Cluster Approximation for the Contact Process

The contact process (CP) is an irreversible lattice model involving nearest neighbor interactions only [1,2]. This model incorporates spontaneous desorption and nearest-neighbor induced adsorption. This stochastic process can be used to mimic epidemic spread as well as catalytic reactions. This model belongs to a general class of nonequilibrium models exhibiting a continuous phase transition. Near the critical point, the system exhibits divergence of spatial and temporal correlations. Such properties, conveniently characterized by critical exponents, can be used to classify different models. The CP belongs to the same universality class as Sclögel’s first model [3], Reggeon field theory [4], directed percolation [5], and the ZGB model [6] of catalysis. Field theoretic renormalization group studies [7,8] provide considerable understanding of the critical behavior of the CP. However, the best estimates for the characteristic exponents were found numerically by Monte Carlo simulations [9] and by series expansion analysis [10,11] . Motivated by the incomplete theoretical understanding, we introduce an approximate approach to the CP. We study the temporal evolution of the density of empty intervals. The corresponding rate equations lead to an infinite hierarchy of equations. By writing the density of pairs of neighboring empty intervals as a product over single interval densities, we obtain a closed set of equations. We use the generating function technique to obtain the steady-state properties of the system. Within this approximation, the system exhibits a discontinuous phase transition from an active state to the empty state. As the system approaches the critical point, the relaxation time, associated with the temporal approach to the final state, diverges. Consequently, at the critical point, an anomalously slow decay towards the final state takes place. We find the corresponding kinetic exponent by scaling techniques, as well as by numeric integration of the rate equations. We compare the cluster approximation predictions with the results of site mean-field theory and with series analysis of this process. Despite the failure to predict a continuous transition, the cluster approximation provides a good approximation for the final density and the empty interval density for a reasonable range of desorption rates. Moreover, the resulting estimates for the critical exponents are closer to the numeric values in comparison with site mean-field theory. The cluster approach is also applicable to generalizations of the contact process, such as the A model and the N3 model. We verify that the resulting critical behavior of these processes is identical with the contact process. Our approach is advantageous since it can be improved systematically by considering the evolution of higher order empty interval densities.