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Quantum Brownian Motion in a Simple Model System

  • W. De Roeck
  • J. Fröhlich
  • A. Pizzo
Article

Abstract

We consider a quantum particle coupled (with strength λ) to a spatial array of independent non-interacting reservoirs in thermal states (heat baths). Under the assumption that the reservoir correlations decay exponentially in time, we prove that the motion of the particle is diffusive at large times for small, but finite λ. Our proof relies on an expansion around the kinetic scaling limit (\({\lambda \searrow 0}\), while time and space scale as λ−2) in which the particle satisfies a Boltzmann equation. We also show an equipartition theorem: the distribution of the kinetic energy of the particle tends to a Maxwell-Boltzmann distribution, up to a correction of O2).

Keywords

Boltzmann Equation Irreducible Component Heat Bath Anderson Model Kinetic Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Araki H., Woods E.J.: Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas. J. Math. Phys. 4, 637 (1963)CrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Bach V., Fröhlich J., Sigal I.: Return to equilibrium. J. Math. Phys. 41, 3985 (2000)zbMATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Brattelli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics: 2. Berlin: Springer-Verlag, 2nd edition, 1996Google Scholar
  4. 4.
    Bryc W.: A remark on the connection between the large deviation principle and the central limit theorem. Stat. Prob. Lett. 18, 44 (1993)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen T.: Localization lengths and Boltzmann limit for the Anderson model at small disorder in dimension 3. J. Stat. Phys. 120(1–2), 279–337 (2005)zbMATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Clark, J., De Roeck, W., Maes, C.: Diffusive behaviour from a quantum master equation. http://arxiv.org/abs/0812.2858v2[math-ph], 2008
  7. 7.
    Dereziński, J.: Introduction to Representations of Canonical Commutation and Anticommutation Relations. Volume 695 of Lecture Notes in Physics. Berlin: Springer-Verlag, 2006Google Scholar
  8. 8.
    Dereziński J., Jakšić V., Pillet C.-A.: Perturbation theory of W *-dynamics, Liouvilleans and KMS-states. Rev. Math. Phys. 15, 447–489 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Erdös L.: Linear Boltzmann equation as the long time dynamics of an electron weakly coupled to a phonon field. J. Stat. Phys. 107(85), 1043–1127 (2002)zbMATHCrossRefGoogle Scholar
  10. 10.
    Erdös L., Salmhofer M., Yau H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit II. the recollision diagrams. Commun. Math. Phys 271, 1–53 (2007)zbMATHCrossRefADSGoogle Scholar
  11. 11.
    Erdös L., Salmhofer M., Yau H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit I. the non-recollision diagrams. Acta Math. 200, 211–277 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Erdös L., Yau H.-T.: Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. Comm. Pure Appl. Math. 53(6), 667–735 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Fröhlich J., Merkli M.: Another return of ‘return to equilibrium’. Commun. Math. Phys. 251, 235–262 (2004)zbMATHCrossRefADSGoogle Scholar
  14. 14.
    Jakšić V., Pillet C.-A.: On a model for quantum friction. III: Ergodic properties of the spin-boson system. Commun. Math. Phys. 178, 627–651 (1996)zbMATHCrossRefADSGoogle Scholar
  15. 15.
    Kang Y., Schenker J.: Diffusion of wave packets in a Markov random potential. J. Stat. Phys. 134, 1005–1022 (2009)zbMATHCrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Ovchinnikov A.A., Erikhman N.S.: Motion of a quantum particle in a stochastic medium. Sov. Phys. -JETP 40, 733–737 (1975)ADSGoogle Scholar
  17. 17.
    Pillet C.-A.: Some results on the quantum dynamics of a particle in a Markovian potential. Commun. Math. Phys. 102, 237–254 (1985)zbMATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Reed, M., Simon, B.: Methods of Modern Mathematical physics, Volume 4. New York: Academic Press, 1972Google Scholar
  19. 19.
    Reed, M., Simon, B.: Methods of Modern Mathematical physics, Volume 2. New York: Academic Press, 1972Google Scholar
  20. 20.
    De Roeck W.: Large deviation generating function for currents in the Pauli-Fierz model. Rev. Math. Phys. 21(4), 549–585 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Silvius A., Parris P., De Bievre S.: Adiabatic-nonadiabatic transition in the diffusive hamiltonian dynamics of a classical Holstein polaron. Phys. Rev. B. 73, 014304 (2006)CrossRefADSGoogle Scholar
  22. 22.
    Spohn H.: Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys. 17, 385–412 (1977)CrossRefMathSciNetADSGoogle Scholar
  23. 23.
    Spohn H.: Kinetic equations from Hamiltonian dynamics; Markovian limits. Rev. Mod. Phys. 53, 569–615 (1980)CrossRefMathSciNetADSGoogle Scholar
  24. 24.
    Tcheremchantsev S.: Markovian Anderson model: Bounds for the rate of propagation. Commun. Math. Phys. 187(2), 441–469 (1997)zbMATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsK.U. LeuvenHeverleeBelgium
  2. 2.Institute for Theoretical PhysicsZürichSwitzerland
  3. 3.Department of MathematicsUniversity of California at DavisDavisUSA

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