Nonlinear behavior at a chemical instability: A detailed renormalization-group analysis of a case model

Abstract

A theory that is suitable for the analysis of the nonlinear behavior at or very near to a far-from-equilibrium instability point is presented. The theory is based on the idea that far-from-equilibrium instabilities, like equilibrium critical systems, have a critical space dimensionality, $d_c$, above which a quasilinear treatment yields the correct values of the critical exponents. The results for physical systems (i.e., space dimensionality $d\le 3$) are obtained by an $\epsilon$ expansion, in a manner analogous to the one used in the dynamical renormalization group. We treat in detail a case model of a chemical instability. The main result is that the critical exponents differ from those predicted by the quasilinear approximation or by master equations (i.e., the so-called "classical" exponents). We show that great care must be taken with the correlation of random forces, since a faulty choice of this correlation can yield results that differ significantly from those obtained by using the correct correlation.