Dynamic correlation functions in the Brusselator

Abstract

The Mori-Zwanzig formalism is applied to analyze the dynamics of the homogeneous (zero-dimensional) Brusselator. A comprehensive description of particle number correlations is given below, near and above the hard instability. The static correlations which must be calculated beforehand, are obtained a) by explicitly solving the master equation (small system size), b) from Monte Carlo simulations (medium system size), c) from a Fokker-Planck-Landau approximation (large system size). The dynamic correlations are characterized by a bare relaxation matrix and a memory kernel. Below the transition the memory effects are small, and the relaxation behaviour can be expressed in terms of second order static correlations. In the limit cycle regime, the bare oscillation period increases with the relative excess of the staticY- over theX-autocorrelations while the decay constant decreases both with the distance from the transition and the system size, indicating the existence of order. The memory effects are generally inversely proportional to the system size. Their main effect for large systems is therefore to modify the decay rate above the transition (the ensemble dephasing).