Stochastic analysis of chemical reactions in multi-component interacting systems at criticality

Abstract

We numerically and analytically investigate the behavior of a non-equilibrium phase transition in the second Schl\"ogl autocatalytic reaction scheme. Our model incorporates both an interaction-induced phase separation and a bifurcation in the reaction kinetics, with these critical lines coalescing at a bicritical point in the macroscopic limit. We construct a stochastic master equation for the reaction processes to account for the presence of mutual particle interactions in a thermodynamically consistent manner by imposing a generalized detailed balance condition, which leads to exponential corrections for the transition rates. In a non-spatially extended (zero-dimensional) setting, we treat the interactions in a mean-field approximation, and introduce a minimal model that encodes the physical behavior of the bicritical point and permits the exact evaluation of the anomalous scaling for the particle number fluctuations in the thermodynamic limit. We obtain that the system size scaling exponent for the particle number variance changes from $\beta_0 = 3/2$ at the standard non-interacting bifurcation to $\beta = 12/7$ at the interacting bicritical point. The methodology developed here provides a generic route for the quantitative analysis of fluctuation effects in chemical reactions occurring in multi-component interacting systems.