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Quantum Brownian Motion: The Functional Integral Approach

H. Grabert, P. Schramm, and G.-L. Ingold

Phys. Rep. 168, 115–207 (1988) DOI: 10.1016/0370-1573(88)90023-3


The quantum mechanical dynamics of a particle coupled to a heat bath is treated by functional integral methods and a generalization of the Feynman-Vernon influence functional is derived. The extended theory describes the time evolution of nonfactorizing initial states and of equilibrium correlation functions. The theory is illuminated through exactly solvable models.

Table of contents

  • Introduction
Part I. General theory
  • Microscopic model and preparation of the initial state
    • The model Hamiltonian
    • Initial states and preparation function
  • Functional integral representation of the density matrix and elimination of the environment
    • Euclidean functional integral
    • Real time functional integral
    • Integration over the environmental coordinates and influence functional
    • Reduced dynamics and propagating function
  • Minimal action paths and damping kernel
    • The potential renormalization
    • Minimal action paths
    • Formulation of the theory in terms of the damping kernel
Part II. Damped harmonic oscillator
  • Time evolution of a damped harmonic oscillator
    • Extremal imaginary time path and reduced equilibrium density matrix
    • Extremal real time paths and minimal effective action
  • Equilibrium correlation functions and response function
    • Linear response of the coordinate to an applied force
    • Coordinate autocorrelation function
    • The fluctuation dissipation theorem
    • Linear response of the momentum to an applied force
    • Further correlation functions
    • Variances
    • Propagating function
    • Effect of initial correlations
  • Ohmic dissipation
  • Relaxation of nonequilibrium initial states
    • Approach to equilibrium
    • Relaxation of expectation values
    • Relaxation of factorizing initial states
    • Coherent and squeezed states
Part III. Free Brownian motion
  • Time evolution of a damped free particle
    • The displacement correlation function
    • The propagating function
  • Ohmic dissipation
  • Frequency-dependent damping
    • Spectral density and damping coefficient
    • The antisymmetrized displacement correlation function
    • The symmetrized displacement correlation function
  • Relaxation of nonequilibrium inital states
    • Time evolution of a Gaussian density matrix
    • Asymptotic spreading of the state
    • Long time behaviour for arbitrary initial states at finite temperatures
    • Long time behaviour for arbitrary initial states at zero temperature
Appendices
  • Elimination of an environmental oscillator
  • Determination of the auxiliary function Ψ(t,t')
  • Microscopic origin of the inhomogeneity in the equation of motion for <q>t


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