Ergodic properties of a generic nonintegrable quantum many-body system in the thermodynamic limit

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 60, Issue 4, 1999, Pages 3949-3968

  Prosen, Tomaž ^a  

^a UNIVERSITY OF LJUBLJANA   (Slovenia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33646976074     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.60.3949     Document Type: Article

References (60)

1. I. Kornfeld, S. Fomin, and Ya. Sinai, Ergodic Theory (Springer, New York, 1982).
2. V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Benjamin, New York, 1968).
3. M. Henon, in Chaotic behaviour of Deterministic Systems, edited by G. Iooss, R.H.G. Helleman, and R. Stora, Les Houches 1981 Proceedings (North-Holland, Amsterdam, 1983).
4. G. Gallavotti and E.G.D. Cohen, Phys. Rev. Lett. 74, 2694 (1995)
5. G. GallavottiE.G.D. CohenJ. Stat. Phys. 80, 931 (1995).
6. E. Fermi, J. Pasta, S. Ulam, and M. Tsingou, Report No. LA-1940, 1955 (unpublished)
7. E. Fermi, Collected Papers (University of Chicago Press, Chicago, 1965), Vol. 2, p. 978.
8. S. Lepri, R. Livi, and A. Politi, Phys. Rev. Lett. 78, 1896 (1997)
9. S. LepriR. LiviA. PolitiPhysica D 119, 140 (1998).
10. D.L. Shepelyansky, Nonlinearity 10, 1331 (1997).
11. B.V. Chirikov and F.M. Izrailev, Dokl. Akad. Nauk SSSR 166, 57 (1967) [Sov. Phys. Dokl. 11, 30 (1966)]
12. B.V. Chirikov, F.M. Izrailev, and V.A. Tayursky, Comput. Phys. Commun. 5, 11 (1973).
13. We would like to demonstrate statistical laws with minimal external randomness. Hence we think of an isolated system left to evolve with no contact with thermal reservoirs of any kind, since we would like to inspect how nonlinear interaction alone can control the ergodic properties of dynamics.
14. For example, one may generally take (Formula presented) to be a a strong limit of trigonometric polynomials, (Formula presented), such as are the projection operators (Formula presented) according to the spectral theorem for unitary operators 11.
15. N. Dunford and J.T. Schwartz, Linear Operators, Part II: Spectral Theory (Interscience, New York, 1963).
16. M.L. Mehta, Random Matrices (Academic Press, London, 1991).
17. M. Gutzwiller, Chaos in Classical and Quantum Mechanics, (Springer-Verlag, New York, 1990)
18. F. Haake, Quantum Signatures of Chaos (Springer-Verlag, Berlin, 1991)
19. Quantum Chaos: Between Order and Disorder, edited by G. Casati and B. Chirikov (Cambridge University Press, Cambridge, 1995).
20. Chaos and Quantum Physics, edited by M.-J. Giannoni, A. Voros, and J. Zinn-Justin,Les Houches 1989 Proceedings (North-Holland, Amsterdam, 1991).
21. T. Prosen, Phys. Rev. E 60, 1658 (1999).
22. S. Graffi and A. Martinez, J. Math. Phys. 37, 5111 (1996).
23. M. Lenci, J. Math. Phys. 37, 5137 (1996).
24. G. Jona-Lasinio and C. Presilla, Phys. Rev. Lett. 77, 4322 (1996)
25. P. Castiglione, G. Jona-Lasinuo, and C. Presilla, J. Phys. A 29, 6169 (1996).
26. V.V. Flambaum, F. Izrailev, and G. Casati, Phys. Rev. E 54, 2136 (1996)
27. V.V. Flambaum, A.A. Gribakina, G.F. Gribakin, and I.V. Ponomarev, e-print, cond-mat/9711213.
28. G. Montambaux, D. Poilblanc, J. Bellissard, and C. Sire, Phys. Rev. Lett. 70, 497 (1993)
29. D. Poilblanc, T. Ziman, J. Bellissard, F. Mila, and G. Montambaux, Europhys. Lett. 22, 537 (1993)
30. T.C. Hsu and J.C. Angles d’Auriac, Phys. Rev. B 47, 14 291 (1993).
31. X. Zotos and P. Prelovšek, Phys. Rev. B 53, 983 (1996)
32. H. Castella, X. Zotos, and P. Prelovšek, Phys. Rev. Lett. 74, 972 (1995).
33. X. Zotos, F. Naef, and P. Prelovšek, Phys. Rev. B 55, 11 029 (1997).
34. R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).
35. P. Mazur, Physica (Amsterdam) 43, 533 (1969).
36. M. Suzuki, Physica (Amsterdam) 51, 277 (1971).
37. Precise conditions on the “generic” class of interactions are yet to be determined; here we show the conjecture for a particular specific example and quote few other results (Sec. VI). However, we should stress that we have in mind only purely deterministic systems and nonrandom interactions, so Hamiltonians (Formula presented) of infinite systems are supposed to be prescribed using a finite amount of information.
38. T. Prosen, Phys. Rev. Lett. 80, 1808 (1998).
39. V.E. Korepin, N. M. Bogoliubov, and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions (Cambridge University Press, Cambridge, 1993).
40. P. Jordan and E. Wigner, Z. Phys. 47, 631 (1928).
41. G. Casati, B.V. Chirikov, F. M. Izrailev, and J. Ford, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, edited by G. Casati and J. Ford, Lecture Notes in Physics Vol. 93 (Springer, New York, 1979).
42. R. Artuso, G. Casati, F. Borgonovi, L. Rebuzzini, and I. Guarneri, Int. J. Mod. Phys. B 8, 207 (1994).
43. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge University Press, Cambridge, 1992).
44. S. Winograd, Arithmetic Complexity of Computations (SIAM, Philadelphia, 1980).
45. O. Bohigas, M.-J. Giannoni, and C. Schmit, Phys. Rev. Lett. 52, 1 (1984).
46. O. Bohigas, in Chaos and Quantum Physics (Ref. 14).
47. A. V. Andreev, O. Agam, B.D. Simons, and B.L. Altshuler, Phys. Rev. Lett. 76, 3947 (1996).
48. M. V. Berry and M. Tabor, Proc. R. Soc. London, Ser. A 156, 375 (1977).
49. M.V. Berry and M. Robnik, J. Phys. A 17, 2413 (1984).
50. T. Prosen and M. Robnik, J. Phys. A 27, 8059 (1994).
51. E.B. Bogomolny, U. Gerland, and C. Schmit, Phys. Rev. E 59, R1315 (1999).
52. D. Braun, G. Montambaux, and M. Pascaud, e-print cond-mat/9712256.
53. M.V. Berry, Proc. R. Soc. London, Ser. A 400, 229 (1985).
54. S. Kirchner, H. G. Evertz, and W. Hanke, e-print, cond-mat/9804148.
55. S. Fujimoto and N. Kawakami, J. Phys. A 31, 465 (1998)
56. N.M.R. Peres, P.D. Sacramento, D.K. Campbell, and J.M.P. Carmelo, e-print, cond-mat/9804169.
57. A simpler and perhaps more powerful version of the method for autonomous Hamiltonian systems, which may be used also to give rigorous upper bounds on charge or spin stiffness, etc., will be presented elsewhere.
58. T. Prosen, J. Phys. A 31, L645 (1998).
59. T. Prosen, G. Usaj, and H. M. Pastawski (unpublished).
60. G. Usaj, H.M. Pastawski, and P.R. Levstein, e-print, cond-mat/9803047.