Examination of the Stochastic Process Underlying a Simple Isomerization Reaction

Abstract

A characterization of the concentration fluctuations in the isomerization reaction $\mathrm{A}+\mathrm{X}\mathrm{\hspace{0.17em}rightleftarrows}\mathrm{B}+\mathrm{X}$ is given based on the Liouville equation for the conditional probability. By separating the relaxation process into four qualitatively distinct time regimes, the approach in time to the birth and death equations of the usual stochastic theory can be followed. It is shown that these equations are approximately solved by a normal distribution which is generated by a modified Langevin‐type equation. In this time domain the stochastic process is stationary and Markov and as the fourth time domain is approached the process also becomes Gaussian as predicted by the fluctuating force theory of this reaction. Since the time scale in this last regime is the order of the time constant for the reaction, it is concluded that the fluctuating force theory is valid only near equilibrium.