On Symmetry-Breaking Instabilities in Dissipative Systems

The theory of hydrodynamic instability has always been an important part of fluid dynamics see, e.g., Chandrasekhar, in Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford, England, 1961) and Non-Equilibrium Thermodynamics, Variation Techniques, and Stability, R. J. Donnelly, R. Herman, and I. Prigogine, Eds. (University of Chicago Press, Chicago, Ill., 1966). Such instabilities involve both convective processes (such as mechanical flow) and dissipative processes (such as viscous dissipation). We investigate the possibility of an instability in purely dissipative systems involving chemical reactions and transport processes such as diffusion, but no hydrodynamic motion. We demonstrate that for well-defined values of the constraints such as the chemical affinities of the over-all reactions and the constants involved, such systems can indeed become unstable. Such an instability is investigated following an example of autocatalytic reactions first proposed by Turing. The major feature of this instability is its symmetry breaking character. Indeed, beyond the transition point the stable steady state is inhomogenous, the diffusion compensating the differences in reaction rates. The existence of such instabilities has far-reaching consequences which are briefly discussed.