Abstract
The renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann’s 1929 article on the quantum ergodic theorem. We have found this almost forgotten article, which until now has been available only in German, to be a treasure chest, and to be much misunderstood. In it, von Neumann studied the long-time behavior of macroscopic quantum systems. While one of the two theorems announced in his title, the one he calls the “quantum H-theorem”, is actually a much weaker statement than Boltzmann’s classical H-theorem, the other theorem, which he calls the “quantum ergodic theorem”, is a beautiful and very non-trivial result. It expresses a fact we call “normal typicality” and can be summarized as follows: for a “typical” finite family of commuting macroscopic observables, every initial wave function ψ0 from a micro-canonical energy shell so evolves that for most times t in the long run, the joint probability distribution of these observables obtained from ψ t is close to their micro-canonical distribution.
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References
P.W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109, 1492 (1958)
V. Bach, J. Fröhlich, I.M. Sigal, Return to equilibrium, J. Math. Phys. 41, 3985 (2000)
C. Bartsch, J. Gemmer, Dynamical typicality of quantum expectation values, Phys. Rev. Lett. 102, 110403 (2009)
G.D. Birkhoff, Proof of the ergodic theorem, Proceedings of the National Academy of Science USA 17, 656 (1931)
P. Bocchieri, A. Loinger, Ergodic Theorem in Quantum Mechanics, Phys. Rev. 111, 668 (1958)
P. Bocchieri, A. Loinger, Ergodic Foundation of Quantum Statistical Mechanics, Phys. Rev. 114, 948 (1959)
P. Bocchieri, G.M. Prosperi, Recent Developments in Quantum Ergodic Theory, in: Statistical Mechanics, Foundations and Applications, edited by T.A. Bak, Proceedings of the IUPAP Meeting, Copenhagen, 1966 (Benjamin, 1967)
L. Boltzmann, Vorlesungen über Gastheorie(Leipzig, Barth, 1896, 1898), 2 vols., English translation by S.G. Brush, Lectures on Gas Theory(Cambridge University Press, 1964)
J. Bourgain, A remark on the uncertainty principle for Hilbertian basis, J. Funct. Anal. 79, 136 (1988)
M.D. Choi, Almost commuting matrices need not be nearly commuting, Proceedings of the American Mathematical Society 102, 529 (1988)
W. De Roeck, C. Maes, K. Netočný, Quantum Macrostates, Equivalence of Ensembles and an H-theorem, J. Math. Phys. 47, 073303 (2006)
J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991)
A. Einstein, Beiträge zur Quantentheorie. Deutsche Physikalische Gesellschaft, Verhandlungen 16, 820 (1914)
I.E. Farquhar, Ergodic Theory in Statistical Mechanics(Interscience Publishers and John Wiley, 1964)
I.E. Farquhar, P.T. Landsberg, On the quantum-statistical ergodic and H-theorems, Proceedings of the Royal Society London A 239, 134 (1957)
M. Fierz, Ergodensatz in der Quantenmechanik, Helv. Phys. Acta 28, 705 (1955)
M. Fierz, Statistische Mechanik, in Theoretical Physics in the 20th Century: A Memorial Volume to Wolfgang Pauli, edited by M. Fierz, V.F. Weisskopf (Interscience, New York, 1960), pp. 161–186
S. Garnerone, T.R. d. Oliveira, P. Zanardi, Typicality in random matrix product states, Phys. Rev. A 81, 032336 (2010)
J. Gemmer, M. Michel, G. Mahler, Quantum Thermodynamics, Lecture Notes in Physics (Springer-Verlag, Berlin, 2004), Vol. 657
S. Goldstein, Boltzmann’s approach to statistical mechanics, in Chance in Physics: Foundations and Perspectives, edited by J. Bricmont, D. Dürr, M.C. Galavotti, G.C. Ghirardi, F. Petruccione, N. Zanghì, Lecture Notes in Physics 574(Springer-Verlag, Berlin, 2001), pp. 39–54, http://arxiv.org/abs/cond-mat/0105242
S. Goldstein, J.L. Lebowitz, On the (Boltzmann) Entropy of Nonequilibrium Systems, Physica D 193, 53 (2004)
S. Goldstein, J.L. Lebowitz, C. Mastrodonato, R. Tumulka, N. Zanghì, Normal Typicality and von Neumann’s Quantum Ergodic Theorem, to appear in Proceedings of the Royal Society London A(2010), http://arxiv.org/abs/0907.0108
S. Goldstein, J.L. Lebowitz, C. Mastrodonato, R. Tumulka, N. Zanghì, On the Approach to Thermal Equilibrium of Macroscopic Quantum Systems, Phys. Rev. E 81, 011109 (2010)
S. Goldstein, J.L. Lebowitz, R. Tumulka, N. Zanghì, Canonical Typicality, Phys. Rev. Lett. 96, 050403 (2006)
S. Goldstein, J.L. Lebowitz, R. Tumulka, N. Zanghì, On the Distribution of the Wave Function for Systems in Thermal Equilibrium, J. Statist. Phys. 125, 1193 (2006)
S. Goldstein, J.L. Lebowitz, R. Tumulka, N. Zanghì, Universal Probability Distribution for the Wave Function of an Open Quantum System, in preparation
M.B. Hastings, Making Almost Commuting Matrices Commute, Commun. Math. Phys. 291, 321 (2009)
M.B. Hastings, T.A. Loring, Almost Commuting Matrices, and Localized Wannier Functions, and the Quantum Hall Effect, J. Math. Phys. 51, 015214 (2010)
V. Jakšić, 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)
R. Jancel, Foundations of Classical and Quantum Statistical Mechanics(Pergamon, Oxford, 1969), Translation by W.E. Jones of R. Jancel: Les Fondements de la Mécanique Statistique Classique et Quantique(Gauthier-Villars, Paris, 1963)
E.C. Kemble, Fluctuations, Thermodynamic Equilibrium and Entropy, Phys. Rev. 56, 1013 (1939)
N.S. Krylov, Works on the foundations of statistical physics, with an afterword by Y. Sinai (University Press, Princeton, 1979)
L.D. Landau, E.M. Lifshitz, Statistical Physics(Course of theoretical physics) (Pergamon, London, 1958), Vol. 5
P.T. Landsberg, Pauli, an ergodic theorem and related matters, Am. J. Phys. 73, 119 (2005)
O.E. Lanford, Entropy and Equilibrium States in Classical Statistical Mechanics, in Lecture Notes in Physics 2, edited by A. Lenard (Springer-Verlag, Berlin, 1973), pp. 1–113
O.E. Lanford, Time evolution of large classical systems, in Lecture Notes in Physics 38, edited by J. Moser (Springer-Verlag, Berlin, 1975), pp. 1–111
J.L. Lebowitz, Microscopic Origins of Irreversible Macroscopic Behavior: An Overview, Physica A 263, 516 (1999)
J.L. Lebowitz, From Time-symmetric Microscopic Dynamics to Time-asymmetric Macroscopic Behavior: An Overview, in Boltzmann’s Legacy, edited by G. Gallavotti, W.L. Reiter, J. Yngvason (European Mathematical Society, Zürich, 2008), pp. 63–88, http://arxiv.org/abs/0709.0724
J.L. Lebowitz, C. Maes, Entropy – A Dialogue, in On Entropy, edited by A. Greven, G. Keller, G. Warnecke (University Press, Princeton, 2003), pp. 269–273
J.L. Lebowitz, H. Spohn, Irreversible Thermodynamics for Quantum Systems Weakly Coupled to Thermal Reservoirs, Adv. Chem. Phys. 38, 109 (1978)
H. Lin, Almost commuting self-adjoint matrices and applications, Fields Institute Communications 13, 193 (1997)
N. Linden, S. Popescu, A.J. Short, A. Winter, Quantum mechanical evolution towards thermal equilibrium, Phys. Rev. E 79, 061103 (2009)
G. Ludwig, Zum Ergodensatz und zum Begriff der makroskopischen Observablen. I, Z. Phys. 150, 346 (1958)
G. Ludwig, Zum Ergodensatz und zum Begriff der makroskopischen Observablen. II, Z. Phys. 152, 98 (1958)
G. Ludwig, Axiomatic Quantum Statistics of Macroscopic Systems (Ergodic Theory), in Physics School Enrico Fermi: Ergodic Theory(Academic Press, 1962), pp. 57–132
M.C. Mackey, The dynamic origin of increasing entropy, Rev. Mod. Phys. 61, 981 (1989)
Normal number, in Wikipedia, the free encyclopedia(accessed December 15, 2009), http://en.wikipedia.org/wiki/Normal_number
V. Oganesyan, D.A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007)
W. Pauli, in a letter to M. Fierz, dated 9 August 1956, quoted from
W. Pauli, M. Fierz, Über das H-Theorem in der Quantenmechanik, Z. Phys. 106, 572 (1937)
P. Pechukas, Sharpening an inequality in quantum ergodic theory, J. Math. Phys. 25, 532 (1984)
S. Popescu, A.J. Short, A. Winter, Entanglement and the foundation of statistical mechanics, Nature Phys. 2, 754 (2006)
G.M. Prosperi, A. Scotti, Ergodicity Conditions in Quantum Mechanics, J. Math. Phys. 1, 218 (1960)
P. Reimann, Typicality for Generalized Microcanonical Ensembles, Phys. Rev. Lett. 99, 160404 (2007)
P. Reimann, Foundation of Statistical Mechanics under Experimentally Realistic Conditions, Phys. Rev. Lett. 101, 190403 (2008)
P. Reimann, Typicality of pure states randomly sampled according to the Gaussian adjusted projected measure, J. Stat. Phys. 132, 921 (2008)
M. Rigol, V. Dunjko, M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452, 854 (2008)
D.W. Robinson, Return to equilibrium, Commun. Math. Phys. 31, 171 (1973)
D. Ruelle, What physical quantities make sense in non-equilibrium statistical mechanics? In Boltzmann’s Legacy, edited by G. Gallavotti, W.L. Reiter, J. Yngvason (European Mathematical Society, Zürich, 2008), pp. 89–97
E. Schrödinger, in a letter to J. von Neumann, dated 25 December 1929, published in E. Schrödinger, Eine Entdeckung von ganz außerordentlicher Tragweite, Schrödingers Briefwechsel zur Wellenmechanik und zum Katzenparadoxon,edited by K. von Meyenn (Springer, Berlin, 2010)
E. Schrödinger, Energieaustausch nach der Wellenmechanik. Annalen der Physik 83, 956 (1927), English translation by J.F. Shearer, W.M. Deans, The Exchange of Energy according to Wave Mechanics, in E. Schrödinger: Collected Papers on Wave Mechanics, Providence, R.I.: AMS Chelsea (1982), pp. 137–146
A. Serafini, O.C.O. Dahlsten, D. Gross, M.B. Plenio, Canonical and micro-canonical typical entanglement of continuous variable systems, J. Phys. A: Math. Theor. 40, 9551 (2007)
Y. Sinai, Introduction to Ergodic Theory, Translated from the Russian by V. Scheffer (University Press, Princeton, 1976)
M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994)
H. Tasaki, From Quantum Dynamics to the Canonical Distribution: General Picture and a Rigorous Example, Phys. Rev. Lett. 80, 1373 (1998)
R.C. Tolman, The Principles of Statistical Mechanics(University Press, Oxford, 1938)
B. Vacchini, K. Hornberger, Quantum linear Boltzmann equation, Phys. Rep. 478, 71 (2009)
L. van Hove, The approach to equilibrium in quantum statistics, Physica 23, 441 (1957)
L. van Hove, The Ergodic Behaviour of Quantum Many-Body Systems, Physica 25, 268 (1959)
J. von Neumann, Thermodynamik quantenmechanischer Gesamtheiten, Göttinger Nachrichten 273 (11 November 1927)
J. von Neumann, Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik, Z. Phys. 57, 30 (1929)
J. von Neumann, Proof of the Quasi-ergodic Hypothesis, Proceedings of the National Academy of Science USA 18, 70 (1932)
J. von Neumann, Mathematische Grundlagen der Quantenmechanik(Springer-Verlag, Berlin, 1932), English translation by R.T. Beyer, published as J. von Neumann, Mathematical Foundation of Quantum Mechanics(University Press, Princeton, 1955)
E.P. Wigner, Random Matrices in Physics, SIAM Rev. 9, 1 (1967)
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Goldstein, S., Lebowitz, J., Tumulka, R. et al. Long-time behavior of macroscopic quantum systems. EPJ H 35, 173–200 (2010). https://doi.org/10.1140/epjh/e2010-00007-7
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DOI: https://doi.org/10.1140/epjh/e2010-00007-7