Bifurcations, propagating solutions, and phase transitions in a nonlinear chemical reaction with diffusion

The Schlögl model of a nonlinear chemical reaction with diffusion is presented as an example of a reaction-diffusion system displaying a nonequilibrium phase transition. It is described by a scalar diffusion equation with a cubic nonlinearity and is used here to show such features of nonlinear systems as bifurcations and spatial dissipative structures, as well as to illustrate some of the simpler mathematical techniques used in their analysis. The model contains analogs of the critical isotherm and of Maxwell’s construction. It shows hysteresis and a limiting behavior interpretable in terms of the thermodynamic limit. More importantly, it provides a description of the dynamics of a phase transition, showing a fluctuation-induced nucleation process and the evolution of a phase transition by the motion of phase boundaries.