Fluctuations and transitions at chemical instabilities. I. A computer simulation approach

Abstract

A stochastic model is designed for the study of the critical behavior of nonequilibrium chemical systems characterized by multiple homogeneous stationary states. In the model, the molecules are distributed on a lattice and reactive collisions are smeared out over small reaction cells which overlap over half their width. The model is studied using Monte Carlo techniques. The typical examples we choose for our study are the two Schlögl’s reaction mechanisms. For Schlögl’s first model, we find a transition in all physical dimensionalities (d=1, 2, and 3), as predicted by a deterministic analysis. For the second model, there is a transition for d≥2 only, in agreement with recent analytical and Monte Carlo calculations. For both models, we find that the deviations of the fluctuations from Poissonian behavior in small volumes increase as the square of the radius, as previously observed. As the critical point is approached, the fluctuations for Schlögl’s first model remain finite, as already predicted by the master equation approach. The relative fluctuations, however, diverge. For d<4, we find ‘‘nonclassical’’ values of the critical exponent, which agree with the prediction of a recent renormalization group analysis. Our results for Schlögl’s second model show a divergence of the fluctuations themselves, also in agreement with previous work. In the limit of homogeneous fluctuations, we recover the usual classical exponents for both a ‘‘magnetic field’’- and a ‘‘temperature’’-like approaches to the critical point. For d<4, nonclassical values are obtained. For the temperature-like approach, these are similar to the values found in the Ising model for equilibrium critical phenomena, as predicted by renormalization group studies.