Physica 81A (1975) 485-508 © North-Holland Publishing Co. MICROSCOPIC THEORY OF BROWNIAN MOTION II. NONLINEAR LANGEVIN EQUATIONS JAMES T. HYNES* Department of Chemistry, University of Colorado, Boulder, Colorado 80302, USA RAYMOND KAPRAL Department of Chemistry, University of Toronto, Toronto, Ontario M5S 1A1, Canada and MICHAEL WEINBERG Department of Physics, Toledo University, Toledo, Ohio 43606, USA Received 9 June 1975 In this article nonlinear Langevin equations for a brownian (B) particle are derived and analyzed. Attention is focussed on the role of nonlinear B particle momentum (P) modes (powers of P). The multimode Mori formalism is used to derive equations of motion for P(t) for different numbers n of modes included in the description. The well-known linear equation of Mori cor- responds to the case n = 1. Friction kernels and random forces in these equations exhibit slow decay and mass ratio (2) expansion anomalies due to mode coupling. The nonlinear Langevin equation obtained for a complete mode set (n = co) is free of these difficulties and is used to examine the first correction [0(24)] to standard 0(22 ) results. Although no closed set of nonlinear Langewin equations exists at order ;t*, a truncated set extends standard momentum correlation function predictions. I. Introduction There has been considerable interest recently in the dynamics of nonlinear fluctuations due to their evident importance in relaxation 1,2), critical phenomena3), and the behavior of systems far from equilibrium4). In the present article, we derive and analyze nonlinear Langevin equations for a classic example of relaxa- * Alfred P. Sloan Fellow. 485