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== The phenomenology of reaction-diffusion systems==

Reaction-diffusion systems in a closed vessel and in the absence of external forces evolve eventually to the state of chemical equilibrium, whereby the constituents involved are distributed uniformly in space and each elementary reactive step is counteracted by its inverse. It has long been realized that the approach to equilibrium can be both in the form of a simple exponential decay, or more involved transient behaviors associated with damped oscillations or non-trivial space dependencies including wave-like patterns. While the former found immediately a series of important applications such as the experimental measurement of the rate constants of chemical reactions, the latter were originally regarded as curiosities. The development of irreversible thermodynamics in the 1950s and onwards provided an explanation of the origin of these two kinds of behaviors by linking them to time evolutions starting close to and far from equilibrium, respectively. Experiment and modeling on laboratory scale reactive systems such as the Belousov-Zhabotinski reaction confirmed this view. Still as long as reaction-diffusion processes were carried out in a closed thermostated reactor there was no way to analyze systematically what was going on since, by virtue of the second law of thermodynamics, the system was bound to reach sooner or later the state of equilibrium. 

A major development that opened new horizons in the experimental study of reaction-diffusion systems and stimulated, in parallel, important theoretical developments has been the systematic use of open reactors, whereby the system is maintained in a nonequilibrium state as long as desired, through the pumping of fresh reactants (the rate of which determines the distance from equilibrium) and the outflow of used products. When spatial homogeneity is maintained within the reactor through stirring in CSTRs (Continuously Stirred Tank Reactors), a rich phenomenology was revealed as key parameters were varied, in addition to the steady state extrapolating the familiar equilibrium like behavior: simple periodic, multi-periodic and chaotic oscillations; multistability, i.e., the coexistence of more than one simultaneously stable states; and excitability, whereby once perturbed a system performs an extended excursion before settling back to its original stable state. A most exciting set of behaviors pertains to space patterning, made possible when stirring is not imposed within the reactor as is the case in many real world situations in chemistry, engineering and biology. Propagating wave fronts, or stabilized ones (like a flame separating fresh and burnt reactants in combustion for instance), are familiar examples. A still different form of spatial organization is the formation of regular steady state patterns arising from the spontaneous symmetry breaking of a spatially uniform state (see Fig. 1).

[[Image:Fig_Turing.jpg|thumb|400px|right|F1|Two-dimensional reaction-diffusion Turing patterns featuring hexagons and bands obtained by numerically integrating the Brusselator model]]