Stochastic Landau equation with time-dependent drift

The stochastic differential equation ${\mathrm{\tau }}_0$ ${\mathrm{\partial }}_{\mathit{t}}$A=ε(t)A-${\mathit{g}}_3$ ${\mathit{A}}^3$+f¯(t), where f¯(t) is Gaussian white noise, is studied for arbitrary time dependence of ε(t). In particular, cases are considered where ε(t) goes through the bifurcation of the deterministic system, which occurs at ε=0. In the limit of weak noise an approximate analytic expression generalizing earlier work of Suzuki [Phys. Lett. A 67, 339 (1978); Prog. Theor. Phys. (Kyoto) Suppl. 64, 402 (1978)] is obtained for the time-dependent distribution function P(A,t). The results compare favorably with a numerical simulation of the stochastic equation for the case of a linear ramp (both increasing and decreasing) and for a periodic time dependence of ε(t). The procedure can be generalized to an arbitrary deterministic part ${\mathrm{\partial }}_{\mathit{t}}$A=D(A,t)+f¯(t), but the deterministic equation may then have to be solved numerically.