A scaling theory for the quasi-deterministic limit

Abstract

Deterministic rate equations are widely used in the study of stochastic, interacting particles systems. This approach assumes that the inherent noise, associated with the discreteness of the elementary constituents, may be neglected when the number of particles $N$ is large. Accordingly, it fails close to the extinction transition, when the amplitude of stochastic fluctuations is comparable with the size of the population. Here we present a general scaling theory of the transition regime for spatially extended systems. Two fundamental models for out-of-equilibrium phase transitions are considered: the Susceptible-Infected-Susceptible (SIS) that belongs to the directed percolation equivalence class, and the Susceptible-Infected-Recovered (SIR) model belonging to the dynamic percolation class. Implementing the Ginzburg criteria we show that the width of the fluctuation-dominated region scales like $N^{-\kappa}$, where $N$ is the number of individuals per site and $\kappa = 2/(d_u-d)$, $d_u$ is the upper critical dimension. Other exponents that control the approach to the deterministic limit are shown to depend on $\kappa$. The theory is supported by the results of extensive numerical simulations for systems of various dimensionalities.