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

Abstract

A projector elimination and an adiabatic elimination of irrelevant degrees-of-freedom are developed for the contraction of state variables in stochastic equations of motion. For multiplicative stochastic equations, a master equation for the probability density of relevant variables A(t) ≡{ A_i(t) } is derived by means of the projector method and is shown to reduce to a Fokker-Planck equation if the stochastic forces S_i(a, t) are Gaussian processes with time correlations of the form ≪S_i(a, t)S_j(a′, t′)>=2[ξ_ij(a, a′)δ₊(t-t′)+ ξ_ji(a′, a)δ₊(t′-t)], where δ₊(t) is the right half of the δ function δ(t), nonvanishing only at t=0+. If ξ_ij(a, a′)= ξ_ji(a′, a), then this reduces to the conventional form 2ξ_ij(a, a′)δ(t-t′). With the aid of stochastic processes of this new type, an adiabatic elimination from the Langevin equations is proposed for a stochastic Haken-Zwanzig model for non-equilibrium phase transitions. A projector elimination from the Langevin equations and an adiabatic elimination from the Fokker-Planck equation are also explored. Calculation is carried out up to second order in the slowness parameter. Three different methods are thus developed with consistent results and are applied to a laser model for illustration.