Stochastic Approach to the Random Activation Energy Model with a Rate-Determining Step. Universal Behavior for the Probability of Concentration Fluctuations

Abstract

We study the concentration fluctuations for a reversible overall reaction with a rate-determining step occurring in a system with static disorder described in terms of the random activation energy model. We assume that the rates of the forward and backward reactions can be expressed as products of random rate coefficients times concentration dependent factors, which can also depend on nonrandom quasiequilibrium constants such as adsorption coefficients or Michaelis-Menten constants. Further, we take the activation energies of the forward and backward processes to have a random component ΔE, which is selected from a frozen Maxwell-Boltzmann distribution. We derive a stochastic evolution equation for the joint probability density of the reaction extent ξ and of the random component ΔE of the activation barrier. The solution of this stochastic evolution equation leads to a general expression for the probability density (ξ,t) of the reaction extent ξ at time t. For a long time, the probability density?(ξ,t) of concentration fluctuations approaches its stationary value? s t (ξ), according to a universal power scaling law, which is independent of the detailed kinetics of the process?(ξ,t) ? s t (ξ) + t - α C(ξ) as t → ∞, where a is a fractal exponent between 0 and 1 and C(ξ) is a concentration dependent amplitude factor. A similar behavior is displayed by systems approaching a nonequilibrium steady state. We generalize our analysis to multiple overall reactions and to systems with dynamic disorder and develop methods for extracting kinetic information from experimental data.