On the second law of thermodynamics and contact geometry

Abstract

In this paper we consider the second law of thermodynamics for a dissipative system and its symmetry property in terms of contact geometry. We first show that the inaccessibility condition of Caratheodory and the assumption of semipositive definite property of the dissipative energy are equivalent to Clausius' inequality. The inaccessibility condition then gives rise to a generalized Gibbs relation (GGR). By means of the GGR a 1-form ω can be defined such that the zero of ω reproduces the GGR. Such 1-form ω has the property ω∧ (dω)n ≠ 0 and ω∧ (d ω)n + 1 = 0. The integral surface of the GGR is an n-dimensional 1-graph space G (Legendre submanifold) of a 1-jet space J1 (En, R), where En is the base space of J1 (En, R) with thermodynamic variables as its coordinates. The (2n + 1)-dimensional J1(En, R) equipped with the 1-form ω is also called a contact bundle K, where the intensive thermodynamic variables are considered as the contact elements to K at every x. Next we construct an isovector field Xf such that the inaccessibility condition is invariant under the contact transformations generated by Xf. Finally, suppose under some specific assumptions the dynamical equations of the thermodynamic variables x can be approximated by the flow equations of a vector field XG on G. We can lift XG to Xf such that the 1-graph space G as well as the inaccessibility condition are preserved under the contact transformations generated by Xf. © 1998 American Institute of Physics.