  A stochastic version of the Brusselator model is proposed and studied via the
system size expansion. The mean-field equations are derived and shown to yield
to organized Turing patterns within a specific parameters region. When
determining the Turing condition for instability, we pay particular attention
to the role of cross diffusive terms, often neglected in the heuristic
derivation of reaction diffusion schemes. Stochastic fluctuations are shown to
give rise to spatially ordered solutions, sharing the same quantitative
characteristic of the mean-field based Turing scenario, in term of excited
wavelengths. Interestingly, the region of parameter yielding to the stochastic
self-organization is wider than that determined via the conventional Turing
approach, suggesting that the condition for spatial order to appear can be less
stringent than customarily believed.
