Cellular automata model for the diffusion equation

We consider a new cellular automata rule for a synchronous random walk on a two-dimensional square lattice, subject to an exclusion principle. It is found that the macroscopic behavior of our model obeys the telegraphists's equation, with an adjustable diffusion constant. By construction, the dynamics of our model is exactly described by a linear discrete Boltzmann equation which is solved analytically for some boundary conditions. Consequently, the connection between the microscopic and the macroscopic descriptions is obtained exactly and the continuous limit studied rigorously. The typical system size for which a true diffusive behavior is observed may be deduced as a function of the parameters entering into the rule. It is shown that a suitable choice of these parameters allows us to consider quite small systems. In particular, our cellular automata model can simulate the Laplace equation to a precision of the order (?/L)⁶, whereL is the size of the system and? the lattice spacing. Implementation of this algorithm on special-purpose machines leads to the fastest way to simulate diffusion on a lattice.