Finite-Size Scaling Studies of Reaction-Diffusion Systems Part III: Numerical Methods

TL;DR

This paper compares Monte Carlo simulations and finite lattice extrapolations to study the universal scaling behavior of a 1D coagulation reaction model, demonstrating the effectiveness of both methods in capturing key properties.

Contribution

It introduces and validates two numerical methods for analyzing reaction-diffusion systems, emphasizing their accuracy and applicability for finite-size scaling studies.

Findings

Scaling exponent and function are universal and independent of initial conditions.

Monte Carlo simulations effectively compute correction terms.

Finite lattice data up to ten sites suffice for reliable extrapolations.

Abstract

The scaling exponent and scaling function for the 1D single species coagulation model $(A+A\rightarrow A)$ are shown to be universal, i.e. they are not influenced by the value of the coagulation rate. They are independent of the initial conditions as well. Two different numerical methods are used to compute the scaling properties: Monte Carlo simulations and extrapolations of exact finite lattice data. These methods are tested in a case where analytical results are available. It is shown that Monte Carlo simulations can be used to compute even the correction terms. To obtain reliable results from finite-size extrapolations exact numerical data for lattices up to ten sites are sufficient.