The analysis outlined in the preceeding sections leads to some basic mechanisms responsible for nonequilibrium transitions. Stability and bifurcation analyses define a number of bifurcation points, primary, secondary or multiple, in the vicinity of which a system switches between qualitatively different types of behavior. Because of its singular character (real part of stability exponent, Re ω, goes to zero), bifurcation becomes extremely sensitive to otherwise small effects, and in particular to statistical fluctuations. The latter are, in turn, deeply affected by bifurcation :long range correlations are generated; the variances are no longer extensive; and the stationary probability distribution (cf. eq.(5.24)) takes an unexpected form, completely different from multi-gaussian or mean-field theoretic forms.
Several problems remain open in this area. In bifurcation theory, a fascinating possibility is to achieve a more and more complete knowledge and classification of the various transitions, including the transition to chaotic behavior. In fluctuation theory, the link between stochastic master equations for nonequilibrium systems and kinetic theory is an obvious problem to be done. Furthermore, many of the properties of these stochastic equations, and in particular the behavior of their solutions past bifurcation remain largely unexplored.