Entropy productions in dissipative systems

Abstract

The present study aims to find connections between two types of entropy production rates for continuous dynamical systems, one defined in the general dissipation context and the other one defined in a thermodynamics context. On one hand, the entropy production rate e p e_p introduced by Andrey and Ruelle in the general dynamics context is based on Shannon’s entropy in the phase space that characterizes volume contraction relating to dissipation. On the other hand, the Helmholtz-Boltzmann entropy production rate Q ˙ / T \dot {Q}/T introduced in the context of mechanical theory of thermodynamics, where Q ˙ \dot {Q} is the rate of heat generations and T T is the temperature, captures the thermodynamic entropy production relating to the mechanical energy dissipation. For certain cases of energy dissipative systems, we show that e p e_p is indeed related to Q ˙ / T \dot {Q}/T . This consistency between the general mathematical notion of dissipation and the thermodynamic entropy production in dissipative systems suggests a unifying representation of nonequilibrium phenomenon in deterministic systems based on the theory of nonlinear dynamical systems.