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Physical Review A

Augmented Langevin evaluation of imaginary-time Feynman path integrals

R. Badii, G. Broggi

Published Nov 1, 1990• 0 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.42.6256

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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.
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.
CONTENTS: Preface Bosonic Path Integrals Nelson's Stochastic Mechanics by Means of Feynman Path Integrals Feymman path-integrals for Brownian Classical Motions (1+1)-dimensional Burger turbulence in the framework of Hopf-Cole Transformation and Feymman Trajectories path-integrals Feynman-Wiener Path Integral Representations for the Quantum Harmonic Oscillator with Time-Dependent Stochastic Frequency: Quantum Brownian Motions and Navier-Stokes Weakly Turbulence Some studies on functional integrals representations for fluid motion with random conditions Statistics turbulence suppression by a strong magnetic field: A Feynman path integral argument A non-linear diffusion equation, Some existence and uniqueness results on weak solutions for random initial conditions and its path integral representation -- a mathematical methods approach The Exact Evalution of the Response Function for Wave Propagation in Random Media: A Functional Integral Approach On the Abstract Method of Integration for Wave Equation and some Applications for Wave Physics Feynman Path-Integral Studies for Quantum Particles Propagators Interacting with Environments The Quantum Fractional Dimensional Brownian Motion: A Path-Integral Study Ohm's Law: A Langevin Path-Integral Study at Classical and Quantum W.K.B. Regime On the Statistical of An Ideal Gas with Mass Fluctuations and the Boltzman Ergodic Theorem: The Combined Boltzman-Tsallis Statistics The Suppression of the Quantum Phenomena by an External Strong White-Noise Magnetic Field: A Path Integral Argument Index.
A formulation of quantum statistical mechanics is discussed in which the Feynman path centroid density in Feynman path integration is recast as the central statistical distribution used to average equilibrium and dynamical quantities. In this formulation, the path integral centroid density occupies the same role as the Boltzmann density in classical statistical mechanics. Therefore, the statistical ensemble of imaginary time path centroid configurations provides the distribution which is used to average the appropriately formulated effective operators and imaginary time correlation functions. An accurate renormalized diagrammatic perturbation theory for the centroid density and centroid-constrained imaginary time propagator will also be described with particular emphasis given to the mathematical advantages arising from the centroid-based formulation. The present paper is concerned with the calculation of equilibrium properties from the centroid perspective, while the companion paper describes a centroid-based formalism for calculating dynamical time correlation functions.
We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.