---

Prog. Theor. Phys. Vol. 64 No. 2 (1980) pp. 500-521

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

Terumitsu Morita, Hazime Mori and Kazuko T. Mashiyama

Department of Physics, Kyushu University 33, Fukuoka 812

(Received February 7, 1980)

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.

URL : http://ptp.ipap.jp/link?PTP/64/500/
DOI : 10.1143/PTP.64.500

[ Full Text PDF : FREE ACCESS (1361K) ] Citation: Plain Text BiBTeX EndNote(RIS) RSS

---

References:

1. H. Mori, T. Morita and K. T. Mashiyama, Prog. Theor. Phys. 63 (1980), 1865[PTP].
   This will be referred to as I.
2. R. Kubo, in Fluctuations, Relaxations and Resonance in Magnetic Systems, ed. D. ter Haar (Oliver & Boyd, Edinburgh, 1962).
3. R. L. Stratonovich, Topics in the Theory of Random Noise, vol. 1 (Gordon and Breach, New York, 1963), pp. 100-103.
4. N. G. van Kampen, Phys. Rep. C 24 (1976), 171[CrossRef].
5. K. Kitahara, W. Horsthemke and R. Lefever, Phys. Lett. A 70 (1979), 377, [CrossRef]and references cited therein.
6. H. Hasegawa and T. Nakagomi, J. Stat. Phys. 21 (1979), 191, and references cited therein.
7. A. Schenzle and H. Brand, Phys. Rev. A 20 (1979), 1628[APS].
8. H. Fujisaka and H. Mori, Prog. Theor. Phys. 56 (1976), 754[PTP].
9. H. Haken, Rev. Mod. Phys. 47 (1975), 67[APS].
10. R. F. Fox, Phys. Rep. 48 (1978), 179[CrossRef].
11. R. F. Fox, J. Math. Phys. 13 (1972), 1196[CrossRef].

Citing Article(s) :

1. Journal of the Physical Society of Japan 57 (1988) pp. 1163-1166 :

   Chaotic Behaviour of Soliton-Like Pulses in a Driven Modified Kuramoto-Sivashinsky Equation

   Hidetoshi Konno and Hideo Soneda

2. Progress of Theoretical Physics Vol. 65 No. 6 (1981) pp. 1820-1834 :

   Anomalous Fluctuations near Nonequilibrium Soft Transitions. I

   Kazuko T. Mashiyama, Kiyohumi Takayoshi and Hazime Mori

3. Progress of Theoretical Physics Vol. 66 No. 1 (1981) pp. 129-142 :

   Adiabatic Elimination by the Eigenfunction Expansion Method

   Kunihiko Kaneko

4. Progress of Theoretical Physics Vol. 66 No. 1 (1981) pp. 154-171 :

   Anomalous Fluctuations near Nonequilibrium Soft Transitions. II

   Kazuko T. Mashiyama and Hazime Mori

5. Progress of Theoretical Physics Vol. 67 No. 1 (1982) pp. 98-117 :

   On the Contraction of Fast Driving Variables from Stochastic Processes

   Hiroshi Hasegawa, Masahiko Mizuno and Mahito Mabuchi

6. Progress of Theoretical Physics Supplement No.69 (1980) pp. 160-173 :

   Fluctuation and Relaxation in Stochastic Systems

   Masuo Suzuki

---

[PTP HOME] [BROWSE] [SEARCH] [SUBSCRIPTION] [SUBMISSION] [REGISTRATION]

Copyright © Progress of Theoretical Physics 2013 All rights reserved.