Optimizing the accuracy of Lattice Monte Carlo algorithms for simulating diffusion

TL;DR

This paper investigates how to optimize Lattice Monte Carlo algorithms for simulating diffusion, demonstrating that certain finite parameter choices yield maximal accuracy and proposing improved move strategies in multiple dimensions.

Contribution

It identifies optimal finite parameter settings for unbiased diffusion algorithms and introduces diagonal moves and boundary handling techniques to enhance accuracy.

Findings

Optimal accuracy achieved at finite parameter values.

Diagonal moves improve multi-dimensional diffusion simulation.

Boundary crossing moves should be projected along boundaries.

Abstract

The behavior of a Lattice Monte Carlo algorithm (if it is designed correctly) must approach that of the continuum system that it is designed to simulate as the time step and the mesh step tend to zero. However, we show for an algorithm for unbiased particle diffusion that if one of these two parameters remains fixed, the accuracy of the algorithm is optimal for a certain finite value of the other parameter. In one dimension, the optimal algorithm with moves to the two nearest neighbor sites reproduces the correct second and fourth moments (and minimizes the error for the higher moments at large times) of the particle distribution and preserves the first two moments of the first-passage time distributions. In two and three dimensions, the same level of accuracy requires simultaneous moves along two axes ("diagonal" moves). Such moves attempting to cross an impenetrable boundary should be projected along the boundary, rather than simply rejected. We also treat the case of absorbing boundaries.