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Title: Legendre submanifolds in contact manifolds as attractors and geometric nonequilibrium thermodynamics

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.
Authors:
 [1]
  1. Institute for Molecular Science, 38 Nishigo-Naka, Myodaiji, Okazaki 444-8585 (Japan)
Publication Date:
OSTI Identifier:
22479581
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 7; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ATTRACTORS; DUALITY; EQUILIBRIUM; GEOMETRY; HAMILTONIANS; PHASE SPACE; STATISTICAL MECHANICS; TENSOR FIELDS; THERMODYNAMICS; VECTOR FIELDS