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

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.

Contact Hamiltonian mechanics

“…Contact Hamiltonian dynamics has been used already in thermodynamics (both equilibrium and not [19][20][21][22][23][24]) and in the description of dissipative systems at the mesoscopic level [25]. Furthermore, it has been recently introduced in the study of mechanical systems exchanging energy with a reservoir [26,27].…”

Contact Hamiltonian Dynamics: The Concept and Its Use

“…However, contact geometry and contact Hamiltonian dynamics appear also in the theory of the optimal control of systems [58,59]. Other areas where contact geometry has been used and which have not been covered in this survey are fluid mechanics [60], electromagnetism [61], electric circuits theory [37,62] and black hole thermodynamics [7,63]. Moreover, recently, a novel application of contact geometry in order to obtain a generally covariant approach to quantum mechanics has been presented in [64].…”

“…Further (and different) approaches to the use of contact Hamiltonian dynamics in irreversible thermodynamics can be found in [12,15,16,[35][36][37].…”

Contact variational integrators

“…By definition of a modified equation, the discrete curve (x j , z j ) j∈Z defined by x j = x(jh) and z j = z(jh) satisfies the discrete system (22) with a defect of order O(h k+1 ). Now consider the action z N =z(N h) wherez is a solution of the higher order Equation (26), with x as before. The discrete Herglotz variational principle implies that z N is critical with respect to variations of x(t), supported on the interval (0, N h), again up to a defect of order O(h k+1 ).…”