Normal typicality and von Neumann's quantum ergodic theorem

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 466, Issue 2123, 2010, Pages 3203-3224

  Goldstein, Sheldon ^a   Lebowitz, Joel L ^a   Mastrodonato, Christian ^b   Tumulka, Roderich ^a   Zanghí, Nino ^b  

^a RUTGERS UNIVERSITY   (United States)

^b INFM   (Italy)

Author keywords

Equilibration; Ergodicity in quantum statistical mechanics; Generic Hamiltonian; Macro state; Thermalization; Typical Hamiltonian

Indexed keywords

HAMILTONIANS; STATISTICAL MECHANICS;

EQUILIBRATION; ERGODICITY; GENERIC HAMILTONIAN; MACRO-STATE; THERMALIZATION; TYPICAL HAMILTONIAN;

QUANTUM THEORY;

EID: 78149297482     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2009.0635     Document Type: Conference Paper

References (32)

1. Bocchieri, P. & Loinger, A. 1958 Ergodic theorem in quantum mechanics. Phys. Rev. 111, 668-670. (doi:10.1103/PhysRev.111.668)
2. Bocchieri, P. & Loinger, A. 1959 Ergodic foundation of quantum statistical mechanics. Phys. Rev. 114, 948-951. (doi:10.1103/PhysRev.114.948)
3. Choi, M. D. 1988 Almost commuting matrices need not be nearly commuting. Proc. Am. Math. Soc. 102, 529-533. (doi:10.2307/2047216)
4. De Roeck, W., Maes, C. & Netočný, K. 2006 Quantum macrostates, equivalence of ensembles and an H-theorem. J. Math. Phys. 47, 073303. (doi:10.1063/1.2217810)
5. Farquhar, I. E. 1961 The present state of ergodic theory. Nature 190, 17-18. (doi:10.1038/190017a0)
6. Farquhar, I. E. 1964 Ergodic theory in statistical mechanics. New York, NY: Interscience Publishers and John Wiley.
7. Farquhar, I. E. & Landsberg, P. T. 1957 On the quantum-statistical ergodic and H-theorems. Proc. R. Soc. Lond. A 239, 134-144. (doi:10.1098/rspa. 1957.0027)
8. Fierz, M. 1955 Ergodensatz in der Quantenmechanik. Helvetica Phys. Acta 28, 705-715.
9. Gemmer, J., Michel, M. & Mahler, G. 2004 Quantum thermodynamics. Lecture Notes in Physics, no.657. Berlin, Germany: Springer-Verlag.
10. Goldstein, S., Lebowitz, J. L., Tumulka, R. & Zanghì, N. 2006 Canonical typicality. Phys. Rev. Lett. 96, 050403. (doi:10.1103/PhysRevLett.96. 050403)
11. Goldstein, S., Lebowitz, J. L., Mastrodonato, C., Tumulka, R. & Zanghì, N. 2010 On the approach to thermal equilibrium of macroscopic quantum systems. Phys. Rev. E 81, 011109. (http://arxiv.org/abs/0911.1724)
12. Goldstein, S., Lebowitz, J. L., Tumulka, R. & Zanghì, N. In press. Long-time behavior of macroscopic quantum systems: commentary accompanying the English translation of John von Neumann's 1929 article on the quantum ergodic theorem. Eur. Phys. J. H. (http://arxiv.org/ abs/1003.2129)
13. Jancel, R. 1969 Foundations of classical and quantum statistical mechanics. Oxford, UK: Pergamon. (Translation by Jones, W. E. 1963 Les Fondements de la Mécanique Statistique Classique e Quantique. Paris, France: Gauthier-Villars.)
14. Landsberg, P. T. 1961 Quantum statistical ergodic and H-theorems for incompletely specified systems. Proc. R. Soc. Lond. A 262, 100-109. (doi:10.1098/rspa.1961.0104)
15. Landsberg, P. T. 2005 Pauli, an ergodic theorem and related matters. Am. J. Phys. 73, 119-121. (doi:10.1119/1.1811622)
16. Linden, N., Popescu, S., Short, A. J. & Winter, A. 2009 Quantum mechanical evolution towards thermal equilibrium. Phys. Rev. E 79, 061103. (doi:10.1103/PhysRevE.79.061103)
17. Ludwig, G. 1958 Zum Ergodensatz und zum Begriff der makroskopischen Observablen. I. Zeitschrift für Physik 150, 346-374. (doi:10.1007/ BF01322328)
18. Normal number.2009 In Wikipedia, the free encyclopedia. See http://en.wikipedia.org/wiki/ Normal-number (accessed 4 June 2009).
19. Pauli, W. 1956 A letter to M. Fierz. 9 August 1956 (quoted from Landsberg (2005)).
20. Pechukas, Ph. 1984 Sharpening an inequality in quantum ergodic theory. J. Math. Phys. 25, 532. (doi:10.1063/1.526202)
21. Popescu, S., Short, A. J. & Winter, A. 2006 Entanglement and the foundation of statistical mechanics. Nat. Phys. 21, 754-758. (doi:10.1038/nphys444)
22. Prosperi, G. M. & Scotti, A. 1960 Ergodicity conditions in quantum mechanics. J. Math. Phys. 1, 218. (doi:10.1063/1.1703654)
23. Reimann, P. 2008 Foundation of statistical mechanics under experimentally realistic conditions. Phys. Rev. Lett. 101, 190403. (doi:10.1103/PhysRevLett. 101.190403)
24. Rigol, M., Dunjko, V. & Olshanii, M. 2008 Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854-858. (doi:10.1038/nature06838)
25. Schrödinger, E. 1927 Energieaustausch nach derWellenmechanik. Annalen der Physik 388, 956-968.
26. (English translation by Shearer, J. F. 1928 and 1982 The exchange of energy according to wave mechanics. In Collected papers on wave mechanics (ed. E. Schrödinger), pp. 137-146. Providence, RI: AMS Chelsea.)
27. Tasaki, H. 1998 From quantum dynamics to the canonical distribution: general picture and a rigorous example. Phys. Rev. Lett. 80, 1373-1376. (doi:10.1103/PhysRevLett.80.1373)
28. Toda, M., Kubo, R. & Saito, N. 1991 Statistical physics I. Berlin, Germany: Springer-Verlag.
29. van Hove, L. 1959 Ergodic behaviour of quantum many-body systems. Physica 25, 268-276. (doi:10.1016/S0031-8914(59)93062-9)
30. von Neumann, J. 1929 Beweis des Ergodensatzes und des H-theorems in der neuen Mechanik. Zeitschrift für Physik 57, 30-70.
31. (English translation by Tumulka, R. in von Neumann, J. In press. Proof of the ergodic theorem and the H-theorem in quantum mechanics. Eur. Phys. J. H (http://arxiv.org/abs/1003.2133)).
32. Wigner, E. P. 1967 Random matrices in physics. SIAM Rev. 9, 1-23. (doi:10.1137/1009001)