Spectral theory of thermal relaxation

Journal of Mathematical Physics

Volumn 38, Issue 4, 1997, Pages 1757-1780

  Jakšić, Vojkan ^a   Pillet, Claude Alain ^b  

^a UNIVERSITY OF OTTAWA   (Canada)

^b UNIVERSITY OF GENEVA   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0042996865     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.531912     Document Type: Article

References (73)

1. V. Jakšić and C.-A. Pillet, "On a Model for Quantum Friction I. Fermi's Golden Rule and Dynamics at Zero Temperature," Ann. Inst. H. Poincaré 62, 47 (1995).
2. V. Jakšić and C.-A. Pillet, "On a Model for Quantum Friction II. Fermi's Golden Rule and Dynamics at Positive Temperature," Commun. Math. Phys. 176, 619 (1996).
3. V. Jakšić and C.-A. Pillet, "On a Model for Quantum Friction III. Ergodic Properties of the Spin-Boson System," Commun. Math. Phys. 178, 627 (1996).
4. V. Jakšić and C.-A. Pillet, "On a Model for Quantum Friction IV. Return to Equilibrium" (in preparation).
5. V. Jakšić and C.-A. Pillet, "From Resonances to Master Equations" (to appear in Ann Inst. H. Poincaré).
6. V. Jakšić and C.-A. Pillet, "Ergodic Properties of the non-Markovian Langevin Equation" (to appear in Lett. Math. Phys.).
7. V. Jakšić and C.-A. Pillet, "Ergodic Properties of Classical Dissipative Systems I" (preprint 1996).
8. V. Jakšić and C.-A. Pillet, "Ergodic Properties of Classical Dissipative Systems II" (in preparation).
9. This experimental observation is known as the "Zeroth law of thermodynamics."
10. The definition of the notion of temperature based on the Gibbs canonical ensemble is one of the pillars of equilibrium statistical mechanics. For a detailed discussion see Chap. 1 in G. E. Uhlenbeck and G. W. Ford, Lectures in Statistical Mechanics (AMS, Providence, 1963).
11. The Langevin equation was first studied by Ornstcin and Uhlenbeck and the resulting random process is known as the Ornstein-Uhlenbeck process. Early works on the Langevin equation are reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, New York, 1964). For a modern exposition, see E. Nelson, Dynamical Theories of Brownian Motion (Princeton U. P., Princeton, NJ, 1976). A review which is close in spirit to ours is J. T. Lewis and L. C. Thomas, "How to make a heat bath," in Functional Integration and Its Applications, edited by A. M. Arthurs (Clarendon, Oxford, 1975).
12. The Langevin equation was first studied by Ornstcin and Uhlenbeck and the resulting random process is known as the Ornstein-Uhlenbeck process. Early works on the Langevin equation are reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, New York, 1964). For a modern exposition, see E. Nelson, Dynamical Theories of Brownian Motion (Princeton U. P., Princeton, NJ, 1976). A review which is close in spirit to ours is J. T. Lewis and L. C. Thomas, "How to make a heat bath," in Functional Integration and Its Applications, edited by A. M. Arthurs (Clarendon, Oxford, 1975).
13. The Langevin equation was first studied by Ornstcin and Uhlenbeck and the resulting random process is known as the Ornstein-Uhlenbeck process. Early works on the Langevin equation are reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, New York, 1964). For a modern exposition, see E. Nelson, Dynamical Theories of Brownian Motion (Princeton U. P., Princeton, NJ, 1976). A review which is close in spirit to ours is J. T. Lewis and L. C. Thomas, "How to make a heat bath," in Functional Integration and Its Applications, edited by A. M. Arthurs (Clarendon, Oxford, 1975).
14. A. Einstein, "Zur Quantentheorie der Strahlung, Physik Zeitschr. 18, 121 (1917). This paper is reprinted in B. L. van der Waerden, Sources of Quantum Mechanics (Dover, New York, 1967). We are not giving a precise account of Einstein's argument. For additional information, see Chap. 21 in A. Pais, "Subtle is the Lord...". The Science and the Life of Albert Einstein (Oxford U. P., Oxford, 1982).
15. A. Einstein, "Zur Quantentheorie der Strahlung, Physik Zeitschr. 18, 121 (1917). This paper is reprinted in B. L. van der Waerden, Sources of Quantum Mechanics (Dover, New York, 1967). We are not giving a precise account of Einstein's argument. For additional information, see Chap. 21 in A. Pais, "Subtle is the Lord...". The Science and the Life of Albert Einstein (Oxford U. P., Oxford, 1982).
16. A. Einstein, "Zur Quantentheorie der Strahlung, Physik Zeitschr. 18, 121 (1917). This paper is reprinted in B. L. van der Waerden, Sources of Quantum Mechanics (Dover, New York, 1967). We are not giving a precise account of Einstein's argument. For additional information, see Chap. 21 in A. Pais, "Subtle is the Lord...". The Science and the Life of Albert Einstein (Oxford U. P., Oxford, 1982).
17. ik are related to the width of the observed spectral lines in V. Weisskopf and E. P. Wigner, "Berechnung der naturlichen Linienbreite auf Grund der Diracschen Lichtheorie," Z. Phys. 63, 54 (1930). For a summary of these early developments, see W. Heitler, The Quantum Theory of Radiation (Oxford U. P., Oxford, 1954).
18. ik are related to the width of the observed spectral lines in V. Weisskopf and E. P. Wigner, "Berechnung der naturlichen Linienbreite auf Grund der Diracschen Lichtheorie," Z. Phys. 63, 54 (1930). For a summary of these early developments, see W. Heitler, The Quantum Theory of Radiation (Oxford U. P., Oxford, 1954).
19. ik are related to the width of the observed spectral lines in V. Weisskopf and E. P. Wigner, "Berechnung der naturlichen Linienbreite auf Grund der Diracschen Lichtheorie," Z. Phys. 63, 54 (1930). For a summary of these early developments, see W. Heitler, The Quantum Theory of Radiation (Oxford U. P., Oxford, 1954).
20. Of course, in more general situations Eq. (1) is modified by the addition of appropriate external forces.
21. R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II. Non-equilibrium Statistical Mechanics, Springer Series in Solid-State Sciences, Vol. 31 (Springer, New York, 1991).
22. An excellent review of the physical aspects of the spin-boson system is A. J. Legget, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Gorg, and W. Zwerger, "Dynamics of the dissipative two-state system," Rev. Mod. Phys. 59, 1 (1987).
23. G. W. Ford, M Kac, and P. Mazur, "Statistical mechanics of assemblies of coupled oscillators," J. Math. Phys. 6, 504 (1965).
24. 3) for all β>0.
25. ℬ(φ).
26. A. Komech, M. Kunze, and H. Spohn, "Long-time asymptotics for a classical particle interacting with a scalar wave field" (preprint 1995).
27. This singular term is discussed in Ref. 26.
28. M. M. Troper, "Ergodic and quasi-deterministic properties of finite-dimensional stochastic systems," J. Stat. Phys. 17, 491 (1977).
29. This terminology has been introduced in D. W. Robinson, "Return to equilibrium," Commun. Math. Phys. 31, 171 (1973). See also D. W. Robinson, "C*- algebras in quantum statistical mechanics," in C*-algebras and their Applications to Statistical Mechanics and Quantum Field Theory, edited by D. Kastler (North-Holland, Amsterdam, 1976).
30. This terminology has been introduced in D. W. Robinson, "Return to equilibrium," Commun. Math. Phys. 31, 171 (1973). See also D. W. Robinson, "C*- algebras in quantum statistical mechanics," in C*-algebras and their Applications to Statistical Mechanics and Quantum Field Theory, edited by D. Kastler (North-Holland, Amsterdam, 1976).
31. 2 Euclidean (Quantum) Field Theory (Princeton U. P., Princeton, 1974).
32. 2 Euclidean (Quantum) Field Theory (Princeton U. P., Princeton, 1974).
33. This operator can be explicitly computed; see Proposition 3.5 in Ref. 7.
34. G. W. Ford and M. Kac, "On the quantum Langevin equation," J. Stat. Phys. 46, 803 (1987).
35. P. Smedt, D. Dür, J. L. Lebowitz, and C. Liverani, "Quantum system in contact with a thermal environment: Rigorous treatment of a simple model," Commun. Math. Phys. 120, 120 (1988); H. Massen, "Return to thermal equilibrium by the solution of a quantum Langevin equation," J. Stat. Phys. 34, 239 (1984).
36. P. Smedt, D. Dür, J. L. Lebowitz, and C. Liverani, "Quantum system in contact with a thermal environment: Rigorous treatment of a simple model," Commun. Math. Phys. 120, 120 (1988); H. Massen, "Return to thermal equilibrium by the solution of a quantum Langevin equation," J. Stat. Phys. 34, 239 (1984).
37. The first master equation was derived by Pauli. See W. Pauli, Festschrift zum 60. Gerburtstag A. Sommerfeld (Hirzel, Leipzig, 1928), and W. Pauli, Pauli Lectures on Physics. Volume. 4. Statistical Mechanics, edited by C. P. Enz (The MIT Press, Cambridge, 1973). The classical review on the subject is F. Haake, Statistical treatment of open systems by generatized master equations, Springer tracts in modern physics 66 (Springer, New York, 1973). Rigorous works on the subject are discussed in E.B. Davies, Quantum Theory of Open System (Academic, New York, 1976).
38. The first master equation was derived by Pauli. See W. Pauli, Festschrift zum 60. Gerburtstag A. Sommerfeld (Hirzel, Leipzig, 1928), and W. Pauli, Pauli Lectures on Physics. Volume. 4. Statistical Mechanics, edited by C. P. Enz (The MIT Press, Cambridge, 1973). The classical review on the subject is F. Haake, Statistical treatment of open systems by generatized master equations, Springer tracts in modern physics 66 (Springer, New York, 1973). Rigorous works on the subject are discussed in E.B. Davies, Quantum Theory of Open System (Academic, New York, 1976).
39. The first master equation was derived by Pauli. See W. Pauli, Festschrift zum 60. Gerburtstag A. Sommerfeld (Hirzel, Leipzig, 1928), and W. Pauli, Pauli Lectures on Physics. Volume. 4. Statistical Mechanics, edited by C. P. Enz (The MIT Press, Cambridge, 1973). The classical review on the subject is F. Haake, Statistical treatment of open systems by generatized master equations, Springer tracts in modern physics 66 (Springer, New York, 1973). Rigorous works on the subject are discussed in E.B. Davies, Quantum Theory of Open System (Academic, New York, 1976).
40. The first master equation was derived by Pauli. See W. Pauli, Festschrift zum 60. Gerburtstag A. Sommerfeld (Hirzel, Leipzig, 1928), and W. Pauli, Pauli Lectures on Physics. Volume. 4. Statistical Mechanics, edited by C. P. Enz (The MIT Press, Cambridge, 1973). The classical review on the subject is F. Haake, Statistical treatment of open systems by generatized master equations, Springer tracts in modern physics 66 (Springer, New York, 1973). Rigorous works on the subject are discussed in E.B. Davies, Quantum Theory of Open System (Academic, New York, 1976).
41. E. B. Davies, "Markovian master equations," Commun. Math. Phys. 39, 91 (1974); Markovian master equations, II." Math. Ann. 219, 147 (1976). For an alternative rigorous approach, see J. V. Pule, "The Bloch Equations," Commun. Math. Phys. 38, 241 (1974). The traditional master equation technique does not specify the Markovian generator uniquely. See H. Spohn and R. Dümcke, "The proper form of the generator in the weak coupling limit. A," Physik. B 34, 419 (1979). This point is further discussed in Ref. 5.
42. E. B. Davies, "Markovian master equations," Commun. Math. Phys. 39, 91 (1974); Markovian master equations, II." Math. Ann. 219, 147 (1976). For an alternative rigorous approach, see J. V. Pule, "The Bloch Equations," Commun. Math. Phys. 38, 241 (1974). The traditional master equation technique does not specify the Markovian generator uniquely. See H. Spohn and R. Dümcke, "The proper form of the generator in the weak coupling limit. A," Physik. B 34, 419 (1979). This point is further discussed in Ref. 5.
43. E. B. Davies, "Markovian master equations," Commun. Math. Phys. 39, 91 (1974); Markovian master equations, II." Math. Ann. 219, 147 (1976). For an alternative rigorous approach, see J. V. Pule, "The Bloch Equations," Commun. Math. Phys. 38, 241 (1974). The traditional master equation technique does not specify the Markovian generator uniquely. See H. Spohn and R. Dümcke, "The proper form of the generator in the weak coupling limit. A," Physik. B 34, 419 (1979). This point is further discussed in Ref. 5.
44. E. B. Davies, "Markovian master equations," Commun. Math. Phys. 39, 91 (1974); Markovian master equations, II." Math. Ann. 219, 147 (1976). For an alternative rigorous approach, see J. V. Pule, "The Bloch Equations," Commun. Math. Phys. 38, 241 (1974). The traditional master equation technique does not specify the Markovian generator uniquely. See H. Spohn and R. Dümcke, "The proper form of the generator in the weak coupling limit. A," Physik. B 34, 419 (1979). This point is further discussed in Ref. 5.
45. L. van Hove, "Master equation and approach to equilibrium for quantum systems," in Fundamental problems in statistical mechanics, compiled by E. G. D. Cohen (North-Holland, Amsterdam, 1962).
46. H. Spohn and R. Dümcke, "Quantum tunneling with dissipation and Ising model over R," J. Stat. Phys. 41, 381 (1981); H. Spohn, "Ground state(s) of the spin-boson Hamiltonian," Commun. Math. Phys. 123, 277 (1989).
47. H. Spohn and R. Dümcke, "Quantum tunneling with dissipation and Ising model over R," J. Stat. Phys. 41, 381 (1981); H. Spohn, "Ground state(s) of the spin-boson Hamiltonian," Commun. Math. Phys. 123, 277 (1989).
48. 2/2 in Eq. (17)]. See A. Arai, "On a model of a harmonic oscillator coupled to a quantized, mass-less, scalar field I," J. Math. Phys. 22, 2539 (1981); "On a model of a harmonic oscillator coupled to a quantized, mass-less, scalar field II," J. Math. Phys. 22, 2549 (1981).
49. 2/2 in Eq. (17)]. See A. Arai, "On a model of a harmonic oscillator coupled to a quantized, mass-less, scalar field I," J. Math. Phys. 22, 2539 (1981); "On a model of a harmonic oscillator coupled to a quantized, mass-less, scalar field II," J. Math. Phys. 22, 2549 (1981).
50. M. Hübner and H. Spohn, "Radiative decay Non-perturbative approaches," Rev. Math. Phys. 7, 363 (1995).
51. C. Gérard, "Asymptotic completeness for the spin-boson model with a particle number cutoff," Rev. Math. Phys. 8, 549 (1996).
52. The natural choice for the set ℰ are the finite particle wave functions localized in the position representation.
53. H. Cycon, R. Froese, W. Kirsch, and B. Simon, Schrodinger Operators (Springer, New York, 1987).
54. This technique was introduced in J. Aguilar and J. M. Combes, "A class of analytic perturbations for one-body Schrödinger Hamiltonians," Commun. Math. Phys. 22, 269 (1971). The technique was used to study the resonances of N-body quantum system in B. Simon, "Resonances in N-body quantum systems with dilation analytic potentials and the foundations of time perturbation theory," Ann. Math. 97, 247 (1973). The Aguilar-Combes technique was applied to non-relativistic QED for the first time in T. Okamoto and K. Yajima, "Complex scaling technique in non-relativistic massive QED," Ann. Inst. H. Poincaré 42, 31 (1985).
55. This technique was introduced in J. Aguilar and J. M. Combes, "A class of analytic perturbations for one-body Schrödinger Hamiltonians," Commun. Math. Phys. 22, 269 (1971). The technique was used to study the resonances of N-body quantum system in B. Simon, "Resonances in N-body quantum systems with dilation analytic potentials and the foundations of time perturbation theory," Ann. Math. 97, 247 (1973). The Aguilar-Combes technique was applied to non-relativistic QED for the first time in T. Okamoto and K. Yajima, "Complex scaling technique in non-relativistic massive QED," Ann. Inst. H. Poincaré 42, 31 (1985).
56. This technique was introduced in J. Aguilar and J. M. Combes, "A class of analytic perturbations for one-body Schrödinger Hamiltonians," Commun. Math. Phys. 22, 269 (1971). The technique was used to study the resonances of N-body quantum system in B. Simon, "Resonances in N-body quantum systems with dilation analytic potentials and the foundations of time perturbation theory," Ann. Math. 97, 247 (1973). The Aguilar-Combes technique was applied to non-relativistic QED for the first time in T. Okamoto and K. Yajima, "Complex scaling technique in non-relativistic massive QED," Ann. Inst. H. Poincaré 42, 31 (1985).
57. M. Hübner and H. Spohn, "Spectral properties of the spin-boson Hamiltonian," Ann. Inst. H. Poincaré 62, 289 (1996).
58. V. Bach, J. Fröhlich, and I. M. Sigal, "Mathematical theory of non-relativistic matter and radiation," Lett. Math. Phys. 34, 183 (1995).
59. W. Thirring, Quantum Mechanics of Large Systems (Springer, New York, 1980).
60. This representation was explicitly identified in H. Araki and E. J. Woods, "Representation of the canonical commutation relations describing a non-relativistic infinite free Bose gas," J. Math. Phys. 4, 637 (1963).
61. ℬ as the algebra of observables of the reservoir at inverse temperature β.
62. The fundamental "spin-statistic" postulate of quantum mechanics requires the wave function of the reservoir to be completely symmetric with respect to the exchange of any two phonons.
63. A self-contained account of this theory from the standpoint of mathematical physics is H. Araki, "Positive cone. Radon-Nikodym theorems, relative Hamiltonian and the Gibbs condition in statistical mechanics. An application of the Tomita-Takesaki theory," in C*-algebras and Their Applications to Statistical Mechanics and Quantum Field Theory, edited by D. Kastler (North Holland, Amsterdam, 1976).
64. A state ω on a C*-algebra is faithful if, for any A ∈. ω(A*A) = 0 implies A = 0. A simple calculation shows that finite volume Gibbs states share this property. It survives the thermodynamic limit, and is a general feature of thermal equilibrium states at positive temperature (Ref. 47). Faithfulness of a state ω has far-reaching consequences for the associated cyclic representation (ℋ,π,Ω) since, in this case, the vector Ω is not only cyclic for π( ), but also for its commutant π( )′, i.e., π( )′Ω is dense in ℋ.
65. In classical as well as in quantum mechanics, the lack of faithfulness of the ground state is the fundamental obstruction to the spectral characterization of return to equilibrium at zero-temperature.
66. A nontechnical introduction to algebraic statistical mechanics is R. Haag, Local Quantum Physics (Springer, New York, 1993). For additional information, see O. Bratelli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics, Volumes I and II (Springer, New York, 1979).
67. A nontechnical introduction to algebraic statistical mechanics is R. Haag, Local Quantum Physics (Springer, New York, 1993). For additional information, see O. Bratelli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics, Volumes I and II (Springer, New York, 1979).
68. M. Fannes, B. Nachtergaele, and A. Verbeure, "The equilibrium states of the spin-boson model," Commun. Math. Phys. 114, 537 (1988).
69. β.
70. β).
71. In a more mathematical perspective, this construction shows that the KMS state (4.25) exists.
72. 3k which counts the number of physical phonons relative to the equilibrium state.
73. -1|(Ψ, script T sign(θ)Ψ)|.