Nonequilibrium phase transitions and a measure of the ordering of motion in a relaxational auto-oscillatory system

Abstract

An open thermodynamic system is considered. Its state is determined by a one-dimensional temperature field T(x,t) and a heat flux I(x,t), on which nonlocal nonlinear external feedback is imposed. The change in the Clausius entropy and its production in response to the excitation of auto-oscillations in the system is calculated on the basis of the results of a dynamical analysis. The use of the relative increment of the total entropy of the system, normalized to the total entropy production, as a measure of the ordering of motion is proposed. The analogy between the formalism of the Andronov-Hopf bifurcation theorem and the Landau-Ginzburg theory of phase transitions is traced in the second part of the paper. It is shown that in the initial stage of auto-oscillations the phase matching condition, which determines the amplitude of the oscillations within the Andronov-Hopf formalism, becomes meaningless because of fluctuations. In this case the amplitude should be regarded as an order parameter, and the actual state of the system should be determined from the requirement of a minimum for the nonlinear part of the increment of entropy production. The proposed approach permits a description of transient regimes and qualitatively accounts for “soft” and “hard” bifurcations as being due to nonequilibrium first-and second-order phase transitions.