Statistical-dynamical theory of nonlinear stochastic processes: II. Time-convolutionless projector method in nonequilibrium open systems

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A new statistical-dynamical theory of nonlinear stochastic processes in nonequilibrium open systems is presented. It is shown by means of the time-convolutionless projector method that multiplicative type stochastic equations of motion for relevant variables A i ( t ) can always be transformed exactly into additive type (Langevin type) stochastic equations of motion for A i ( t ) and the corresponding master equation for the probability distribution. The Langevin type equation consists of a drift term and a fluctuating force. The fluctuating force is shown to give a new kind of additive stochastic process which has a quite different feature from the ordinary additive processes. The importance of Langevin equations of this type in the multiplicative stochastic process is pointed out from the statistical-dynamical viewpoint. A new cumulant expansion of the master equation in powers of stochastic forces is also found. The application of the dynamic renormalization group method to steady states far from thermal equilibrium is discussed from the statistical-dynamical standpoint developed in the present paper. The present theory is applied to renormalize nonlinear stochastic equations by eliminating the short-wavelength modes and thus to derive nonlinear Langevin equations with renormalized kinetic coefficients and the corresponding master equation. The renormalized coefficients are then used not only to find recursion relations but also to calculate explicitly a cutoff dependence of the kinetic coefficients. As an actual example, fluctuations in the generalized time-dependent Ginzburg-Landau model are investigated near its nonequilibrium tricritical point.