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Journal of the Physical Society of Japan

A Derivation of Generalized Semi-Classical Langevin Equations Based on the Feynman Path-Integral Approach

Natsuki Hashitsume, Michiyo Mori, Tomoko Takahashi

Published Jun 15, 1986• 11 citations
Publisher

Abstract

A new derivation of the Langevin equation for a system in contact with a heat reservoir in equilibrium is given based on the concept of path probability inherent in the Feynman approach to quantum mechanics. The derived Langevin equation is a generalization of the Magalinski type, i.e., it is a second order differential equation and contains a shift term, a retarded damping term, a destruction term and a random force term in addition to the usual external potential force term. It is a semi-classical equation in the sense that it is a classical equation but contains a quantal response function of the Kubo formula type and the second moment of the random force is given by a quantal symmetrized correlation function. The fluctuation-dissipation theorem of the second kind is exactly valid between the response and the correlation functions.

DOI
10.1143/jpsj.55.1887

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A new derivation of the Langevin equation for a system in contact with a heat reservoir in equilibrium is given based on the concept of path probability inherent in the Feynman approach to quantum mechanics. The derived Langevin equation is a generalization of the Magalinski type, i.e., it is a second order differential equation and contains a shift term, a retarded damping term, a destruction term and a random force term in addition to the usual external potential force term. It is a semi-classical equation in the sense that it is a classical equation but contains a quantal response function of the Kubo formula type and the second moment of the random force is given by a quantal symmetrized correlation function. The fluctuation-dissipation theorem of the second kind is exactly valid between the response and the correlation functions.
In a previous paper a new derivation of the Langevin equations for a system coupled with a heat reservoir was given based on the Feynman path-integral representation of the transition probability, i.e. the diagonal part of a reduced density matrix of the system. The derived Langevin equations are of the form of the Lagrangian equations of motion. In this article the temporal change of the reduced density matrix as a whole, or more precisely of the Wigner phase-space distribution, is treated, and the Langevin equations equivalent to those mentioned above but in the form of the canonical equation of motion are derived in the second order perturbational approximation with respect to interactions between the system and the reservoir.
The macroscopic description of a quantum particle with passive dissipation and moving in an arbitrary external potential is formulated in terms of the generalized Langevin equation. The coupling with the heat bath corresponds to two terms: a mean force characterized by a memory function \ensuremath{\mu}(t) and an operator-valued random force. Explicit expressions are given for the correlation and commutator of the random force. The random force is never Markovian. It is shown that \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(z), the Fourier transform of the memory function, must be a positive real function, analytic in the upper half-plane and with Re[\ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\omega}+i0+{0}^{+}0+)] a positive distribution on the real axis. This form is then derived for the independent-oscillator model of a heat bath. It is shown that the most general quantum Langevin equation can be realized by this simple model. A critical comparison is made with a number of other models that have appeared in the literature.
An alternative formulation of the stochastic representation of the Feynman path-integral theory is introduced. Stationary (imaginary-time) paths are generated as solutions of suitable Zwanzig-Ramshaw augmented Langevin equations. The required equilibrium distribution is obtained for a freely chosen functional form of the fluctuations and its associated dissipative term, in the presence of a generalized reversible drift. The method is both theoretically and numerically advantageous with respect to the standard Langevin and hybrid algorithms.
In this article is proposed a method of constructing the quantal Boltzmann-Langevin equation, i.e. the stochastic kinetic equation for a suitably chosen statistical operator with a fluctuating force term. This method is based on the use of the generalized phase-space method, which makes calculations very simple. As an example, the method of spin coherent states is applied to a model of a localized single spin placed under actions of its magnetic environment, and the Bloch-Langevin equation, i.e. the well-known Bloch equation for a spin magnetic moment with a fluctuating force term, is derived. The drift and the relaxation terms contained in the Boltzmann-Langevin and the Bloch-Langevin equations are completely in agreement with those previously obtained. An expression of the fluctuation-dissipation theorem generalized so as to include non-linear dissipation terms is given.
A unified framework of stochastic differential equations for quantum systems, formulated within Non-Equilibrium Thermo Field Dynamics (NETFD), is applied to a model of non-linear damped oscillator. The quantum stochastic Liouville equation and the quantum Langevin equations (both of Ito and Stratonovich type), which are consistent with the corresponding master equation, are written down explicitly in the case of a non-conventional treatment. This solves Kubo's third problem: how one can obtain the correlations of random force operators for the Langevin equation compatible with the master equation derived by the non-conventional treatment of the damping theory.
In Chapter I of this dissertation, we present a pedagogical introduction to the basic concepts of stochastic theory and review the progress made as well as the outstanding unsolved problems in the field. For the rest of this dissertation, we use a generalized quantum Langevin equation (GLE) approach to investigate various properties of quantum dissipative systems. In Chapter II, calculations for the displacement and random force correlation functions for Brownian motion are generalized to the case of an arbitrary heat bath for a damped harmonic quantum oscillator. The mean square displacement of such an oscillator is then evaluated for both Ohmic and blackbody radiation heat baths, to determine the effects of many parameters on the localization of the oscillator. In Chapter III, the formalism is extended to the Brownian motion of a charged particle in an external magnetic field as well as in a potential. The influence of the magnetic field on the memory function and random force is determined, with the blackbody radiation heat bath analyzed as a special case. For a charged harmonic oscillator, the generalized susceptibility is obtained, which enables us to derive the symmetrized position correlation functions using the fluctuation-dissipation theorem. In addition, we obtain the free energy of the system by generalizing the "remarkable formula" of Ford, Lewis, and O'Connell. Explicit calculations are performed for Ohmic and blackbody radiation heat baths. Furthermore, the effect of dissipation on the localization of the oscillator in an Ohmic heat bath at zero temperature is shown to differ qualitatively from that without the magnetic field. Finally, we formulate retarded Green's functions and symmetrized position correlation functions for the oscillator, reach some general conclusions, and make explicit calculations for the Ohmic heat bath. For the special case of Brownian motion at both zero and nonzero temperatures, we prove two general asymptotic relations between the retarded Green's functions and the displacement correlation functions, which we use to evaluate the long-time behaviors of the latter from those of the former, for both the Ohmic heat bath and a rather general class of heat baths discussed extensively in the literature.