On the second law of thermodynamics and contact geometry

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 omega can be defined such that the zero of omega reproduces the GGR. Such 1-form omega has the property omega Lambda(d omega)(n) not equal 0 and omega Lambda(d omega)(n+1)=0. The integral surface of the GGR is an n-dimensional 1-graph space G (Legendre submanifold) of a 1-jet space J(1)(E-n,R), where E-n is the base space of J(1)(E-n,R) with thermodynamic variables as its coordinates. The (2n+1)-dimensional J(1)(E-n,R) equipped with the 1-form omega 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 X-f such that the inaccessibility condition is invariant under the contact transformations generated by X-f. 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 X-G on G. We can lift X-G to X-f such that the 1-graph space G as well as the inaccessibility condition are preserved under the contact transformations generated by X-f. (C) 1998 American Institute of Physics.