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Progress of Theoretical Physics

Statistical-Mechanical Theory of Random Frequency Modulations and Generalized Brownian Motions

Michio Tokuyama, Hazime Mori

Published Feb 1, 1976• 245 citations
Publisher

Abstract

It is shown that the Heisenberg equation of motion can be transformed exactly into the reduced form dA(t)/dt = [ iΩ-∫t0ψ(s)ds] ·A(t)+g(t), where A(t) is an arbitrary column vector of dynamical variables, Ω is a frequency matrix determining collective oscillations, and g(t) is a fluctuating force. ψ(t) differs from the memory function and is related to the time-correlation function of g(t) by ψ(t) = (g(t), g*(0))·(A(t), A*(0))-1. This equation of motion is shown to give a statistical-mechanical theory of random frequency modulations, including Kubo's theory of the stochastic Liouville equation, and to lead to a new theory of generalized Brownian motions which complements a memory-function theory given by Mori.

DOI
10.1143/ptp.55.411

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It is shown that the Heisenberg equation of motion can be transformed exactly into the reduced form dA(t)/dt = [ iΩ-∫t0ψ(s)ds] ·A(t)+g(t), where A(t) is an arbitrary column vector of dynamical variables, Ω is a frequency matrix determining collective oscillations, and g(t) is a fluctuating force. ψ(t) differs from the memory function and is related to the time-correlation function of g(t) by ψ(t) = (g(t), g*(0))·(A(t), A*(0))-1. This equation of motion is shown to give a statistical-mechanical theory of random frequency modulations, including Kubo's theory of the stochastic Liouville equation, and to lead to a new theory of generalized Brownian motions which complements a memory-function theory given by Mori.
In the frames of classical mechanics the generalized Langevin equation is derived for an arbitrary mechanical subsystem coupled to the harmonic bath of a solid. A time-acting temperature operator is introduced for the quantum Klein-Kramers and Smoluchowski equations, accounting for the effect of the quantum thermal bath oscillators. The model of Brownian emitters is theoretically studied and the relevant evolutionary equations for the probability density are derived. The Schrodinger equation is explained via collisions of the target point particles with the quantum force carriers, transmitting the fundamental interactions between the point particles. Thus, electrons and other point particles are no waves and the wavy chapter of quantum mechanics originated for the force carriers. A stochastic Lorentz-Langevin equation is proposed to describe the underlaying Brownian-like motion of the point particles in quantum mechanics. Considering the Brownian dynamics in the frames of the Bohmian mechanics, the density functional Bohm-Langevin equation is proposed, and the relevant Smoluchowski-Bohm equation is derived. A nonlinear master equation is proposed by proper quantization of the classical Klein-Kramers equation. Its equilibrium solution in the exact canonical Gibbs density operator, while the well-known Caldeira-Leggett equation is simply a linearization at high temperature. In the case of a free quantum Brownian particles, a new law for the spreading of the wave packet it discovered, which represents the quantum generalization of the classical Einstein law of Brownian motion. A new projector operator is proposed for the collapse of the wave function of a quantum particle moving in a classical environment. Its application results in dissipative Schrodinger equations, as well as in a new form of dissipative Liouville equation in classical mechanics.
A stochastic approach to the quantum dynamics randomly modulated in time by a discrete state non-Markovian noise, which possesses an arbitrary nonexponential distribution of the residence times, is developed. The formally exact expression for the Laplace-transformed quantum propagator averaged over the stationary realizations of such N -state non-Markovian noise is obtained. The theory possesses a wide range of applications. It includes some previous Markovian and non-Markovian theories as particular cases. In the context of the stochastic theory of spectral line shape and relaxation, the developed approach presents a non-Markovian generalization of the Kubo-Anderson theory of sudden modulation. In particular, the exact analytical expression is derived for the spectral line shape of optical transitions described by a Kubo oscillator with randomly modulated frequency which undergoes jumplike non-Markovian fluctuations in time.
For every system with a passably general Hamiltonian of a quantum harmonic oscillator coupled to infinitely many scalar bosons, we estimate Heisenberg pictures of the annihilation and creation operators, describing the quantum harmonic oscillator, from only data on the autocorrelation function of the operators such that the pictures restore their autocorrelation functions to original ones, by reconsidering Mori's theory of generalized Brownian motion. And we then investigate the Liouville space and Liouville operator associated with the estimation.
The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto ccc-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.
The theory of stochastic motion is formulated from a new point of view. It is shown that the fundamental equations of the theory reduce to Schrödinger's equation for specific values of certain parameters. A generalized Fokker-Planck-Kolmogorov equation is obtained; with other values of the parameters, certain approximations reduce this to the Smoluchowski equation for Brownian movement. In particular, the potential function in the Schrödinger equation differs in the two cases. The usual uncertainty relations appear in a natural way in the theory, but in a broader context. A single theory thus covers both similarities and differences between quantum-mechanical and Brownian motion. Furthermore, possibilities for broadening nonrelativistic quantum mechanics are brought out and, as an example, the possible corrections due to non-Markoffian terms are briefly studied.
The basic equations for the statistical system of waves in a randomly fluctuating medium are presented in the same form as in quantum mechanics, assuming a Markov process for the temporal change of the medium. A simple model is chosen for the Fokker-Planck equation of the medium which gives rise to a fluctuation of the Gaussian process, and several results are given to illustrate how the methods used in quantum mechanics or quantum field theory can be applied almost without change. Thus, the physical variables are represented by linear operators, and their equations of motion are determined by an equation similar to the Heisenberg equation of motion. The system has two stationary states (corresponding to the vacuum states 〈0∣〈0|〈0∣ and ∣0〉|0〉∣0〉 in quantum field theory), and the rather standard methods in field theory can be used for the evaluation of the Green's functions. When the temporal changes of the medium are sufficiently large compared with those of the waves, an adiabatic approximation is possible, and the isomorphic transformation (corresponding to the unitary transformation) is employed to lead to the result that the statistical system of the waves for this model is in perfect correspondence with the waves of bosons which interact with each other only through a two-body potential.