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Prog. Theor. Phys. Vol. 63 No. 6 (1980) pp. 1904-1916

Reduced Equations of Motion for Generalized Fluxes and Forces in the Continued-Fraction Expansion

Katsuhiko Nagano, Takashi Karasudani, Hisao Okamoto and Hazime Mori

Department of Physics, Kyushu University, Fukuoka 812

(Received October 27, 1979)

Abstract:

With the aid of Mori's projection operator method, it is shown that the mechanical equations of motion for flux variables can be transformed rigorously into the reduced form da_j/dt=iω_j·a_j(t)-\int_0^t[ψ_j(t-s)+φ_j(t-s)]·a_j(s)+g_j(t)+f_j(t), where a_j(t) is a column matrix of generalized fluxes standing for the j-th order time derivative of the column matrix of state variables a₀(t), ψ₀(t)≡0, g₀(t)≡0 and ω_j is a frequency matrix. f_j(t) is Mori's j-th order random force and φ_j(t) is the corresponding memory matrix. A new fluctuating force g_j(t) and the corresponding memory matrix ψ_j(t) represent the effects of lower-order fluxes. The temporal behaviour of g_j(t) is quite different from that of f_j(t), and the Laplace transform of ψ_j(t) is represented by an inverted continued fraction of finite order whose poles are all purely imaginary. The Laplace transform of time-correlation matrices of a_j(t) and a_k(0) are given in terms of these memory-spectrum matrices ψ_j[z] and φ_j[z].

URL : http://ptp.ipap.jp/link?PTP/63/1904/
DOI : 10.1143/PTP.63.1904

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References:

1. H. Mori, Prog. Theor. Phys. 33 (1965), 423[IPAP].
2. H. Mori, Prog. Theor. Phys. 34 (1965), 395[IPAP].
3. T. Karasudani, K. Nagano, H. Okamoto and H. Mori, Prog. Theor. Phys. 61 (1979), 850[IPAP].
4. M. Tokuyama and H. Mori, Prog. Theor. Phys. 55 (1976), 411[IPAP].
5. H. Mori, Lecture Note for Young Physicists' Summer School on Material Science (1977), (in Japanese).
6. L. Sjögren, Ann. of Phys. 113 (1978), 304[CrossRef].

Citing Article(s) :

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   Magnetic Properties of Optical Quantum Transition Line Shapes and Line Widths of Electron–Piezoelectric Potential Phonon Interacting Materials under Circularly Oscillating Fields

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2. Progress of Theoretical Physics Vol. 82 No. 2 (1989) pp. 299-303 :

   A Simple Algebraic Technique in the Generalized Continued-Fraction Representation

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3. Progress of Theoretical Physics Vol. 102 No. 4 (1999) pp. 789-801 :

   The Direct Optical Transition Line Shape Function from the Equilibrium Density Projection Operator Technique

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