Uniform states dissipative structures

Fig. 11. Bifurcation diagram for an even wave number and

While pattern formation, epitomized by beautiful snowflakes, was first liberated from theology by Kepler in 1611, it is only relatively recently some progress has been made in its scientific investigation [4]. Pattern formation refers to spatial structures, usually characterized by a band of wavelengths, that replace a uniform state above a critical distance from equilibrium. Patterns are nonequilibrium structures resulting from several dissipative processes characterized by diffusion constants. They are quantified by their wavevector, q, and their frequency, co. 

Domain I represents the domain wherein the thermodynamic solution is stable. In domain II, with the parameters noted in Fig. 1, the thermodynamic branch has become unstable owing to fluctuations in the chemical composition of the system. Beyond the thermodynamic threshold (transition point), fluctuations increase uniformly in this domain (II), eventually resulting in a new steady state which corresponds to regular spatial distributions of X and Y (Fig. 2). This state represents a low entropy, dissipative structure localized in space and whose "natural" boundaries are determined by the system itself. The spatial localization of the resultant dissipative structure demonstrates the symmetry-breaking nature of the instability. It appears that the form which the dissipative structure takes depends on the type of initial perturbation thus, the system possesses a primitive "memory effect" since the initial perturbation determines the form of the dissipative structure established. 

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