Contraction of State Variables in Non-Equilibrium Open Systems. I

Abstract

A projector elimination of irrelevant macrovariables is formulated for dissipative systems far from thermal equilibrium whose time evolution is described by autonomous deterministic equations with appropriate initial ensembles. Reduced equations of motion for relevant macrovariables and the corresponding master equation are obtained in terms of the fluctuating forces generated by the elimination. For a complete set of slowly-varying variables, the second-order terms in their expansion in powers of a slowness parameter lead to a Langevin and a Fokker-Planck equation of a generalized type. The formulation is then applied to the Haken-Zwanzing model for non-equilibrium phase transitions, and a prototype procedure for obtaining a stochastic Ginzburg-Landau type equation for the order parameter is clarified. Thus it turns out that the mode couplings of the order parameter to irrelevant macrovariables generate fluctuations of the order parameter in the ordered phase, but the variance due to these fluctuations does not diverge even near the instability point. In the critical region, the fluctuations have the critical dimensionality dc=2.