  In systems removed from equilibrium, intrinsic microscopic fluctuations
become correlated over distances comparable to the characteristic macroscopic
length over which the external constraint is exerted. In order to investigate
this phenomenon, we construct a microscopic model with simple stochastic
dynamics using lattice gas automaton rules that satisfy local detailed balance.
Because of the simplicity of the automaton dynamics, analytical theory can be
developed to describe the space and time evolution of the density fluctuations.
The exact equations for the pair correlations are solved explicitly in the
hydrodynamic limit. In this limit, we rigorously derive the results obtained
phenomenologically by fluctuating hydrodynamics. In particular, the spatial
algebraic decay of the equal-time fluctuation correlations predicted by this
theory is found to be in excellent agreement with the results of our lattice
gas automaton simulations for two different types of boundary conditions.
Long-range correlations of the type described here appear generically in
dynamical systems that exhibit large scale anisotropy and lack detailed
balance.
