Emergence and Evolution of Patterns

We consider macroscopic systems driven away from thermodynamic equilibrium by an imposed gradient in temperature, velocity, or concentration. For a sufficiently small imposed gradient, a system will assume (asymptotically) the symmetry of the boundary conditions. However, when the imposed gradient exceeds a critical value, a system will spontaneously break the symmetry of the boundary conditions and form a spatial pattern. In two dimensions the pattern could be traveling waves (e.g., spirals) or an array of squares, stripes, or hexagons. With larger imposed gradients, the patterns can become more complex and even disordered in both space and time. We discuss the general principles of pattern formation in systems driven away from equilibrium and illustrate the principles with examples from experiments.