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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 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., Comment: 39 pages

We draw connections between contact topology and Maxwell fields in vacuo on 3-dimensional closed Riemannian submanifolds in 4-dimensional Lorentzian manifolds. This is accomplished by showing that contact topological methods can be applied to reveal topological features of a class of solutions to Maxwell's equations. This class of Maxwell fields is such that electric fields are parallel to magnetic fields. In addition these electromagnetic fields are composed of the so-called Beltrami fields. We employ several theorems resolving the Weinstein conjecture on closed manifolds with contact structures and stable Hamiltonian structures, where this conjecture refers to the existence of periodic orbits of the Reeb vector fields. Here a contact form is a special case of a stable Hamiltonian structure. After showing how to relate Reeb vector fields with electromagnetic 1-forms, we apply a theorem regarding contact manifolds and an improved theorem regarding stable Hamiltonian structures. Then a closed field line is shown to exist, where field lines are generated by Maxwell fields. In addition, electromagnetic energies are shown to be conserved along the Reeb vector fields.

We study a nonequilibrium mean field Ising model in the low temperature phase regime, where metastable equilibrium states develop a cuspidal (spinodal) singularity. We focus on celebrated Glauber dynamics, and design a contact Hamiltonian flow which captures some of its rough features in this regime. We prove, however, that there is an inevitable discrepancy between the scaling laws for the relaxation time in the Glauber and the contact Hamiltonian dynamical systems., Comment: 2 figures

Thermodynamics provides a unified perspective of thermodynamic properties of various substances. To formulate thermodynamics in the language of sophisticated mathematics, thermodynamics is described by a variety of differential geometries, including contact and symplectic geometries. Meanwhile affine geometry is a branch of differential geometry and is compatible with information geometry, where information geometry is known to be compatible with thermodynamics. By combining above, it is expected that thermodynamics is compatible with affine geometry, and is expected that several affine geometric tools can be introduced in the analysis of thermodynamic systems. In this paper affine geometric descriptions of equilibrium and nonequilibrium thermodynamics are proposed. For equilibrium systems, it is shown that several thermodynamic quantities can be identified with geometric objects in affine geometry, and that several geometric objects can be introduced in thermodynamics. Examples of these include: specific heat is identified with the affine fundamental form, a flat connection is introduced in thermodynamic phase space. For nonequilibrium systems, two classes of relaxation processes are shown to be described in the language of an extension of affine geometry. Finally this affine geometric description of thermodynamics for equilibrium and nonequilibrium systems is compared with a contact geometric description.