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Journal of the Physical Society of Japan

On the Generalized Langevin Equation Approach to a Quantum Many-Body System

Hiroshi Ueyama

Published Mar 15, 1990• 1 citation
Publisher

Abstract

An application of the method of quantum Langevin equation to a many-body system far from equilibrium is considered. A coupled set of Langevin equations is derived exactly. By approximate reductions of variables, the generalized Langevin equation of the center of mass of electrons in solid is derived, that has previously been derived on heuristic grounds.

DOI
10.1143/jpsj.59.1025

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UNLABELLED: The Langevin equation is universally applied to describe noise in physical systems of all kinds. To derive it, rather than postulate it, one has to introduce a noise source such as an environment or "reservoir". The solution requires that at some initial time the reservoir is in equilibrium and uncorrelated with the system. This is unphysical and, moreover, in the quantum mechanical case leads to inconsistencies. The way to overcome this anomaly is to consider the equilibrium of system and noise source combined. Then the resulting time-dependent equations for the correlation of various operators can be given. To find out whether they can be reproduced by a Langevin equation, the result is applied to the explicitly solvable system of a harmonic oscillator in a harmonic bath. CONCLUSION: Langevin cannot describe quantum noise but belongs to the realm of mesoscopic physics.

The Journal of Physical Chemistry B•Sep 28, 2005•7 citations
An application of the method of quantum Langevin equation to a many-body system far from equilibrium is considered. A coupled set of Langevin equations is derived exactly. By approximate reductions of variables, the generalized Langevin equation of the center of mass of electrons in solid is derived, that has previously been derived on heuristic grounds.
We treat the quantum transport of an interacting system of electrons, impurities, and phonons, in a time-dependent electric field, by using the quantum generalized Langevin equation (GLE), in which the system is shown to be equivalent to a quantum particle in a heat bath. We follow here the philosophy of Ford, Lewis, and O'Connell, who have demonstrated the usefulness of the GLE approach to heat-bath problems. The center of mass of the electrons acts like a quantum particle, while the relative electrons and phonons play the role of heat bath. They are coupled through the electron-impurity and electron-phonon interactions. After eliminating the heat-bath variables, the equation of motion for the quantum particle is written in a form of a quantum generalized Langevin equation, with a memory term which reflects the retarded effects of the heat bath on the quantum particle. The evaluation of the memory term immediately leads to a result for the susceptibility from which we can calculate the conductivity directly, in contrast to Kubo-type calculations which require the evaluation of correlation functions as an intermediate step. As a demonstration of the directness of our approach, we show that the usual random-phase-approximation conductivity results are easily derived. In addition, we derive an expression for the memory function, which incorporates higher-order electron-impurity scattering.
We treat the quantum transport of an interacting system of electrons, impurities and phonons. The dynamic system is in a time-dependent electric field. We analyzed quantum transport process by using the quantum generalized Langevin equation, in which the system is shown to be equivalent to a quantum particle in a heat bath. After eliminating the heat-bath variables, the equation of motion for the quantum particle is written in a form of quantum generalized Langevin equation, with a memory term which reflects the retarded effects of the heat bath on the quantum particle.
From microscopic models, a Langevin equation can, in general, be derived only as an approximation. Two possible conditions to validate this approximation are studied. One is, for a linear Langevin equation, that the frequency of the Fourier transform should be close to the natural frequency of the system. The other is by the assumption of "slow" variables. We test this method by comparison with an exactly soluble model and point out its limitations. We base our discussion on two approaches. The first is a direct, elementary treatment of Senitzky. The second is via a generalized Langevin equation as an intermediate step.
UNLABELLED: The Langevin equation is universally applied to describe noise in physical systems of all kinds. To derive it, rather than postulate it, one has to introduce a noise source such as an environment or "reservoir". The solution requires that at some initial time the reservoir is in equilibrium and uncorrelated with the system. This is unphysical and, moreover, in the quantum mechanical case leads to inconsistencies. The way to overcome this anomaly is to consider the equilibrium of system and noise source combined. Then the resulting time-dependent equations for the correlation of various operators can be given. To find out whether they can be reproduced by a Langevin equation, the result is applied to the explicitly solvable system of a harmonic oscillator in a harmonic bath. CONCLUSION: Langevin cannot describe quantum noise but belongs to the realm of mesoscopic physics.