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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. II. The Langevin Equations in the Canonical Form

Natsuki Hashitsume, Kaori Naito, Atsuko Washiwo

Published Feb 15, 1990• 3 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.59.464

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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.
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.
We establish a direct connection between the Feynman-Vernon path integral formalism for open quantum systems and the Wiener path integral used in classical stochastic dynamics. By considering a generalized influence functional in the strong decoherence limit, we demonstrate that integrating over the quantum coherence length leads to a derivation of stochastic Langevin dynamics. Specifically, we show that the quantum Feynman measure transforms into the stochastic Wiener measure. Applying this framework to the Wigner function representation, we show that the system follows a stochastic path interpretable via classical probability theory. Finally, we address the inverse problem: constructing an equivalent quantum influence functional from a given classical Langevin equation.
The master equation for a linear open quantum system in a general environment is derived using a stochastic approach. This is an alternative derivation to that of Hu, Paz and Zhang, which was based on the direct computation of path integrals, or to that of Halliwell and Yu, based on the evolution of the Wigner function for a linear closed quantum system. We first show by using the influence functinal formalism that the reduced Wigner function for the open system coincides with a distribution function resulting from averaging both over the initial conditions and the stochastic source of a formal Langevin equation. The master equation for the reduced Wigner function can then be deduced as a Fokker-Planck equation obtained from the formal Langevin equation.
We extend the master-equation treatment of dynamical evolution of a system-plus-reservoir configuration including the propagation of initial correlations as a noise source. Specializing into the quantum harmonic oscillator coupled to a fermionic heat bath, we develop a model for the diffusion matrix in the space of diagonal density operators. It can be shown that mean values of observables undergo Langevin-like motion and, in particular, that the mean value and dispersion of the oscillator quanta approach the canonical equilibrium values. A final interpretation of the characteristics and role of the noise source is given.
Abstract The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.