Single Particle Cellular Automata Models for Simulation of the Master Equation

Abstract

We present a hierarchy of cellular automata models for simulation of non-reacting single specie particle dynamical systems governed by the master equation. In these models lattice site values are interpreted to be representing single particles moving over an arbitrary regular spatial lattice with any one of the finite number of velocities belonging to a discrete velocity set at any time step. The particle interaction rules of these models lead to construction of state transition probability (STP) tables similar to those encountered in discrete Markov processes. Consequently, these models mimic the dynamics represented by the master equation when the STP tables allow the process to be homogeneous in time. The lattice site values in these models are globally coupled; making them inherently sequential simulation tools. Within the framework of single particle representation (SPR) interpretation of lattice site values, a generalized model development methodology has been formalized and demonstrated by developing a synchronous two-dimensional single speed SPR-LGA model over square spatial lattice for simulation of single specie non-reacting dynamical systems. In this model particles are permitted to have any one of the four velocities belonging to the discrete velocity set {(±1, 0), (0, ±1)} at any time step. Using this model simulations have been carried out on system of particles enclosed in a container at equilibrium. Simulation results are found to be in excellent agreement with the available theoretical results.