EFFECT OF FLUCTUATIONS ON BIFURCATION PHENOMENA

Bifurcation theoretic problems involving applications in physics, chemistry, biology deal typically with equations ofevolution with a limited number of macroscopic observables. Such equations (deterministic rate laws) are usually taken for granted on the basis of vague references to the laws of large numbers in probability theory, according to which statistical averages provide a satisfactory description of the time course of macrovariables. Therefore, everything happens as though, in any small volume element A V , the rate of change of a macrovariable were the result of a collective effect determined by the values of all the variables, which, in turn, sense their effect. We call this notion the mean-field view of bifurcation phenomena. This lumping together of all but the macroscopic degrees of freedom ignores fluctuations from average behavior. Our purpose in this paper is to assess the influence of these fluctuations in a variety of problems involving the onset of cooperative behavior associated with bifurcation. Consider a typical bifurcation phenomenon shown in FIGURE I , (see e.g., the paper by M. Herschkowitz-Kaufman and T. Erneux in this volume). At X = A,, a stable state, branch (a) loses its stability and is succeeded for X > A, by two simultaneously stable branches ( b ) and ( c ) . At the critical point X = A,, the system has to choose between coalescing solutions. But nothing in the equations of evolution justifies preference for any particular choice. It is not unreasonable to expect that fluctuations will play an important role in this critical choice. Suppose next that we have a range of X > A, values in which two bifurcating states ( b ) and ( c ) coexist in macroscopic amounts. A particular system will exist for the most part in the neighborhood of either ( b ) or (c) . Yet the statistical average will give a value on a branch (a”), which is close to the unstable branch (a’). This value is completely irrelevant t o the observed behavior. Again, one would be justified in expecting that fluctuations around this average would play a prominent role. We shall illustrate the progressive breakdown of the deterministic description associated with bifurcation and the concomitant existence of simultaneously stable states. We shall present some results obtained from a master equation for analyzing model systems in which spatial coordinates can be ignored. We shall deal, successively, with the exact form of a steady-state solution in the large volume limit, the coexistence region of simultaneously stable states, and the transitions between these states. Finally we shall present some general comments on the nature of the solutions of master equations around bifurcation points leading to limit