Abstract
We present a fully nonlinear stochastic theory of chemical reactions closely below, at and above instability points. As explicit example we treat the Prigogine-Lefever-Nicolis model of two interacting kinds of molecules, generalizing it to two and three dimensions, to a mode continuum, and taking into account fluctuations. Adopting a description by means of birth and death processes we establish the master-equation, and proceed to the Fokker-Planck equation. This is transformed to new coordinates connected with unstable and stable modes. After adiabatic elimination of the stable modes, we obtain a functional Fokker-Planck equation for a continuous set of “unstable” modes. This final equation can now be treated by standard methods. In one dimension our results reveal striking analogies to the Ginzburg-Landau theory of superconductivity, to the continuous mode laser and to small-band excitations at hydrodynamical instabilities, while in three dimensions, in thin layers a hexagonal structure similar to Bénard cells occurs.
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Haken, H. Statistical physics of bifurcation, spatial structures, and fluctuations of chemical reactions. Z Physik B 20, 413–420 (1975). https://doi.org/10.1007/BF01313213
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DOI: https://doi.org/10.1007/BF01313213