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arXiv · 2403.07334

From the Fokker-Planck equation to a contact Hamiltonian system

Shin-itiro Goto
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Abstract

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 Riemannian manifolds. 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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Shin-itiro Goto. 2024-03-12. From the Fokker-Planck equation to a contact Hamiltonian system. https://doi.org/10.1088/1751-8121/ad6225

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