On the invariant quantities in an entropy balance and their impact on the stability of some physical systems

Abstract: In a wide class of physical systems, diffeomorphisms in the state space leave the amount of entropy produced per unit time inside the bulk of the system unaffected [M. Polettini et al., 12th Joint European Thermodynamics Conference, Brescia, Italy, July 1-5, 2013]. This invariance implies that if relaxation towards some final ('relaxed') state occurs, then the necessary condition for the stability of the relaxed state against slowly evolving perturbations is the same for all the systems of this class, regardless of both detailed dynamics of the system, amplitude of fluctuations around mean values and possible occurrence of periodic oscillations in the relaxed state. Our discussion invokes no Onsager symmetry, no detailed model of heat transport and production, and no approximation of local thermodynamic equilibrium. This necessary condition of stability is the constrained minimization of time- and path-ensemble-averaged amount of entropy produced per unit time inside the bulk of the system; for each system, constraints are provided by the equations of motion. (Thermodynamic equilibrium corresponds to identically vanishing amount of entropy produced in the bulk, and satisfies trivially this condition). Our class of physical systems includes both macroscopic and mesoscopic systems: the problems of stability of these systems are dual to each other. As particular cases, we retrieve some necessary criteria for stability of relaxed states available in the literature concerning current filaments and plasmoids in a Dense Plasma Focus, flashing ratchets, Brownian motors and dynamo-affected, convection-ruled shells of Sun-like rotating stars - in the latter case, for example, we retrieve the scaling of the average magnetic field to the reciprocal of Rossby number. Finally, we argue that the validity of the popular "maximum entropy production principle" is questionable in these systems.