David A. Kessler, Nadav M. Shnerb
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.
View original:
http://arxiv.org/abs/1201.5306
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