Simulating nondiffusive dynamics in reaction-diffusion systems

Abstract

Reaction-diffusion systems are a class of mathematical models with broad applications across physics, chemistry and biology. While very versatile, a limitation of these systems is that they assume diffusion as the only means of motion. To allow for broader application, especially in the realm of condensed matter physics, there is a need to take into account other transport processes. This thesis aims to implement an existing Monte Carlo algorithm for efficient simulation of single-species reaction-diffusion systems on a lattice, and subsequently extend it to also handle nondiffusive dynamics. The considered nondiffusive model is a simple toy model of correlated motion that preserves center of mass, and the nondiffusive nature of this kinetic model is both justified theoretically and demonstrated by simulation. It is then used in conjunction with single-species annihilation with the goal of finding how the particle density scales over time with such dynamics. The implementation of the base algorithm shows behavior consistent with theory and previous computational results. In order to show the nondiffusivity of the correlated motion model, the time evolution of the mean squared displacement is examined and it is found to be subdiffusive and slightly slower than expected. Whether this slower evolution is due to limitations in the theory or the computational model is unknown. When simulating two-particle single species annihilation, the particle density is observed to quickly reach the same steady state of 2/11 inverse length units independent of parameter settings, and no apparent power laws are found in the approach to this steady state. What these results suggest is that there exist certain invariant configurations of the lattice in which no more reactions can occur. This leaves open the opportunity for more reasearch into the equilibrium states of various system geometries and types of reaction.