The behavior of a lattice Monte Carlo (LMC) 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. We discuss the relation between optimally accurate LMC algorithms and a particular case of lattice Boltzmann (LB) algorithms for simulating diffusion and compare the computational efficiency of optimal LMC and optimal LB algorithms.
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