Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics

TL;DR

This paper develops contact geometric methods to describe vector fields on dually flat spaces, with applications to electric circuits and nonequilibrium statistical mechanics, providing new tools for stability analysis and invariant measures.

Contribution

It introduces contact geometric descriptions of vector fields on dually flat spaces and proposes methods to lift these fields to contact manifolds, with applications to physical models.

Findings

Explicit invariant measures in contact manifolds

Stability analysis around Legendre submanifolds

Application to electric circuit and statistical mechanics models

Abstract

Contact geometry has been applied to various mathematical sciences, and it has been proposed that a contact manifold and a strictly convex function induce a dually flat space that is used in information geometry. Here, such a dually flat space is related to a Legendre submanifold in a contact manifold. In this paper contact geometric descriptions of vector fields on dually flat spaces are proposed on the basis of the theory of contact Hamiltonian vector fields. Based on these descriptions, two ways of lifting vector fields on Legendre submanifolds to contact manifolds are given. For some classes of these lifted vector fields, invariant measures in contact manifolds and stability analysis around Legendre submanifolds are explicitly given. Throughout this paper, Legendre duality is explicitly stated. In addition, to show how to apply these general methodologies to applied mathematical disciplines, electric circuit models and some examples taken from nonequilibrium statistical mechanics are analyzed.