the known three) non-holonomic geometrizations from Udriste’s et al. work. The coefficients of the curvature tensor field, of the Ricci tensor field, and of the scalar curvature function still remain rational functions. In addition, we define and study a new holonomic Riemannian geometric model associated, in a canonical way, to the Gibbs–Helmholtz equation from Classical Thermodynamics. Using a specific coordinate system, we define a parameterized hypersurface in R4 as the “graph” of the entropy function. The main geometric invariants of this hypersurface are determined and some of their properties are derived. Using this geometrization, we characterize the equivalence between the Gibbs–Helmholtz entropy and the Boltzmann–Gibbs–Shannon, Tsallis, and Kaniadakis entropies, respectively, by means of three stochastic integral equations. We prove that some specific (infinite) families of normal probability distributions are solutions for these equations. This particular case offers a glimpse of the more general “equivalence problem” between classical entropy and statistical entropy.

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“…There are few dynamical studies beyond setting up a model, though Gromov and Cairnes [16] consider dynamics for a diatomic gas, and Liu et al [21] consider periodic motion in some restricted settings, and a recent paper of Entov and Polterovich [10] discuss trajectories with special properties. It was also found by Bravetti and Tapias [5] that there is an invariant measure defined on the open dense subset of the open submanifold where the Hamiltonian is non-zero, and Bravetti et al [4] consider the type of dynamics on the complement, that is where H = 0.…”

Section: Introductionmentioning

confidence: 99%

Singularities, Bifurcations and Catastrophes

Montaldi

2021

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Suitable for advanced undergraduates, postgraduates and researchers, this self-contained textbook provides an introduction to the mathematics lying at the foundations of bifurcation theory. The theory is built up gradually, beginning with the well-developed approach to singularity theory through right-equivalence. The text proceeds with contact equivalence of map-germs and finally presents the path formulation of bifurcation theory. This formulation, developed partly by the author, is more general and more flexible than the original one dating from the 1980s. A series of appendices discuss standard background material, such as calculus of several variables, existence and uniqueness theorems for ODEs, and some basic material on rings and modules. Based on the author's own teaching experience, the book contains numerous examples and illustrations. The wealth of end-of-chapter problems develop and reinforce understanding of the key ideas and techniques: solutions to a selection are provided.

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Contact topology and non-equilibrium thermodynamics

Cited by 5 publications

References 29 publications

From the Fokker–Planck equation to a contact Hamiltonian system

From the Fokker–Planck equation to a contact Hamiltonian system

Holonomic and Non-Holonomic Geometric Models Associated to the Gibbs–Helmholtz Equation

Singularities, Bifurcations and Catastrophes

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