Determination of the Stochastic Potential in Chemical Systems with Imposed Fluxes

4 Determination of the Stochastic Potential in Chemical Systems with Imposed Fluxes 

Consider the chemical system in (11.1) with the species being either ions or neutrals the system is in a reaction chamber in a non-equilibrium stationary state. We impose a flux of species Ai, J = k A[, into the reaction chamber with 5+ and Q- held constant and thereby move the chemical system to a different non-equilibrium stationary state with different concentrations of the reacting species Ai, Bi, A, B2- This procedure allows the sampling of different combinations of the reacting species by means of the imposition of different fluxes of these reactants. These combinations represent different non-stationary states in the absence of imposed fluxes, but with the imposed fluxes they are stationary states and hence measurements may be made without constraints of time. If we would attempt to measure concentrations in non-stationary states then the measurement technique would have to be fast compared to the time scale of change of the concentrations due to chemical reactions. 

Now we impose a flux of a chemical species present in the system and inquire on the effect of that imposition on the stochastic potential of the system. For that we need to go from the deterministic kinetic equations to a stochastic 

This procedure is easy for a one-variable system because we know the solution of the stationary master equation to this approximation. For example, for the one-variable Schlogl model we have the elementary reaction steps 

Let the imposed flux be J = k X. The stationary solution of the lowest order eikonal approximation of the master equation for the system (11.15) with the imposed flux is 

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