A note on the bimodal stationary distributions in hard chemical instabilities

Abstract

Over the past years there has been considerable activity in the application of birth and death stochastic equations to the analysis of nonlinear chemical reaction schemes (see Refs. 1-4 for reviews). Of particular interest are systems that exhibit multiple stationary states, as transitions between these states are analogous to first-order phase transitions/3-5) It has been claimed that the stationary distributions of such master equations are bimodal/1-5~ and that this bimodality is the stochastic analog of the deterministic hysteresis effect. It is the aim of this note to argue that the stationary distribution of the master equation is effectively unimodal when the thermodynamic limit is properly taken. Thus the hysteresis effect is shown to be of a kinetic origin, and not a property of the stationary character of the stochastic system/6~ The effective unimodality can be established for a general kinetic equation of the form