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Nonlinear, multiplicative Langevin equations for a complete set of slowvariables in equilibrium systems are generally derived on the basis of theseparation of time scales. The form of the equations is universal andequivalent to that obtained by Green. An equation with a nonlinear frictionterm for Brownian motion turns out to be an example of the general results. Akey method in our derivation is to use different discretization schemes in apath integral formulation and the corresponding Langevin equation, which alsoleads to a consistent understanding of apparently different expressions for thepath integral in previous studies.

Universal Form of Stochastic Evolution for Slow Variables in Equilibrium Systems