From the Fokker-Planck equation to a contact Hamiltonian system

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.

arXiv comment39 pages

References in corpus (13)

• Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics
• Thermodynamic Unification of Optimal Transport: Thermodynamic Uncertainty Relation, Minimum Dissipation, and Thermodynamic Speed Limits
• Geometry of Thermodynamic Processes
• Geometric thermodynamics for the Fokker-Planck equation: Stochastic thermodynamic links between information geometry and optimal transport
• Asymptotic solutions of a nonlinear diffusive equation in the framework of $\kappa$-generalized statistical mechanics
• Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics
• Legendre submanifolds in contact manifolds as attractors and geometric nonequilibrium thermodynamics
• Dirac structures in nonequilibrium thermodynamics
• Supersymmetric Fokker-Planck strict isospectrality
• Moment generating function bound from detailed fluctuation theorem
• Nonequilibrium thermodynamic process with hysteresis and metastable states -- A contact Hamiltonian with unstable and stable segments of a Legendre submanifold
• Diffusion equations from master equations -- A discrete geometric approach --
• Contact geometric approach to Glauber dynamics near a cusp and its limitation