The formation of complex patterns in many non-equilibrium systems, ranging from solidifying alloys to multiphase flow, nonlinear chemical reactions and the growth of bacterial colonies, involves the propagation of an interface that is unstable to diffusive motion. Most existing theoretical treatments of diffusive instabilities are based on mean-field approaches, such as the use of reaction-diffusion equations, that neglect the role of fluctuations. Here we show that finite fluctuations in particle number can be essential for such an instability to occur. We study, both analytically and with computer simulations, the planar interface separating different species in the simple two-component reaction A + B → 2A (which can also serve as a simple model of bacterial growth in the presence of a nutrient). The interface displays markedly different dynamics within the reaction-diffusion treatment from that when fluctuations are taken into account. Our findings suggest that fluctuations can provide a new and general pattern-forming mechanism in non-equilibrium growth.
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