Investigations of reaction-diffusion systems that faithfully incorporate internal stochastic reaction noise and dynamically generated spatial correlations have been a centerpiece of nonequilibrium statistical physics in the past two decades. Considerable progress in their mathematical description has been achieved through exact solutions in one dimension, e.g., via mapping to non-Hermitian spin chains and/or the empty-interval approach; and in higher dimensions by means of field theory representations of the fundamental stochastic master equations and subsequent analysis of the asymptotic long-time large-scale properties by means of the dynamical renormalization group. Naturally, these analytical studies have been guided, supported, and complemented through large-scale individual-based Monte Carlo computer simulations of stochastic cellular automaton models.
In this talk, I will focus less on these mathematical tools and technical details, but will provide an overview of the basic physical mechanisms that we are now understood to induce drastic deviations from the behavior predicted by simple mean-field rate equation approximations. I shall discuss simple paradigmatic model systems: (1) the formation of depletion zones in diffusin-limited single-species pair annihilation A+A -> 0; (2) the emergence of species segregation and sharp reaction fronts in two-species pair annihilation A+B ->0, as well as some interesting generalizations; and (3) the fluctuating steady state in the species coexistence regime for Lotka-Volterra predator-prey interaction, A -> 0,
A+B -> A+A, B -> B+B, which in spatially extended systems is
by spreading and competing activity fronts that induce persistent stochastic population oscillations.