skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Exact analytical solutions to the master equation of quantum Brownian motion for a general environment

Abstract

Research Highlights: > We study the model of a quantum oscillator linearly coupled to a bath of oscillators. > We derive the master equation and solutions for general spectra and temperatures. > We generalize to cases with an external force and arbitrary number of oscillators. > Other derivations have incorrect diffusion and force response for nonlocal damping. > We give exact results for ohmic, sub-ohmic and supra-ohmic environments. - Abstract: We revisit the model of a quantum Brownian oscillator linearly coupled to an environment of quantum oscillators at finite temperature. By introducing a compact and particularly well-suited formulation, we give a rather quick and direct derivation of the master equation and its solutions for general spectral functions and arbitrary temperatures. The flexibility of our approach allows for an immediate generalization to cases with an external force and with an arbitrary number of Brownian oscillators. More importantly, we point out an important mathematical subtlety concerning boundary-value problems for integro-differential equations which led to incorrect master equation coefficients and impacts on the description of nonlocal dissipation effects in all earlier derivations. Furthermore, we provide explicit, exact analytical results for the master equation coefficients and its solutions in a wide variety of cases,more » including ohmic, sub-ohmic and supra-ohmic environments with a finite cut-off.« less

Authors:
 [1]
  1. Max-Planck-Institut fuer Gravitationsphysik (Albert-Einstein-Institut), Am Muehlenberg 1, 14476 Golm (Germany)
Publication Date:
OSTI Identifier:
21579896
Resource Type:
Journal Article
Journal Name:
Annals of Physics (New York)
Additional Journal Information:
Journal Volume: 326; Journal Issue: 5; Other Information: DOI: 10.1016/j.aop.2010.12.003; PII: S0003-4916(10)00217-4; Copyright (c) 2010 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Journal ID: ISSN 0003-4916
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANALYTICAL SOLUTION; BOUNDARY-VALUE PROBLEMS; BROWNIAN MOVEMENT; DAMPING; DIFFUSION; FLEXIBILITY; INTEGRO-DIFFERENTIAL EQUATIONS; MARKOV PROCESS; OSCILLATORS; SPECTRA; SPECTRAL FUNCTIONS; ELECTRONIC EQUIPMENT; EQUATIONS; EQUIPMENT; FUNCTIONS; MATHEMATICAL SOLUTIONS; MECHANICAL PROPERTIES; STOCHASTIC PROCESSES; TENSILE PROPERTIES

Citation Formats

Fleming, C.H., E-mail: hfleming@physics.umd.edu, Roura, Albert, and Hu, B.L., E-mail: hub@physics.umd.edu. Exact analytical solutions to the master equation of quantum Brownian motion for a general environment. United States: N. p., 2011. Web. doi:10.1016/j.aop.2010.12.003.
Fleming, C.H., E-mail: hfleming@physics.umd.edu, Roura, Albert, & Hu, B.L., E-mail: hub@physics.umd.edu. Exact analytical solutions to the master equation of quantum Brownian motion for a general environment. United States. https://doi.org/10.1016/j.aop.2010.12.003
Fleming, C.H., E-mail: hfleming@physics.umd.edu, Roura, Albert, and Hu, B.L., E-mail: hub@physics.umd.edu. Sun . "Exact analytical solutions to the master equation of quantum Brownian motion for a general environment". United States. https://doi.org/10.1016/j.aop.2010.12.003.
@article{osti_21579896,
title = {Exact analytical solutions to the master equation of quantum Brownian motion for a general environment},
author = {Fleming, C.H., E-mail: hfleming@physics.umd.edu and Roura, Albert and Hu, B.L., E-mail: hub@physics.umd.edu},
abstractNote = {Research Highlights: > We study the model of a quantum oscillator linearly coupled to a bath of oscillators. > We derive the master equation and solutions for general spectra and temperatures. > We generalize to cases with an external force and arbitrary number of oscillators. > Other derivations have incorrect diffusion and force response for nonlocal damping. > We give exact results for ohmic, sub-ohmic and supra-ohmic environments. - Abstract: We revisit the model of a quantum Brownian oscillator linearly coupled to an environment of quantum oscillators at finite temperature. By introducing a compact and particularly well-suited formulation, we give a rather quick and direct derivation of the master equation and its solutions for general spectral functions and arbitrary temperatures. The flexibility of our approach allows for an immediate generalization to cases with an external force and with an arbitrary number of Brownian oscillators. More importantly, we point out an important mathematical subtlety concerning boundary-value problems for integro-differential equations which led to incorrect master equation coefficients and impacts on the description of nonlocal dissipation effects in all earlier derivations. Furthermore, we provide explicit, exact analytical results for the master equation coefficients and its solutions in a wide variety of cases, including ohmic, sub-ohmic and supra-ohmic environments with a finite cut-off.},
doi = {10.1016/j.aop.2010.12.003},
url = {https://www.osti.gov/biblio/21579896}, journal = {Annals of Physics (New York)},
issn = {0003-4916},
number = 5,
volume = 326,
place = {United States},
year = {2011},
month = {5}
}