Fundamental aspects of quantum Brownian motion

Chaos

Volumn 15, Issue 2, 2005, Pages

  Hänggi, Peter ^a   Ingold, Gert Ludwig ^a  

^a UNIVERSITY OF AUGSBURG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 31144462179     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1853631     Document Type: Article

References (110)

1. A. Einstein, " Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen., " Ann. Phys. 17, 549-560 (1905).
2. H. Nyquist, " Thermal agitation of electric charge in conductors., " Phys. Rev. 32, 110 -113 (1928).
3. J. B. Johnson, " Thermal agitation of electricity in conductors., " Phys. Rev. 32, 97 -109 (1928).
4. H. B. Callen and T. A. Welton, " Irreversibility and generalized noise., " Phys. Rev. 83, 34 -40 (1951).
5. H. Grabert, P. Schramm, and G.-L. Ingold, " Quantum Brownian motion: The functional integral approach., " Phys. Rep. 168, 115-207 (1988).
6. T. Dittrich, P. Hänggi. G.-L. Ingold, B. Kramer, G. Schön, and W. Zwerger, Quantum Transport and Dissipation (Wiley-VCH, Weinheim, 1998).
7. M. Grifoni and P. Hänggi, " Driven quantum tunneling., " Phys. Rep. 304, 229-358 (1998).
8. V. B. Magalinskiǐ, " Dynamical model in the theory of the Brownian motion., " Sov. Phys. JETP 0038-5646 9, 1381-1382 (1959) V. B. Magalinskiǐ [J. Exp. Theor. Phys. 1063-7761 36, 1942-1944 (1959)]; R. Benguria and M. Kac, " Quantum Langevin equation., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.46.1 46, 1 -4 (1981); G. W. Ford and M. Kac, " On the quantum Langevin equation., " J. Stat. Phys. 0022-4715 10.1007/BF01011142 46, 803-810 (1987); G. W. Ford, J. T. Lewis, and R. F. O'Connell, " Quantum Langevin equation., " Phys. Rev. A 37, 4419 -4428 (1988).
9. V. B. Magalinskiǐ, " Dynamical model in the theory of the Brownian motion., " Sov. Phys. JETP 0038-5646 9, 1381-1382 (1959) V. B. Magalinskiǐ [J. Exp. Theor. Phys. 1063-7761 36, 1942-1944 (1959)]; R. Benguria and M. Kac, " Quantum Langevin equation., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.46.1 46, 1 -4 (1981); G. W. Ford and M. Kac, " On the quantum Langevin equation., " J. Stat. Phys. 0022-4715 10.1007/BF01011142 46, 803-810 (1987); G. W. Ford, J. T. Lewis, and R. F. O'Connell, " Quantum Langevin equation., " Phys. Rev. A 37, 4419 -4428 (1988).
10. V. B. Magalinskiǐ, " Dynamical model in the theory of the Brownian motion., " Sov. Phys. JETP 0038-5646 9, 1381-1382 (1959) V. B. Magalinskiǐ [J. Exp. Theor. Phys. 1063-7761 36, 1942-1944 (1959)]; R. Benguria and M. Kac, " Quantum Langevin equation., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.46.1 46, 1 -4 (1981); G. W. Ford and M. Kac, " On the quantum Langevin equation., " J. Stat. Phys. 0022-4715 10.1007/BF01011142 46, 803-810 (1987); G. W. Ford, J. T. Lewis, and R. F. O'Connell, " Quantum Langevin equation., " Phys. Rev. A 37, 4419 -4428 (1988).
11. V. B. Magalinskiǐ, " Dynamical model in the theory of the Brownian motion., " Sov. Phys. JETP 0038-5646 9, 1381-1382 (1959) V. B. Magalinskiǐ [J. Exp. Theor. Phys. 1063-7761 36, 1942-1944 (1959)]; R. Benguria and M. Kac, " Quantum Langevin equation., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.46.1 46, 1 -4 (1981); G. W. Ford and M. Kac, " On the quantum Langevin equation., " J. Stat. Phys. 0022-4715 10.1007/BF01011142 46, 803-810 (1987); G. W. Ford, J. T. Lewis, and R. F. O'Connell, " Quantum Langevin equation., " Phys. Rev. A 37, 4419 -4428 (1988).
12. V. B. Magalinskiǐ, " Dynamical model in the theory of the Brownian motion., " Sov. Phys. JETP 0038-5646 9, 1381-1382 (1959) V. B. Magalinskiǐ [J. Exp. Theor. Phys. 1063-7761 36, 1942-1944 (1959)]; R. Benguria and M. Kac, " Quantum Langevin equation., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.46.1 46, 1 -4 (1981); G. W. Ford and M. Kac, " On the quantum Langevin equation., " J. Stat. Phys. 0022-4715 10.1007/BF01011142 46, 803-810 (1987); G. W. Ford, J. T. Lewis, and R. F. O'Connell, " Quantum Langevin equation., " Phys. Rev. A 37, 4419 -4428 (1988).
13. E. Nelson, " Derivation of the Schrödinger equation from Newtonian mechanics., " Phys. Rev. 150, 1079 -1085 (1966).
14. H. Grabert, P. Hänggi, and P. Talkner, " Is quantum mechanics equivalent to a classical stochastic process? " Phys. Rev. A 19, 2440 -2445 (1979).
15. M. Roncadelli, " Connection between Langevin quantization and classical mechanics., " Europhys. Lett. 0295-5075 28, 379-384 (1994); H. Kleinert and S. V. Shabanov, " Quantum Langevin equation from forward-backward path integral., " Phys. Lett. A 0375-9601 10.1016/0375-9601(95)00169-4 200, 224-232 (1995); L. Diósi and W. T. Strunz, " The non-Markovian stochastic Schrödinger equation for open systems., " Phys. Lett. A 0375-9601 10.1016/S0375-9601(97)00717-2 235, 569-573 (1997); W. T. Strunz, L. Diósi, and N. Gisin, " Open system dynamics with non-Markovian quantum trajectories., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.82.1801 82, 1801 -1805 (1999); H. P. Breuer, B. Kappler, and F. Petruccione, " Stochastic wave function method for non-Markovian quantum master equations., " Phys. Rev. A 1050-2947 10.1103/PhysRevA.59.1633 59, 1633 -1643 (1999); J. T. Stockburger and H. Grabert, " Non-Markovian quantum state diffusion., " Chem. Phys. 0301-0104 10.1016/S0301-0104(01)00307-X 268, 249-256 (2001); J. Shao, " Decoupling quantum dissipation interaction via stochastic fields., " J. Chem. Phys. 120, 5053-5056 (2004).
16. M. Roncadelli, " Connection between Langevin quantization and classical mechanics., " Europhys. Lett. 0295-5075 28, 379-384 (1994); H. Kleinert and S. V. Shabanov, " Quantum Langevin equation from forward-backward path integral., " Phys. Lett. A 0375-9601 10.1016/0375-9601(95)00169-4 200, 224-232 (1995); L. Diósi and W. T. Strunz, " The non-Markovian stochastic Schrödinger equation for open systems., " Phys. Lett. A 0375-9601 10.1016/S0375-9601(97)00717-2 235, 569-573 (1997); W. T. Strunz, L. Diósi, and N. Gisin, " Open system dynamics with non-Markovian quantum trajectories., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.82.1801 82, 1801 -1805 (1999); H. P. Breuer, B. Kappler, and F. Petruccione, " Stochastic wave function method for non-Markovian quantum master equations., " Phys. Rev. A 1050-2947 10.1103/PhysRevA.59.1633 59, 1633 -1643 (1999); J. T. Stockburger and H. Grabert, " Non-Markovian quantum state diffusion., " Chem. Phys. 0301-0104 10.1016/S0301-0104(01)00307-X 268, 249-256 (2001); J. Shao, " Decoupling quantum dissipation interaction via stochastic fields., " J. Chem. Phys. 120, 5053-5056 (2004).
17. M. Roncadelli, " Connection between Langevin quantization and classical mechanics., " Europhys. Lett. 0295-5075 28, 379-384 (1994); H. Kleinert and S. V. Shabanov, " Quantum Langevin equation from forward-backward path integral., " Phys. Lett. A 0375-9601 10.1016/0375-9601(95)00169-4 200, 224-232 (1995); L. Diósi and W. T. Strunz, " The non-Markovian stochastic Schrödinger equation for open systems., " Phys. Lett. A 0375-9601 10.1016/S0375-9601(97)00717-2 235, 569-573 (1997); W. T. Strunz, L. Diósi, and N. Gisin, " Open system dynamics with non-Markovian quantum trajectories., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.82.1801 82, 1801 -1805 (1999); H. P. Breuer, B. Kappler, and F. Petruccione, " Stochastic wave function method for non-Markovian quantum master equations., " Phys. Rev. A 1050-2947 10.1103/PhysRevA.59.1633 59, 1633 -1643 (1999); J. T. Stockburger and H. Grabert, " Non-Markovian quantum state diffusion., " Chem. Phys. 0301-0104 10.1016/S0301-0104(01)00307-X 268, 249-256 (2001); J. Shao, " Decoupling quantum dissipation interaction via stochastic fields., " J. Chem. Phys. 120, 5053-5056 (2004).
18. M. Roncadelli, " Connection between Langevin quantization and classical mechanics., " Europhys. Lett. 0295-5075 28, 379-384 (1994); H. Kleinert and S. V. Shabanov, " Quantum Langevin equation from forward-backward path integral., " Phys. Lett. A 0375-9601 10.1016/0375-9601(95)00169-4 200, 224-232 (1995); L. Diósi and W. T. Strunz, " The non-Markovian stochastic Schrödinger equation for open systems., " Phys. Lett. A 0375-9601 10.1016/S0375-9601(97)00717-2 235, 569-573 (1997); W. T. Strunz, L. Diósi, and N. Gisin, " Open system dynamics with non-Markovian quantum trajectories., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.82.1801 82, 1801 -1805 (1999); H. P. Breuer, B. Kappler, and F. Petruccione, " Stochastic wave function method for non-Markovian quantum master equations., " Phys. Rev. A 1050-2947 10.1103/PhysRevA.59.1633 59, 1633 -1643 (1999); J. T. Stockburger and H. Grabert, " Non-Markovian quantum state diffusion., " Chem. Phys. 0301-0104 10.1016/S0301-0104(01)00307-X 268, 249-256 (2001); J. Shao, " Decoupling quantum dissipation interaction via stochastic fields., " J. Chem. Phys. 120, 5053-5056 (2004).
19. M. Roncadelli, " Connection between Langevin quantization and classical mechanics., " Europhys. Lett. 0295-5075 28, 379-384 (1994); H. Kleinert and S. V. Shabanov, " Quantum Langevin equation from forward-backward path integral., " Phys. Lett. A 0375-9601 10.1016/0375-9601(95)00169-4 200, 224-232 (1995); L. Diósi and W. T. Strunz, " The non-Markovian stochastic Schrödinger equation for open systems., " Phys. Lett. A 0375-9601 10.1016/S0375-9601(97)00717-2 235, 569-573 (1997); W. T. Strunz, L. Diósi, and N. Gisin, " Open system dynamics with non-Markovian quantum trajectories., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.82.1801 82, 1801 -1805 (1999); H. P. Breuer, B. Kappler, and F. Petruccione, " Stochastic wave function method for non-Markovian quantum master equations., " Phys. Rev. A 1050-2947 10.1103/PhysRevA.59.1633 59, 1633 -1643 (1999); J. T. Stockburger and H. Grabert, " Non-Markovian quantum state diffusion., " Chem. Phys. 0301-0104 10.1016/S0301-0104(01)00307-X 268, 249-256 (2001); J. Shao, " Decoupling quantum dissipation interaction via stochastic fields., " J. Chem. Phys. 120, 5053-5056 (2004).
20. M. Roncadelli, " Connection between Langevin quantization and classical mechanics., " Europhys. Lett. 0295-5075 28, 379-384 (1994); H. Kleinert and S. V. Shabanov, " Quantum Langevin equation from forward-backward path integral., " Phys. Lett. A 0375-9601 10.1016/0375-9601(95)00169-4 200, 224-232 (1995); L. Diósi and W. T. Strunz, " The non-Markovian stochastic Schrödinger equation for open systems., " Phys. Lett. A 0375-9601 10.1016/S0375-9601(97)00717-2 235, 569-573 (1997); W. T. Strunz, L. Diósi, and N. Gisin, " Open system dynamics with non-Markovian quantum trajectories., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.82.1801 82, 1801 -1805 (1999); H. P. Breuer, B. Kappler, and F. Petruccione, " Stochastic wave function method for non-Markovian quantum master equations., " Phys. Rev. A 1050-2947 10.1103/PhysRevA.59.1633 59, 1633 -1643 (1999); J. T. Stockburger and H. Grabert, " Non-Markovian quantum state diffusion., " Chem. Phys. 0301-0104 10.1016/S0301-0104(01)00307-X 268, 249-256 (2001); J. Shao, " Decoupling quantum dissipation interaction via stochastic fields., " J. Chem. Phys. 120, 5053-5056 (2004).
21. M. Roncadelli, " Connection between Langevin quantization and classical mechanics., " Europhys. Lett. 0295-5075 28, 379-384 (1994); H. Kleinert and S. V. Shabanov, " Quantum Langevin equation from forward-backward path integral., " Phys. Lett. A 0375-9601 10.1016/0375-9601(95)00169-4 200, 224-232 (1995); L. Diósi and W. T. Strunz, " The non-Markovian stochastic Schrödinger equation for open systems., " Phys. Lett. A 0375-9601 10.1016/S0375-9601(97)00717-2 235, 569-573 (1997); W. T. Strunz, L. Diósi, and N. Gisin, " Open system dynamics with non-Markovian quantum trajectories., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.82.1801 82, 1801 -1805 (1999); H. P. Breuer, B. Kappler, and F. Petruccione, " Stochastic wave function method for non-Markovian quantum master equations., " Phys. Rev. A 1050-2947 10.1103/PhysRevA.59.1633 59, 1633 -1643 (1999); J. T. Stockburger and H. Grabert, " Non-Markovian quantum state diffusion., " Chem. Phys. 0301-0104 10.1016/S0301-0104(01)00307-X 268, 249-256 (2001); J. Shao, " Decoupling quantum dissipation interaction via stochastic fields., " J. Chem. Phys. 120, 5053-5056 (2004).
22. P. Talkner, " The failure of the quantum regression hypothesis., " Ann. Phys. (N.Y.) 167, 390-436 (1986); " Untersuchungen irreversibler Prozesse in quantenmechanischen Systemen., " Ph.D. thesis, Universität Stuttgart, 1979.
23. P. Talkner, " The failure of the quantum regression hypothesis., " Ann. Phys. (N.Y.) 167, 390-436 (1986); " Untersuchungen irreversibler Prozesse in quantenmechanischen Systemen., " Ph.D. thesis, Universität Stuttgart, 1979.
24. M. Planck, " Eine neue Strahlungshypothese., " Verh. Dtsch. Phys. Ges. 13, 138-148 (1911).
25. H. Grabert, U. Weiss, and P. Talkner, " Quantum theory of the damped harmonic oscillator., " Z. Phys. B: Condens. Matter 55, 87-94 (1984).
26. P. S. Riseborough, P. Hänggi, and U. Weiss, " Exact results for a damped quantum-mechanical harmonic oscillator., " Phys. Rev. A 31, 471 -478 (1985).
27. H. Grabert, Projection Operator Techniques in Nonequilibrium Statistical Mechanics, Springer Tracts in Modern Physics (Springer, Berlin, 1982), Vol. 95, pp. 1-164.
28. F. Haake, Statistical Treatment of Open Systems by Generalized Master Equations, Springer Tracts in Modern Physics (Springer, Berlin, 1973), Vol. 66, pp. 98-168.
29. H. Spohn, " Kinetic equations from Hamiltonian dynamics: Markovian limits., " Rev. Mod. Phys. 52, 569 -615 (1980).
30. R. Alicki, " General theory and applications to unstable particles., " in Quantum Dynamical Semigroups and Applications, Lect. Notes Phys. (Springer, Berlin, 1987), Vol. 286, Chaps. and.
31. G.-L. Ingold and H. Grabert, " Sluggish decay of preparation effects in low temperature quantum systems., " Lect. Notes Math. 1442, 219-230 (1990).
32. R. Jung, G.-L. Ingold, and H. Grabert, " Long-time tails in quantum Brownian motion., " Phys. Rev. A 32, 2510 -2512 (1985).
33. A. Hanke and W. Zwerger, " Density of states of a damped quantum oscillator., " Phys. Rev. E 52, 6875 -6878 (1995).
34. G.-L. Ingold, " Path integrals and their application to dissipative quantum systems., " Lect. Notes Phys. 611, 1-53 (2002).
35. H. Mori, " Transport, collective motion and Brownian motion., " Prog. Theor. Phys. 0033-068X 33, 423-455 (1965); K. Kawasaki, " Simple derivations of generalized linear and nonlinear Langevin equations., " J. Phys. A 0305-4470 10.1088/0305-4470/6/9/004 6, 1289-1295 (1973); S. Nordholm and R. Zwanzig, " Systematic derivation of exact generalized Brownian-motion theory., " J. Stat. Phys. 13, 347-371 (1975).
36. H. Mori, " Transport, collective motion and Brownian motion., " Prog. Theor. Phys. 0033-068X 33, 423-455 (1965); K. Kawasaki, " Simple derivations of generalized linear and nonlinear Langevin equations., " J. Phys. A 0305-4470 10.1088/0305-4470/6/9/004 6, 1289-1295 (1973); S. Nordholm and R. Zwanzig, " Systematic derivation of exact generalized Brownian-motion theory., " J. Stat. Phys. 13, 347-371 (1975).
37. H. Mori, " Transport, collective motion and Brownian motion., " Prog. Theor. Phys. 0033-068X 33, 423-455 (1965); K. Kawasaki, " Simple derivations of generalized linear and nonlinear Langevin equations., " J. Phys. A 0305-4470 10.1088/0305-4470/6/9/004 6, 1289-1295 (1973); S. Nordholm and R. Zwanzig, " Systematic derivation of exact generalized Brownian-motion theory., " J. Stat. Phys. 13, 347-371 (1975).
38. H. Grabert, P. Hänggi, and P. Talkner, " Microdynamics and nonlinear stochastic processes of gross variables., " J. Stat. Phys. 22, 537-552 (1980).
39. P. Hänggi, " Generalized Langevin equations: A useful tool for the perplexed modeller of nonequilibrium fluctuations? " Lect. Notes Phys. 484, 15-22 (1997).
40. I. R. Senitzky, " Dissipation in quantum mechanics. The harmonic oscillator., " Phys. Rev. 119, 670 -679 (1960).
41. P. Ullersma, " An exactly solvable model for Brownian motion., " Physica (Utrecht) 0031-8914 10.1016/0031-8914(66)90102-9 32, 27-55 (1966); P. Ullersma, Physica (Amsterdam) 0031-8914 10.1016/0031-8914(66)90103-0 32, 56-73 (1966); P. Ullersma, Physica (Amsterdam) 0031-8914 10.1016/0031-8914(66)90104-2 32, 74-89 (1966); P. Ullersma, Physica (Amsterdam) 32, 90-96 (1966).
42. P. Ullersma, " An exactly solvable model for Brownian motion., " Physica (Utrecht) 0031-8914 10.1016/0031-8914(66)90102-9 32, 27-55 (1966); P. Ullersma, Physica (Amsterdam) 0031-8914 10.1016/0031-8914(66)90103-0 32, 56-73 (1966); P. Ullersma, Physica (Amsterdam) 0031-8914 10.1016/0031-8914(66)90104-2 32, 74-89 (1966); P. Ullersma, Physica (Amsterdam) 32, 90-96 (1966).
43. P. Ullersma, " An exactly solvable model for Brownian motion., " Physica (Utrecht) 0031-8914 10.1016/0031-8914(66)90102-9 32, 27-55 (1966); P. Ullersma, Physica (Amsterdam) 0031-8914 10.1016/0031-8914(66)90103-0 32, 56-73 (1966); P. Ullersma, Physica (Amsterdam) 0031-8914 10.1016/0031-8914(66)90104-2 32, 74-89 (1966); P. Ullersma, Physica (Amsterdam) 32, 90-96 (1966).
44. P. Ullersma, " An exactly solvable model for Brownian motion., " Physica (Utrecht) 0031-8914 10.1016/0031-8914(66)90102-9 32, 27-55 (1966); P. Ullersma, Physica (Amsterdam) 0031-8914 10.1016/0031-8914(66)90103-0 32, 56-73 (1966); P. Ullersma, Physica (Amsterdam) 0031-8914 10.1016/0031-8914(66)90104-2 32, 74-89 (1966); P. Ullersma, Physica (Amsterdam) 32, 90-96 (1966).
45. R. Zwanzig, " Nonlinear generalized Langevin equations., " J. Stat. Phys. 9, 215-220 (1973).
46. P. Hänggi, P. Talkner, and M. Borkovec, " Reaction-rate theory: Fifty years after Kramers., " Rev. Mod. Phys. 62, 251 -341 (1990).
47. A. O. Caldeira and A. J. Leggett, " Quantum tunnelling in a dissipative system., " Ann. Phys. (N.Y.) 0003-4916 10.1103/PhysRevLett.46.211 149, 374-456 (1983); A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 153, 445 (1984).
48. A. O. Caldeira and A. J. Leggett, " Quantum tunnelling in a dissipative system., " Ann. Phys. (N.Y.) 0003-4916 10.1103/PhysRevLett.46.211 149, 374-456 (1983); A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 153, 445 (1984).
49. U. Weiss, Quantum Dissipative Systems, 2nd ed. (World Scientific, Singapore, 1999).
50. U. Eckern, G. Schön, and V. Ambegaokar, " Quantum dynamics of a superconducting tunnel junction., " Phys. Rev. B 30, 6419 -6431 (1984).
51. G. Schön and A. D. Zaikin, " Quantum coherent effects, phase transitions, and the dissipative dynamics of ultra small tunnel junctions., " Phys. Rep. 198, 237-413 (1990).
52. G.-L. Ingold and Yu. V. Nazarov, " Charge tunneling rates in ultrasmall junctions., " in Single Charge Tunneling, NATO ASI Series B, edited by, H. Grabert, and, M. H. Devoret, (Plenum, New York, 1992), Vol. 294, pp. 21-107.
53. U. Zürcher and P. Talkner, " Quantum-mechanical harmonic chain attached to heat baths II. Nonequilibrium properties., " Phys. Rev. A 42, 3278 -3290 (1990).
54. D. Dvira, A. Nitzan, and P. Hänggi, " Thermal conductance through molecular wires., " J. Chem. Phys. 119, 6840-6855 (2003).
55. S. Camalet, J. Lehmann, S. Kohler, and P. Hänggi, " Current noise in ac-driven nanoscale conductors., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.90.210602 90, 210602 (2003); S. Camalet, S. Kohler, and P. Hänggi, " Shot-noise control in ac-driven nanoscale conductors., " Phys. Rev. B 70, 155326 (2004).
56. S. Camalet, J. Lehmann, S. Kohler, and P. Hänggi, " Current noise in ac-driven nanoscale conductors., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.90.210602 90, 210602 (2003); S. Camalet, S. Kohler, and P. Hänggi, " Shot-noise control in ac-driven nanoscale conductors., " Phys. Rev. B 70, 155326 (2004).
57. P. C. Hemmer, L. C. Maximon, and H. Wergeland, " Recurrence time of a dynamical system., " Phys. Rev. 111, 689 -694 (1958).
58. P. Mazur and E. Montroll, " Poincaŕ cycles, ergodicity, and irreversibility in assemblies of coupled harmonic oscillators., " J. Math. Phys. 1, 70-84 (1960).
59. S. A. Adelman, " Quantum generalized Langevin equation approach to gassolid collisions., " Chem. Phys. Lett. 40, 495-499 (1976).
60. A. Schmid, " On a quasiclassical Langevin equation., " J. Low Temp. Phys. 49, 609-626 (1982).
61. S. K. Banik, B. C. Bag, and D. S. Ray, " Generalized quantum Fokker-Planck, diffusion and Smoluchowski equations with true probability distribution functions., " Phys. Rev. E 1063-651X 10.1103/PhysRevE.65.051106 65, 051106 (2002); D. Banerjee, B. C. Bag, S. K. Banik, and D. S. Ray, " Solution of quantum Langevin equation: Approximations, theoretical and numerical aspects., " J. Chem. Phys. 120, 8960-8972 (2004).
62. S. K. Banik, B. C. Bag, and D. S. Ray, " Generalized quantum Fokker-Planck, diffusion and Smoluchowski equations with true probability distribution functions., " Phys. Rev. E 1063-651X 10.1103/PhysRevE.65.051106 65, 051106 (2002); D. Banerjee, B. C. Bag, S. K. Banik, and D. S. Ray, " Solution of quantum Langevin equation: Approximations, theoretical and numerical aspects., " J. Chem. Phys. 120, 8960-8972 (2004).
63. U. Eckern, W. Lehr, A. Menzel-Dorwarth, F. Pelzer, and A. Schmid, " The quasiclassical Langevin equation and its application to the decay of a metastable state and to quantum fluctuations., " J. Stat. Phys. 59, 885-934 (1990).
64. P. Hänggi and H. Thomas, " Stochastic processes: Time evolution, symmetries and linear response., " Phys. Rep. 88, 207-319 (1982).
65. J. Shao and P. Hänggi, " Decoherent dynamics of a two-level system coupled to a sea of spins., " Phys. Rev. Lett. 81, 5710 -5713 (1998).
66. P. Schramm, R. Jung, and H. Grabert, " A closer look at the quantum Langevin equation: Fokker-Planck equation and quasiprobabilities., " Phys. Lett. 107A, 385-389 (1985).
67. H. Grabert, P. Talkner, and P. Hänggi, " Microdynamics and time-evolution of macroscopic non-Markovian systems., " Z. Phys. B 0340-224X 26, 389-395 (1977); H. Grabert, P. Talkner, P. Hänggi, and H. Thomas, " Microdynamics and time-evolution of macroscopic non-Markovian systems II., " Z. Phys. B 29, 273-280 (1978).
68. H. Grabert, P. Talkner, and P. Hänggi, " Microdynamics and time-evolution of macroscopic non-Markovian systems., " Z. Phys. B 0340-224X 26, 389-395 (1977); H. Grabert, P. Talkner, P. Hänggi, and H. Thomas, " Microdynamics and time-evolution of macroscopic non-Markovian systems II., " Z. Phys. B 29, 273-280 (1978).
69. K. M. F. Romero, P. Talkner, and P. Hänggi, " Is the dynamics of open quantum systems always linear? " Phys. Rev. A 69, 052109 (2004).
70. C. Zerbe and P. Hänggi, " Brownian parametric quantum oscillator with dissipation., " Phys. Rev. E 52, 1533 -1543 (1995).
71. S. Kohler, T. Dittrich, and P. Hänggi, " Floquet-Markovian description of parametrically driven, dissipative harmonic oscillator., " Phys. Rev. E 55, 300 -313 (1997).
72. J. Łuczka, " Spin in contact with thermostat: Exact reduced dynamics., " Physica A 167, 919-934 (1990).
73. N. G. van Kampen, " A soluble model for quantum mechanical dissipation., " J. Stat. Phys. 78, 299-310 (1995).
74. L.-D. Chang and S. Chakravarty, " Dissipative dynamics of a two-state system coupled to a heat bath., " Phys. Rev. B 0163-1829 10.1103/PhysRevB.31.154 31, 154 -164 (1985); F. Sols and F. Guinea, " Bulk and surface diffusion of heavy particles in metals: A path-integral approach., " Phys. Rev. B 0163-1829 10.1103/PhysRevB.36.7775 36, 7775 -7785 (1987); P. Hedegård and A. O. Caldeira, " Quantum dynamics of a particle in a fermionic environment., " Phys. Scr. 35, 609-622 (1987).
75. L.-D. Chang and S. Chakravarty, " Dissipative dynamics of a two-state system coupled to a heat bath., " Phys. Rev. B 0163-1829 10.1103/PhysRevB.31.154 31, 154 -164 (1985); F. Sols and F. Guinea, " Bulk and surface diffusion of heavy particles in metals: A path-integral approach., " Phys. Rev. B 0163-1829 10.1103/PhysRevB.36.7775 36, 7775 -7785 (1987); P. Hedegård and A. O. Caldeira, " Quantum dynamics of a particle in a fermionic environment., " Phys. Scr. 35, 609-622 (1987).
76. L.-D. Chang and S. Chakravarty, " Dissipative dynamics of a two-state system coupled to a heat bath., " Phys. Rev. B 0163-1829 10.1103/PhysRevB.31.154 31, 154 -164 (1985); F. Sols and F. Guinea, " Bulk and surface diffusion of heavy particles in metals: A path-integral approach., " Phys. Rev. B 0163-1829 10.1103/PhysRevB.36.7775 36, 7775 -7785 (1987); P. Hedegård and A. O. Caldeira, " Quantum dynamics of a particle in a fermionic environment., " Phys. Scr. 35, 609-622 (1987).
77. R. P. Feynman, " Space-time approach to nonrelativistic quantum mechanics., " Rev. Mod. Phys. 20, 367 -387 (1948); R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
78. R. P. Feynman, " Space-time approach to nonrelativistic quantum mechanics., " Rev. Mod. Phys. 20, 367 -387 (1948); R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
79. R. P. Feynman and F. L. Vernon, " The theory of a general quantum system interacting with a linear dissipative system., " Ann. Phys. (N.Y.) 24, 118-173 (1963).
80. P. Hanggi, H. Grabert, G.-L. Ingold, and U. Weiss, " Quantum theory of activated events in presence of long-time memory., " Phys. Rev. Lett. 55, 761 -764 (1985).
81. H. Grabert, P. Olschowski, and U. Weiss, " Quantum decay rates for dissipative systems at finite temperatures., " Phys. Rev. B 36, 1931 -1951 (1987).
82. G.-L. Ingold, " Anwendung von Funktionalintegralen auf Transport- und Relaxationsphänomene in dissipativen Quantensystemen., " Ph.D. thesis, Universität Stuttgart, 1988.
83. H. Hofmann und G.-L. Ingold, " Dissipative transport across a parabolic barrier., " Phys. Lett. B 264, 253-258 (1991).
84. J. Ankerhold, H. Grabert, and G.-L. Ingold, " Dissipative quantum systems with potential barrier. General theory and parabolic barrier., " Phys. Rev. E 1063-651X 10.1103/PhysRevE.51.4267 51, 4267 -4281 (1995); J. Ankerhold and H. Grabert, " Dissipative quantum systems with a potential barrier. II. Dynamics near the barrier top., " Phys. Rev. E 1063-651X 10.1103/PhysRevE.52.4704 52, 4704 -4723 (1995); Dissipative quantum systems with a potential barrier. III. Steady state nonequilibrium flux and reaction rate., " Phys. Rev. E 55, 1355 -1374 (1997).
85. J. Ankerhold, H. Grabert, and G.-L. Ingold, " Dissipative quantum systems with potential barrier. General theory and parabolic barrier., " Phys. Rev. E 1063-651X 10.1103/PhysRevE.51.4267 51, 4267 -4281 (1995); J. Ankerhold and H. Grabert, " Dissipative quantum systems with a potential barrier. II. Dynamics near the barrier top., " Phys. Rev. E 1063-651X 10.1103/PhysRevE.52.4704 52, 4704 -4723 (1995); Dissipative quantum systems with a potential barrier. III. Steady state nonequilibrium flux and reaction rate., " Phys. Rev. E 55, 1355 -1374 (1997).
86. J. Ankerhold, H. Grabert, and G.-L. Ingold, " Dissipative quantum systems with potential barrier. General theory and parabolic barrier., " Phys. Rev. E 1063-651X 10.1103/PhysRevE.51.4267 51, 4267 -4281 (1995); J. Ankerhold and H. Grabert, " Dissipative quantum systems with a potential barrier. II. Dynamics near the barrier top., " Phys. Rev. E 1063-651X 10.1103/PhysRevE.52.4704 52, 4704 -4723 (1995); Dissipative quantum systems with a potential barrier. III. Steady state nonequilibrium flux and reaction rate., " Phys. Rev. E 55, 1355 -1374 (1997).
87. J. Ankerhold and H. Grabert, " Quantum tunneling and the semiclassical real time dynamics of the density matrix., " Europhys. Lett. 47, 285-291 (1999).
88. P. Hänggi and W. Hontscha, " Periodic orbit approach to the quantum-Kramers-rate., " Ber. Bunsenges. Phys. Chem. 0005-9021 95, 379-385 (1991); " Unified approach to the quantum-Kramers reaction rate., " J. Chem. Phys. 88, 4094-4095 (1988).
89. P. Hänggi and W. Hontscha, " Periodic orbit approach to the quantum-Kramers-rate., " Ber. Bunsenges. Phys. Chem. 0005-9021 95, 379-385 (1991); " Unified approach to the quantum-Kramers reaction rate., " J. Chem. Phys. 88, 4094-4095 (1988).
90. P. S. Riseborough, P. Hänggi, and E. Freidkin, " Quantum tunneling in dissipative media: Intermediate-coupling-strength results., " Phys. Rev. A 32, 489 -499 (1985).
91. H. Grabert, U. Weiss, and P. Hanggi, " Quantum tunneling in dissipative systems at finite temperatures., " Phys. Rev. Lett. 52, 2193 -2196 (1984).
92. J. Ankerhold, P. Pechukas, and H. Grabert, " Strong friction limit in quantum mechanics: The quantum Smoluchowski equation., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.87.086802 87, 086802 (2001); J. Ankerhold, " Quantum decay rates for driven barrier potentials in the strong friction limit., " Phys. Rev. E 1063-651X 10.1103/PhysRevE.64.060102 64, 060102 (2001); L. Machura, M. Kostur, P. Hänggi, P. Talkner, and J. Łuczka, " Consistent description of quantum Brownian motors operating at strong friction., " Phys. Rev. E 70, 031107 (2004).
93. J. Ankerhold, P. Pechukas, and H. Grabert, " Strong friction limit in quantum mechanics: The quantum Smoluchowski equation., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.87.086802 87, 086802 (2001); J. Ankerhold, " Quantum decay rates for driven barrier potentials in the strong friction limit., " Phys. Rev. E 1063-651X 10.1103/PhysRevE.64.060102 64, 060102 (2001); L. Machura, M. Kostur, P. Hänggi, P. Talkner, and J. Łuczka, " Consistent description of quantum Brownian motors operating at strong friction., " Phys. Rev. E 70, 031107 (2004).
94. J. Ankerhold, P. Pechukas, and H. Grabert, " Strong friction limit in quantum mechanics: The quantum Smoluchowski equation., " Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.87.086802 87, 086802 (2001); J. Ankerhold, " Quantum decay rates for driven barrier potentials in the strong friction limit., " Phys. Rev. E 1063-651X 10.1103/PhysRevE.64.060102 64, 060102 (2001); L. Machura, M. Kostur, P. Hänggi, P. Talkner, and J. Łuczka, " Consistent description of quantum Brownian motors operating at strong friction., " Phys. Rev. E 70, 031107 (2004).
95. G. W. Ford and R. F. O'Connell, " Inconsistency of the rotating wave approximation with the Ehrenfest theorem., " Phys. Lett. A 215, 245-246 (1996).
96. H. Grabert, " Non-linear relaxation and fluctuations of damped quantum systems., " Z. Phys. B: Condens. Matter 42, 161-172 (1982).
97. G. W. Ford and R. F. O'Connell, " There is no quantum regression theorem., " Phys. Rev. Lett. 77, 798 -801 (1996).
98. M. D. Kostin, " Friction and dissipative phenomena in quantum mechanics., " J. Stat. Phys. 12, 145 (1975).
99. J. Messer, " Friction in quantum mechanics., " Acta Phys. Austriaca 50, 75-91 (1979), and references therein.
100. H. Dekker, " Classical and quantum mechanics of the damped harmonic oscillator., " Phys. Rep. 80, 1-112 (1981).
101. R. Fürth, " Über einige Beziehungen zwischen klassischer Statistik und Quantenmechanik., " Z. Phys. 81, 143-162 (1933).
102. I. F́nyes, " Eine wahrscheinlichkeitstheoretische Begründung und Interpretation der Quantenmechanik., " Z. Phys. 132, 81-106 (1952).
103. W. Weizel, " Ableitung der Quantentheorie aus einem klassischen, kausal determinierten Modell., " Z. Phys. 0044-3328 134, 264-285 (1953); " Ableitung der Quantentheorie aus einem klassischen Modell. II., " Z. Phys. 135, 270-273 (1953).
104. W. Weizel, " Ableitung der Quantentheorie aus einem klassischen, kausal determinierten Modell., " Z. Phys. 0044-3328 134, 264-285 (1953); " Ableitung der Quantentheorie aus einem klassischen Modell. II., " Z. Phys. 135, 270-273 (1953).
105. L. F. Favella, " Brownian motions and quantum mechanics., " Ann. Inst. Henri Poincare, Sect. A 7, 77-94 (1967).
106. G. C. Ghirardi, C. Omero, and A. Rimini, " Stochastic interpretation of quantum mechanics-Critical review., " Riv. Nuovo Cimento 1, 1-34 (1978).
107. M. S. Wang and W. K. Liang, " Comment on repeated measurements in stochastic mechanics., " Phys. Rev. D 48, 1875 -1877 (1993).
108. R. D. Astumian and P. Hänggi, " Brownian motors., " Phys. Today 0031-9228 10.1016/S0370-1573(01)00081-3 55 (11), 33 -39 (2002);
109. P. Reimann and P. Hänggi, " Quantum features of Brownian motors and stochastic resonance., " Chaos 1054-1500 10.1063/1.166345 8, 629-642 (1998);
110. P. Reimann, M. Grifoni, and P. Hänggi, " Quantum ratchets., " Phys. Rev. Lett. 79, 10 -13 (1997).