A Variational Principle in Thermodynamics

Abstract

We deal with those thermodynamical transport processes in which the current densities are proportional to the gradient of the intensive quantities. We introduce a 'potential function' to the intensive quantities so that by its help a Lagrangian and a variational principle of classical type can be constructed. The analogon of the energy momentum tensor, i.e. the thermodynamical tensor, and the canonically conjugated quantities are derived.

Gyarmati’s Variational Principle of Dissipative Processes

“…Gyarmati's principle is based on the fact that the generalization of the dissipation functions that were introduced by Rayleigh and Onsager for special cases always exists locally in continua [3,4,[49][50][51][52] in the linear theory with Onsager's reciprocal relations. These functions are defined as:…”

“…After the above results were reached, a new variational technique was proposed [2][3][4]49,79] that gives the transport equations as Euler-Lagrange equations for the potential functions introduced suitably. Nevertheless, this technique may be assumed as an entirely different variational principle; it is undeniably an offshoot.…”

“…The latter can be calculated from transversality conditions and is obtained after a first integration of the Euler-Lagrange equations [1]. The other method introduces some potential functions the Euler-Lagrange equations concern, and the customary transport equations result after eliminating them [2][3][4]79].…”

“…The conditions of the mathematical theorem were not assumed (among which functions, the extremum was looked for), moreover, the derivation of a parabolic equation does not mean that it has to be the Euler-Lagrange equation, e.g., the variational problem may result in biparabolic Euler-Lagrange equations with transversality conditions, and the first integration of them results in the parabolic equations [1]. Another procedure is based on some potential functions [2][3][4].…”

Can a Lorentz Invariant Equation Describe Thermal Energy Propagation Problems?

“…Since, for the cases when the Lagrangian contains second order time derivatives the HamiltonianH must be expressed as follows (Gambár & Márkus, 1994;Márkus & Gambár, 1991),…”