On Measures of Stability and Relative Stability in Systems with Multiple Stationary States

Abstract

Nonequilibrium systems driven far from equilibrium, which have nonlinear rate mechanisms with feedback loops, may become unstable and may have available multiple stationary states. It is of interest, therefore, to consider the stability of a given stationary state and the relative stability of two or more stationary states for given conditions [1–5]. Kobatake has considered this problem [6] for the transitions between two stationary states in a porous charged membrane. He suggested that the integral of the (inexact) differential, δ x p, of the entropy production with respect to the forces x may serve as a criterion for relative stability. He used the experiments of U. Franck, displayed on a graph of current vs. voltage across a glass filter separating solutions of the same NaCl concentration but at different j pressures. Kobatake analyzed the complex hydrodynamics equations applicable to the system and displayed his theoretical results on a plot of the same (normalized) coordinates [6], In Fig.l, we reproduce the two plots from Kobatake’S paper, super-imposed on each other and arbitrarily fitted to one another such that the maxima of the middle curves, and the location on the abscissa of the experimental and theoretical coexistence point, coincide. The area under the theoretical curves is the integral/6xP; hence, the equivalence of the two shaded areas on each curve corresponds to the coexistence condition $$\int_{{{x_{1}}}}^{{{x_{3}}}} {{\delta _{x}}p = 0}$$, where x1 and x3 are the extreme values of I* in the integration.KeywordsStationary StateRelative StabilityMaster EquationPassage TimeEntropy ProductionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.