Legendre submanifolds in contact manifolds as attractors and geometric nonequilibrium thermodynamics

TL;DR

This paper demonstrates that Legendre submanifolds in contact manifolds serve as attractors for certain contact Hamiltonian vector fields, providing a geometric framework for modeling thermodynamic relaxation processes and linking contact geometry with information geometry.

Contribution

It introduces a class of contact Hamiltonian vector fields as models for thermodynamic relaxation, interpreting equilibrium and nonequilibrium states geometrically, and connects contact geometry with information geometry.

Findings

Legendre submanifolds act as attractors in phase space.

Contact Hamiltonian vector fields model relaxation to equilibrium.

A geometric link between contact geometry and information geometry is established.

Abstract

It has been proposed that equilibrium thermodynamics is described on Legendre submanifolds in contact geometry. It is shown in this paper that Legendre submanifolds embedded in a contact manifold can be expressed as attractors in phase space for a certain class of contact Hamiltonian vector fields. By giving a physical interpretation that points outside the Legendre submanifold can represent nonequilibrium states of thermodynamic variables, in addition to that points of a given Legendre submanifold can represent equilibrium states of the variables, this class of contact Hamiltonian vector fields is physically interpreted as a class of relaxation processes, in which thermodynamic variables achieve an equilibrium state from a nonequilibrium state through a time evolution, a typical nonequilibrium phenomenon. Geometric properties of such vector fields on contact manifolds are characterized after introducing a metric tensor field on a contact manifold. It is also shown that a contact manifold and a strictly convex function induce a lower dimensional dually flat space used in information geometry where a geometrization of equilibrium statistical mechanics is constructed. Legendre duality on contact manifolds is explicitly stated throughout.