On linear dynamical equations of state for isotropic media. I. General formalism

In this paper we investigate the relation (dynamical equation of state) among the (mechanical hydrostatic) pressure P, the volume ¿ and the temperature T of an isotropic medium with an arbitrary number, say n, of scalar internal degrees of freedom. It is assumed that the irreversible processes in the medium are due to volume viscosity and to changes in the internal variables and that these processes can be described with the aid of non-equilibrium thermodynamics. It is shown that an explicit form for the dynamical equation of state may be obtained if in the neighbourhood of some state of thermodynamic equilibrium it is permissible to consider the equilibrium pressure and the thermodynamic affinities (conjugate to the internal degrees of freedom) as linear functions of ¿, T and the internal variables and if, moreover, the phenomenological coefficients, which occur in the equations for the irreversible processes, may be considered as constants. This dynamical equation of state has the form of a linear relation among P, ¿, T, the first n derivatives with respect to time of P and of T and the first n+1 derivatives with respect to time of ¿