Incremental least action principle in the framework of thermodynamics of irreversible processes

Abstract

In this paper, an incremental least action principle is proposed using the framework of the thermodynamics of irreversible processes. First, we establish that an extensive thermodynamic potential defined over an infinitesimal volume satisfies a differential conservation law with close similarities to the Liouville theorem but extended to irreversible processes. This property is applied to a generalized thermodynamic potential depending on equilibrium variables and nonequilibrium flows. It allows the formulation of an absolute integral invariant (AII) that is shown to have a broader field of application than the Poincaré-Cartan integral invariant of dynamic systems. Once integrated over a finite volume, it naturally defines an integral functional that fulfills an incremental least action principle. The Fréchet derivative of the Euler-Lagrange equations associated with the functional is calculated, and its self-adjointness is shown to be equivalent to the symmetry of the classical Tisza and Onsager matrices which link respectively extensive variables to intensive variables and nonequilibrium flows to generalized forces. Finally, the proposed AII and least action principle are formulated for the case of a simple physical process (heat conduction), to illustrate (i) its link with the extended irreversible thermodynamics and (ii) its applications to numerical simulations.