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Journal of the Physical Society of Japan

A Simple Derivation of Langevin Equations for Nonequilibrium States

Masakazu Ichiyanagi

Published May 1, 1983• 3 citations
Publisher

Abstract

In the spirit of the linear response theory, a simple derivation is given of a Langevin equation for collective modes of a macroscopic system. Unlike in the usual treatment, we separate a weak interaction of individual particles with collective modes which is allowed to depend on time. In nonequilibrium cases, we have shown that a damping constant of the collective modes can be expressed in terms of a time correlation function, in which its average is taken with respect to a nonequilibrium density matrix rather than a usual canonical density matrix. A microscopic expression for a Langevin random force is obtained in terms of fluctuations of a density matrix.

DOI
10.1143/jpsj.52.1541

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In the spirit of the linear response theory, a simple derivation is given of a Langevin equation for collective modes of a macroscopic system. Unlike in the usual treatment, we separate a weak interaction of individual particles with collective modes which is allowed to depend on time. In nonequilibrium cases, we have shown that a damping constant of the collective modes can be expressed in terms of a time correlation function, in which its average is taken with respect to a nonequilibrium density matrix rather than a usual canonical density matrix. A microscopic expression for a Langevin random force is obtained in terms of fluctuations of a density matrix.
Equations for correlation functions in nonequilibrium systems are derived using a generalization of the diagrammatic Keldysh method. The four field operators of the correlation function are ordered on a time contour that consists of four axes extending from -\ensuremath{\infty} to +\ensuremath{\infty} and from +\ensuremath{\infty} to -\ensuremath{\infty} (doubled Keldysh contour). For those correlation functions that may be considered as correlation functions of one-particle Green-functions' fluctuations, a closed system of equations is obtained in the case of weak interaction of the particles with the scattering system. Its structure may be related to the Langevin approach: on the left-hand side a linear operator acts on each of two pairs of the correlation-function arguments, each of these two operators being just the operator of the linearized quantum kinetic equation. The expression for the right-hand side of this equation is derived. It has the meaning of the correlation function of extraneous Langevin sources, and the equation may be called the quantum Boltzmann-Langevin equation. In the quasi-classical limit it reduces to the usual Boltzmann-Langevin equation for the particles' distribution-function fluctuations. Possible applications of the method are discussed.
A quantum mechanical form of the Langevin equation is derived from an explicit consideration of the molecule-medium interaction, as advocated by Simons in 1978, and by using two identities in the interaction picture. This can be easily reduced to the classical regime, and further simplified to the macroscopic Langevin equation by considering the stochastic Langevin force autocorrelation function. One of the so-called Einstein relations appears as a byproduct. By following the methodology proposed by Simons, an exact expression for the momentum autocorrelation function is obtained. The latter can be used to calculate the zero-frequency macroscopic diffusion coefficient that is observed to satisfy the second Einstein relation. The formalism described above gives rise to the possibility of explicitly computing the transport characteristics such as friction constant and diffusion coefficient from the corresponding quantum statistical mechanical expressions. A discussion on the Langevin equation becomes complete only when the corresponding Fokker-Planck equation is obtained. Therefore, the probability of the evolution of states with a particular absolute magnitude of linear momentum from those of another momentum eigenvalue is quantum mechanically defined. This probability appears as a special average value of a projection operator and as a special projection operator correlation function. A classical identity is introduced that is shown to be valid also for the quantum mechanically defined probability function. By using this identity, the so-called Fokker-Planck equation for the evolution probability is easily established.
In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser. We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.
In this article we propose a new theory for collective modes of a many-body system which contains linear response theory and supplies us with systematic methods for the analysis of Brownian motion. The essential idea is to develop a method to find the possible justification for Zubarev's nonequilibrium ensemble from the new standpoint which is to draw attention to the usage of the dynamical invariance. The present theory has many interesting features which are relevant to irreversible processes. In this article we are more interested in problems of principle than in problems of applicability.
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.
A time-dependent variational principle is given for nonequilibrium density matrix of a quantum system. It allows a derivation of dynamical equations for thermodynamical forces which are the Lagrangian parameters of the present theory. If the system is near equilibrium, these equations determine the transport coefficients which satisfy the reciprocity relations. For the understanding of irreversibility, the introduction of the van Hove parameter turns out to be essential, which bases upon a distinct separation of microscopic and macroscopic scales in time.