Mean-field theory for the long-range contact process with diffusion

The effect of diffusion in the one-dimensional long-range contact process is investigated by mean-field calculations. Recent works have shown that diffusion decreases the effectiveness of long-range interactions, affecting the character of the phase transition: for higher values of the diffusion coefficient, stronger long-range interactions are required to enable phase coexistence and first-order behavior. Here we apply a generalized mean-field approximation for the master equation of the model that considers states of an aggregate of L lattice sites. The phase diagram of the model for values of L up to 10 is obtained, and for some values of the diffusion rate extrapolations to infinite-sized systems are given. For low-diffusive systems, approximations with L≥3 are able to reveal the suppression of the phase coexistence induced by diffusion, however, in the high-diffusion regime, larger values of L are necessary to correctly account for the higher range of correlations. We present a very efficient method to study the mean-field equations and determine the nature of the phase transitions that may be of general utility.