Fluctuations at chemical instabilities

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Abstract
We consider the effects of fluctuations on chemical systems that have multiple steady states. The systems of interest have two stable steady states and one unstable steady state (a kinetic saddle point). As parameters vary, two or three of the steady states coalesce. We consider experiments beginning near the deterministic separatrix and formulate a stochastic first exit problem. The deterministic separatrix is surrounded by a band. We calculate the first exit probability u (x) and mean exit time T (x) from this band, conditioned on initial position. Fluctuation formalisms connecting the Langevin equation and deterministic kinetic equations are discussed. We use the diffusion approximation so that u (x) and T (x) satisfy (backward) diffusion equations. Approximate solutions of the diffusion equations are constructed by an asymptotic method that involves various incomplete special functions. Two applications are discussed: (1) the spontaneous asymmetric synthesis model of F.C. Frank; (2) fluctuation effects on substrate inhibited reactions in open vessels.