Autocatalytic dissipative structures

The phenomenon of self organization occurs at nonstabHities of the sta tionary state and leads to the formation of temporal and spatio temporal dissipative structures. Remember that oscillating instabilities of stationary states of dynamic systems can be observed for the intermediate nonlinear stepwise reactions only, when no fewer than two intermediates are involved (see Section 3.5) and at least one of the elementary steps is kinet icaUy irreversible. The minimal sufficient requirements for the scheme of a process with temporal instabilities are not yet strictly formulated. However, in aU known examples of such reactions, the rate of the kineti caUy irreversible elementary reaction at one of the intermediate steps is at least in a quadratic dependence on the intermediate concentrations. Among these reactions are autocatalytic steps. 

While Belousov was describing his e)q)eriments into oscillatory chemical reactions, Ilya Prigogine in Brussels was developing theoretical models of nonequilibrium thermodynamics and ended with the notion of "structure dissipative" for which he was awarded the 1977 Nobel Prize in Chemistry. The concept of "Dissipative Structure" is ejq)licitly mentioned in the Nobel quotation "The 1977 Nobel Prize in Chemistry has been awarded to Professor Ilya Prigogine, Brussels, for his contributions to non-equilibrium thermodynamics, particularly the theory of dissipative structures". In the first half of the 1950s, Glansdorff and Balescu defined with Prigogine the thermodynamic criteria necessary for oscillatory behavior in dissipative systems [7]. Nicohs and Lefever then applied these to models of autocatalytic reactions [8]. 

Physical processes Order-disorder structures, ordered-phase transitions, symmetry breaking, spontaneous magnetization, non-equilibrium crystallization phenomena, percolation, electrodeposition, formation of dissipative structures, turbulence and instabilities in fluid dynamics, and diffusion-limited aggregation process. Biological processes Excitation in muscles, pulsation of heart, calcium waves, natural fold-up of protein molecules, deposition of lipid bilayers, auto-regulation of homeostasis morphogenesis, hyper-cycles and autocatalytic networks, etc. 

The appearance of two stable steady states X, X3 allows the system to exist in two phases with different densities X and X3 of the species X. It may even happen that these two phases coexist in the same system separated by a phase boundary. The whole situation is very similar to the phenomenon of phase transitions in equilibrium systems such as gas-liquid or liquid-solid systems. According to this similarity, the phenomenon of different phases in a nonequilibrium system is called a nonequilibrium phase transition or a "dissipative structure". Clearly, the inclusion of coexistence between X and X3 and of phase boundaries into our theory requires the introduction of additional diffusion terms into the equation of motion (6.5) in order to account for spatial variations of X. The analogies between our autocatalytic system (for v = 2) and equilibrium phase transitions have been worked out by F. SCHLOGL (1972) on a phenomenological and by JANSSEN (1974) on a stochastic level. 

Figure 19.4 A simple autocatalytic reaction scheme in which Xl and Xd are produced with equal preference. However, in an open system, this leads to a dissipative structure in which Xl Xd, a state of broken symmetry. A bifurcation diagram shows some general features of transitions to dissipative structures...

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