Fluctuations in coagulating systems

Abstract

In this paper we consider coagulation processes in large but finite systems, and study the time-dependent behavior of the (nonequilibrium) fluctuations in the cluster size distribution. For this purpose we apply van Kampen'sΩ-expansion to a master equation describing coagulation processes, and derive an approximate (Fokker-Planck) equation for the probability distribution of the fluctuations. First we consider two exactly soluble models, corresponding to the choicesK(i, j) =i + jandK(i, j)=1 for the rate constants in the Fokker-Planck equation. For these models and monodisperse initial conditions we calculate the probability distribution of the fluctuations and the equal-time and two-time correlation functions. For general initial conditions we study the behavior of the fluctuations at large cluster sizes, and in the scaling limit. Next we consider, in general, homogeneous rate constants, with the propertyK(i, j) =a-λK(ai,aj)for alla>0, and we give asymptotic expressions for the equal-time correlation functions at large cluster sizes, and in the scaling limit. In the scaling limit we find that the fluctuations show relatively simple scaling behavior for all homogeneous rate constantsK(i, j).