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

概要

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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) Sj (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.
理論物理学刊行会の論文
1980-08-25