Connection between deterministic and stochastic descriptions of nonlinear systems

Using the recent statistical mechanical theory for nonlinear irreversible processes of Grabert, Graham and Green, we re-examine the fluctuations in an electric circuit containing a nonlinear dissipative resistance. We explicitly establish the relationship between the nonlinear thermal fluctuations and the deterministic irreversible transport law. In particular, we verify their choice of the metric in frames where the macrovariables are sums of molecular variables. In a second part, we discuss the connection between the stochastic differential equation description and the deterministics. We show that neither the Ito-drift nor the Stratonovitch-drift coincides with the deterministic flow. By use of the example of driven Brownian motion with nonuniform damping we further demonstrate the usefulness of the nonlinear transport theory for the problem of adiabatic elimination in stochastic nonlinear equilibrium systems.