Legendre submanifolds in contact manifolds as attractors and geometric nonequilibrium thermodynamics

摘要

It has been proposed that equilibrium thermodynamics is described on Legendresubmanifolds in contact geometry. It is shown in this paper that Legendresubmanifolds embedded in a contact manifold can be expressed as attractors inphase space for a certain class of contact Hamiltonian vector fields. By givinga physical interpretation that points outside the Legendre submanifold canrepresent nonequilibrium states of thermodynamic variables, in addition to thatpoints of a given Legendre submanifold can represent equilibrium states of thevariables, this class of contact Hamiltonian vector fields is physicallyinterpreted as a class of relaxation processes, in which thermodynamicvariables achieve an equilibrium state from a nonequilibrium state through atime evolution, a typical nonequilibrium phenomenon. Geometric properties ofsuch vector fields on contact manifolds are characterized after introducing ametric tensor field on a contact manifold. It is also shown that a contactmanifold and a strictly convex function induce a lower dimensional dually flatspace used in information geometry where a geometrization of equilibriumstatistical mechanics is constructed. Legendre duality on contact manifolds isexplicitly stated throughout.