Abstract
We consider a heavy quantum particle with an internal degree of freedom moving on the d-dimensional lattice \({{\mathbb Z}^d}\) (e.g., a heavy atom with finitely many internal states). The particle is coupled to a thermal medium (bath) consisting of free relativistic bosons (photons or Goldstone modes) through an interaction of strength λ linear in creation and annihilation operators. The mass of the quantum particle is assumed to be of order λ−2, and we assume that the internal degree of freedom is coupled “effectively” to the thermal medium. We prove that the motion of the quantum particle is diffusive in d ≥ 4 and for λ small enough.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alicki R., Fannes M.: Quantum Dynamical Systems. Oxford University Press, Oxford (2001)
Araki H., Woods E.J.: Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas. J. Math. Phys. 4, 637 (1963)
Bach V., Fröhlich J., Sigal I.: Return to equilibrium. J. Math. Phys. 41, 3985 (2000)
Brattelli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics: 2. Berlin: Springer-Verlag, 2nd edition, 1996
Bricmont, J., Kupiainen, A.: Diffusion in Energy Conserving Coupled Maps, http://arxiv.org/abs/1102.3831
Bryc W.: A remark on the connection between the large deviation principle and the central limit theorem. Stat. and Prob. Lett. 18, 253–256 (1993)
Bunimovich L.A., Sinai Ya.G.: Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun Math. Phys. 78, 479–497 (1981)
Clark, J., De Roeck,W., Maes, C.: Diffusive behaviour from a quantum master equation. http://arXiv.org/abs0812.2858v3 [math-ph], 2008
Davies E.B.: Linear Operators and their spectra. Cambridge University Press, Cambridge (2007)
Davies E.B.: Markovian master equations. Commun. Math. Phys. 39, 91–110 (1974)
Dereziński J.: Introduction to Representations of Canonical Commutation and Anticommutation. Relations Volume 695 of Lecture Notes in Physics. Springer-Verlag, Berlin (2006)
Dereziński J., Früboes R.: Fermi golden rule and open quantum systems. In: Attal, S., Joye, A., Pillet, C.-A. (eds) Lecture notes Grenoble Summer School on Open Quantum Systems. Lecture Notes in Mathematics. Vol. 118, pp. 67–116. Springer, Berlin (2003)
Dereziński J., Jakšić V.: Return to equilibrium for Pauli-Fierz systems. Ann. H. Poincaré 4, 739–793 (2003)
De Roeck W.: Large deviation generating function for currents in the Pauli-Fierz model. Rev. Math. Phys. 21(4), 549–585 (2009)
De Roeck W., Fröhlich J., Pizzo A.: Quantum Brownian motion in a simple model system. Commun. Math. Phys. 293(2), 361–398 (2010)
De Roeck, W., Spehner, D.: Derivation of the quantum master equation for massive tracer particles. In preparation
Disertori M., Spencer T., Zirnbauer M.: Quasi-diffusion in a 3d supersymmetric hyperbolic sigma model. Commun. Math. Phys. 300, 435–486 (2010)
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)
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)
Erdös L., Salmhofer M., Yau H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit i. the non-recollision diagrams. Acta Mathematica 200, 211–277 (2008)
Erdös L., Yau H.-T.: Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. Commun. Pure Appl. Math. 53(6), 667–735 (2000)
Evans, L.C.: Partial Differential Equations (Graduate Studies in Mathematics, V. 19) GSM/19. Providence, RI: Amer. Math. Soc., 1998
Frigerio A.: Stationary states of quantum dynamical semigroups. Commun. Math. Phys. 63(3), 269–276 (1978)
Fröhlich J., Merkli M.: Another return of ‘return to equilibrium’. Commun. Math. Phys. 251, 235–262 (2004)
Holevo A.S.: A note on covariant dynamical semigroups. Rep. Math. Phys. 32, 211–216 (1992)
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)
Kang Y., Schenker J.: Diffusion of wave packets in a Markov random potential. J. Stat. Phys. 134, 1005–1022 (2008)
Kato, T.: Perturbation Theory for Linear Operators. Berlin: Springer, second edition 1976
Knauf A.: Ergodic and topological properties of coulombic periodic potentials. Commun. Math. Phys. 110(1), 89–112 (1987)
Komorowski T., Ryzhik L.: Diffusion in a weakly random hamiltonian flow. Commun. Math. Phys 263, 277–323 (2006)
Lebowitz J., Spohn H.: Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs. Adv. Chem. Phys. 39, 109–142 (1978)
Ovchinnikov A.A., Erikhman N.S.: Motion of a quantum particle in a stochastic medium. Sov. Phys.-JETP 40, 733–737 (1975)
Spohn H.: An algebraic condition for the approach to equilibrium of an open N-level system. Lett. Math. Phys. 2, 33–38 (1977)
Vacchini B., Hornberger K.: Quantum linear boltzmann equation. Phys. Rept. 478, 71–120 (2009)
Van Hove L.: Quantum-mechanical perturbations giving rise to a statistical transport equation. Physica 21, 517–540 (1955)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H.-T. Yau
Rights and permissions
About this article
Cite this article
De Roeck, W., Fröhlich, J. Diffusion of a Massive Quantum Particle Coupled to a Quasi-Free Thermal Medium. Commun. Math. Phys. 303, 613–707 (2011). https://doi.org/10.1007/s00220-011-1222-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-011-1222-0