Time-series expansion for reaction processes.

There has been considerable recent interest in understanding the kinetics of simple reaction models. Thus far, there are relatively few theoretical methods available to study these reactions and these approaches are usually limited to specific situations. Typically, the transport mechanism of the reactants is specified and the particles interact upon contact. Hence, the use of the representation number formalism is natural and has the advantage that a linear rate equation emerges [1,2]. From a theoretical viewpoint, exact methods [3] as well as perturbative methodsK [4] can be applied to this rate equation. From a numerical viewpoint, the exact solution to any reaction scheme can be evaluated for the first few time steps. If this can be carried to a sufficiently high order, then one might hope to extract useful information about the long-time kinetic behavior of the reaction. There have been two noteworthy studies which have followed this general approach [5,6]. Dickman and Jensen applied the series expansion formalism to nonequilibrium interacting particle systems, while Song and Poland investigated several generic reaction-diffusion processes. While this latter approach appears to be promising, the series calculation was carried out by hand. Furthermore, the analysis methods used sometimes lacked internal consistency, leading to an uncertain interpretation of the results. In this study, we describe a systematic approach to obtain time power series for general types of reaction processes. This exact enumeration method holds in any spatial dimension; however, it is hard to implement numerically for dimensions higher than one. Therefore our discussion will be restricted to the one-dimensional case only. In section II, the theoretical description of reactions in terms of creation and annihilation operators is presented. We then discuss how to convert the theoretical formalism into a computational algorithm simply by considering the time evolution of small clusters with periodic boundary conditions. In section III, we develop a simple and relatively direct method to determine the asymptotic behavior of basic physical quantities from the results of the time series expansion. We first illustrate the failure of the straightforward ratio method for analyzing these types of series. This leads us to suggest a direct application of Padé approximants as a method for determining the asymptotic behavior. We exploit the exactly soluble case of single species annihilation to demonstrate the relative utility of the various series analysis methods. In section IV, the time series expansion is applied to ballistic annihilation and to three-species diffusive annihilation. In both cases, analysis of the series provides useful estimates for the asymptotic kinetic behavior.