Journal of Statistical Physics, Vol. 55, Nos. 3/4, 1989 Stochastically Perturbed Landau-Ginzburg Equations Roberto Benzi, l Giovanni Jona-Lasinio, 2 and Alfonso Sutera 3 Received August 2, 1988; revision received December 2, 1988 We analyze several aspects of a reaction-diffusion equation in two space dimen- sions with cubic nonlinearity, stochastically perturbed by white noise in time and in space. This equation needs renormalization, and physical implications of this circumstance arc discussed. In particular, for sufficiently large coupling con- stant the effective potential becomes a double well and rare transitions from one minimum to the other are possible. These, however, are revealed only by large- scale fluctuations which exhibit a bimodal distribution. Fluctuations below a critical scale have unimodal distribution and do not "see" the double well. This phenomenon is connected with the singular character of local fluctuations in two or more space dimensions. The theoretical results are confirmed by numeri- cal simulations. The possible physical relevance of our results is illustrated in connection with the analysis o,f certain observations of atmospheric fields. KEY WORDS: Stochastic P.D.E.; renormalization; large (small) scale fluctuations; atmospheric bimodality. 1. JINTRODUCTION Landau-Ginzburg (LG) equations ~see (2.1)] are widely used to model a variety of different physical situations. Although they were originally motivated by the study of the order parameter near the critical point in the theory of superconductivity, m their use has been extended to describe B6nard cells near the transition to the convection mode, (2~ the effect of orbital forcing on the earth's climates, (3) the nonlinear optical bistability in laser:s, t4) crystallization phenomena, ~5) etc. Generally speaking, LG equa- tions are phenomenological equations representing an approximate macro- scopic description of the physical phenomenon under study. Their justifica- tion at a deeper level would require a microscopic analysis, which is usually a very difficult task. Therefore, what one very often does is to simulate or i Dipartimento di Fisica, Universitfi di Roma "Tor Vergata," 1-00173 Rome, Italy. 2 Dipartimento di Fisica, Universit/t di Roma "La Sapienza," 1-00185 Rome, Italy. 3 Department of Geology and Geophysics, Yale University, New Haven, Connecticut 06511. 505 0022-4715/89/0500-0505506.00/0 9 1989 Plenum Publishing Corporation