Bose-Einstein condensation, fluctuations, and recurrence relations in statistical mechanics

American Journal of Physics

Volumn 71, Issue 7, 2003, Pages 661-669

  Mullin, W J ^a   Fernandez J P ^a  

^a UNIVERSITY OF MASSACHUSETTS   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0042591802     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1544520     Document Type: Article

References (71)

1. M. H. Andersen, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, "Observation of Bose-Einstein condensation in a dilute atomic vapor," Science 269, 198-201 (1995); K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, "Bose-Einstein condensation in a gas of sodium atoms," Phys. Rev. Lett. 75, 3969-3973 (1995): C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, "Bose-Einstein condensation of lithium: Observation of limited condensate number," ibid. 78, 985-989 (1997).
2. M. H. Andersen, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, "Observation of Bose-Einstein condensation in a dilute atomic vapor," Science 269, 198-201 (1995); K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, "Bose-Einstein condensation in a gas of sodium atoms," Phys. Rev. Lett. 75, 3969-3973 (1995): C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, "Bose-Einstein condensation of lithium: Observation of limited condensate number," ibid. 78, 985-989 (1997).
3. M. H. Andersen, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, "Observation of Bose-Einstein condensation in a dilute atomic vapor," Science 269, 198-201 (1995); K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, "Bose-Einstein condensation in a gas of sodium atoms," Phys. Rev. Lett. 75, 3969-3973 (1995): C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, "Bose-Einstein condensation of lithium: Observation of limited condensate number," ibid. 78, 985-989 (1997).
4. For a review, see C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2002).
5. W. J. Mullin, "Bose-Einstein condensation in a harmonic potential," J. Low Temp. Phys. 106, 615-641 (1997).
6. K. Damle, T. Senthil, S. N. Majumdar, and S. Sachdev, "Phase transition of a Bose gas in a harmonic potential," Europhys. Lett. 36, 6-12 (1996).
7. R. P. Feynman, Statistical Mechanics (W. A. Benjamin, Reading, MA, 1972).
8. M. Toda, R. Kubo, and N. Saitô, Statistical Physics I (Springer-Verlag, Berlin, 1983)
9. For example, Ref. 5 or K. Huang, Statistical Mechanics, 2nd ed. (Wiley, New York, 1987).
10. R London, Superfluids (Dover, New York, 1964), Vol. II, p. 203.
11. W. Ketterle and N. J. van Druten, "Bose-Einstein condensation of a finite number of particles trapped in one or three dimensions," Phys. Rev. A 54, 656-660 (1996); N. J. van Druten and W. Ketterle, "Two-step condensation of the ideal Bose gas in highly anisotropic traps," Phys. Rev. Lett. 79, 549-552 (1997).
12. W. Ketterle and N. J. van Druten, "Bose-Einstein condensation of a finite number of particles trapped in one or three dimensions," Phys. Rev. A 54, 656-660 (1996); N. J. van Druten and W. Ketterle, "Two-step condensation of the ideal Bose gas in highly anisotropic traps," Phys. Rev. Lett. 79, 549-552 (1997).
13. F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, "Quasipure Bose-Einstein condensate immersed in a Fermi sea," Phys. Rev. Lett. 87, 080403-1-080403-4 (2001).
14. A. Görlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, "Realization of Bose-Einstein condensates in lower dimensions," Phys. Rev. Lett. 87, 130402-4 (2001).
15. D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov, "Bose-Einstein condensation in quasi-2D trapped gases," Phys. Rev. Lett. 84, 2551-2555 (2000).
16. D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, "Regimes of quantum degeneracy in trapped 1D gases," Phys. Rev. Lett. 85, 3745-3749 (2000).
17. N. V. Prokof'ev, O. A. Ruebenacker, and B. V. Svistunov, "Critical point of a weakly interacting two-dimensional Bose gas," Phys. Rev. Lett. 87, 270402-1-270402-4 (2001).
18. I. Fujiwara, D. ter Haar, and H. Wergeland, "Fluctuations in the population of the ground state of Bose systems," J. Stat. Phys. 2, 329-347 (1970).
19. R. M. Ziff, G. E. Uhlenbeck, and M. Kac, "The ideal gas revisited," Phys. Rep. 32, 169-248 (1977).
20. S. Grossmann and M. Holthaus, "Microcanonical fluctuations of Bose system's ground state occupation number," Phys. Rev. E 54, 3495-3498 (1996).
21. S. Grossmann and M. Holthaus, "Maxwell's demon at work: two types of Bose condensate fluctuations in power-law traps," Opt. Express 1, 262-271 (1997); "Fluctuations of the particle number in a trapped Bose-Einstein condensate," Phys. Rev. Lett. 79, 3557-3560 (1997).
22. S. Grossmann and M. Holthaus, "Maxwell's demon at work: two types of Bose condensate fluctuations in power-law traps," Opt. Express 1, 262-271 (1997); "Fluctuations of the particle number in a trapped Bose-Einstein condensate," Phys. Rev. Lett. 79, 3557-3560 (1997).
23. S. Grossmann and M. Holthaus, "From number theory to statistical mechanics; Bose-Einstein condensation in isolated traps," Proc. Heraeus-Sem. 1997, Chaos, Solitons Fractals 10, 795-804 (1999).
24. H. D. Politzer, "Condensate fluctuations of a trapped, ideal Bose gas," Phys. Rev. A 54, 5048-5054 (1996).
25. P. Navez, D. Bitouk, M. Gajda, Z. Idiaszek, and K. Rza̧żewski, "Fourth statistical ensemble for the Bose-Einstein condensate," Phys. Rev. Lett. 79, 1789-1792 (1997).
26. C. Weiss and M. Wilkens, "Particle number counting statistics in ideal Bose gases," Opt. Express 1, 272-283 (1997).
27. M. Wilkens and C. Weiss, "Particle number fluctuations in an ideal Bose gas," J. Mod. Opt. 44, 1801-1814 (1997).
28. M. Gajda and K. Rza̧żewski, "Fluctuations of the Bose-Einstein condensate," Phys. Rev. Lett. 78, 2686-2689 (1997).
29. L. Lemmens, F. Brosens, and J. Devreese, "Statistical mechanics and path integrals for a finite number of bosons," Solid State Commun. 109, 615-620 (1999).
30. Z. Idiaszek, M. Gajda, P. Navez, M. Wilkens, and K. Rza̧żewski, "Fluctuations of the weakly interacting Bose-Einstein condensate," Phys. Rev. Lett. 82, 4376-4379 (1999).
31. F. Illuminati, P. Navez, and M. Wilkens, "Thermodynamic identities and particle number fluctuations in weakly interacting Bose-Einstein condensates," J. Phys. B 32, L461-L464 (1999).
32. D. ter Haar, Elements of Statistical Mechanics (Rinehart, New York, 1954).
33. F. C. Auluck and D. S. Kothari, "Statistical mechanics and the partitions of numbers," Proc. Cambridge Philos. Soc. 42, 272-277 (1946).
34. H. N. V. Temperley, "Statistical mechanics and the partition of numbers I. The transition of liquid helium," Proc. R. Soc. London, Ser. A 199, 361-375 (1949).
35. V. S. Nanda, "Bose-Einstein condensation and the partition theory of numbers," Proc. Natl. Inst. Sci. (India) 19, 681-690 (1953).
36. K. Schönhammer and V. Meden, "Fermion-boson transmutation and comparison of statistical ensembles in one dimension," Am. J. Phys. 64, 1168-1176 (1996).
37. G. E. Andrews, The Theory of Partitions (Addison-Wesley, Reading, MA, 1976).
38. F. Brosens, J. T. Devreese, and L. F. Lemmens, "Canonical Bose-Einstein condensation in a parabolic well," Solid State Commun. 100, 123-127 (1996).
39. F. Brosens, J. T. Devreese, and L. F. Lemmens, "Thermodynamics of coupled identical oscillators within the path-integral formalism," Phys. Rev. E 55, 227-236 (1997).
40. K. C. Chase, A. Z. Mekjian, and L. Zamick, "Canonical and microcanonical ensemble approaches to Bose-Einstein condensation: The thermodynamics of particles in harmonic traps," Eur. Phys. J. B 8, 281-285 (1999).
41. M. Toda, "On the relation between Fermions and Bosons," J. Phys. Soc. Jpn. 7, 230 (1952).
42. K. Schönhammer, "Thermodynamics and occupation numbers of a Fermi gas in the canonical ensemble," Am. J. Phys. 68, 1032-1037 (2000).
43. R. M. May, "Quantum statistics of ideal gases in two dimensions," Phys. Rev. 135, A1515-A1518 (1964).
44. R. Aldrovandi, "Two-dimensional quantum gases," Fortschr. Phys. 40, 631-649 (1992).
45. S. Viefers, F. Ravndal, and T. Haugset, "Ideal quantum gases in two dimensions," Am. J. Phys. 63, 369-376 (1995).
46. M. Lee, "Equivalence of ideal gases in two dimensions and Landen's relations," Phys. Rev. E 55, 1518-1520 (1997).
47. H. Schmidt. "A simple derivation of distribution functions for Bose and Fermi statistics," Am. J. Phys. 57, 1150-1151 (1989).
48. T. Sakai, "Gibbs' canonical ensemble and the distribution law in statistical mechanics," Proc. Phys. Math. Soc. Jpn. 22, 193-207 (1940).
49. F. Ansbacher and W. Ehrenberg, "The derivation of statistical expressions from Gibbs' canonical ensemble," Philos. Mag. 40, 626-631 (1949).
50. P. T. Landsberg, Thermodynamics (Interscience, New York, 1961).
51. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965). Section 9.3 presents a somewhat similar derivation of the distribution functions via the canonical ensemble.
52. P. Borrmann and G. Franke, "Recursion formulas for quantum statistical partition functions," J. Chem. Phys. 98, 2484-2485 (1993).
53. J. Tempere and J. T. Devreese, "Canonical Bose-Einstein condensation of interacting bosons in two dimensions," Solid State Commun. 101, 657-659 (1997).
54. F. Brosens, J. T. Devreese, and L. F. Lemmens, "Correlations in a confined gas of harmonically interacting spin-polarized fermions," Phys. Rev. E 58, 1634-1643 (1998).
55. B. Eckhardt, "Eigenvalue statistics in quantum ideal gases," in Emerging Applications in Number Theory, edited by D. Hejkal et al. (Springer, New York, 1997), pp. 187-199.
56. P. Borrmann, J. Hartig, O. Mülken, and E. R. Hilf, "Calculation of thermodynamic properties of finite Bose-Einstein systems," Phys. Rev. A 60, 1519-1521 (1999).
57. M. Holzmann and W. Krauth, "Transition temperature of the homogeneous weakly interacting Bose gas," Phys. Rev. Lett. 83, 2687-2690 (1999).
58. S. Pratt, "Canonical and microcanonical calculations for Fermi systems," Phys. Rev. Lett. 84, 4255-4259 (2000).
59. F. Philippe, J. Arnaud, and L. Chusseau, "Statistics of non-interacting bosons and fermions in microcanonical, canonical, and grand-canonical ensembles: A survey," math-ph/0211029.
60. D. I. Ford, "A note on the partition function for systems of independent particles," Am. J. Phys. 39, 215-230 (1971).
61. H.-J. Schmidt and J. Schnack, "Partition functions and symmetric polynomials," Am. J. Phys. 70, 53-57 (2002); "Symmetric polynomials in physics," cond-mat/0209397.
62. H.-J. Schmidt and J. Schnack, "Partition functions and symmetric polynomials," Am. J. Phys. 70, 53-57 (2002); "Symmetric polynomials in physics," cond-mat/0209397.
63. T. Matsubara, "Quantum-statistical theory of liquid Helium," Prog. Theor. Phys. 6, 714-730 (1951).
64. W. J. Mullin, "The loop-gas approach to Bose-Einstein condensation for trapped particles," Am. J. Phys. 68, 120-128 (2000); "Permutation cycles in the Bose-Einstein condensation of a trapped gas," Physica B 284-288, 7-8 (2000).
65. W. J. Mullin, "The loop-gas approach to Bose-Einstein condensation for trapped particles," Am. J. Phys. 68, 120-128 (2000); "Permutation cycles in the Bose-Einstein condensation of a trapped gas," Physica B 284-288, 7-8 (2000).
66. P. Grüter, D. M. Ceperley, and F. Laloë, "Critical temperature of Bose-Einstein condensation of hard-sphere gases," Phys. Rev. Lett. 79, 3549-3552 (1997).
67. M. Holzmann, P. Grüter, and F. Laloë, "Bose-Einstein condensation in interacting gases," Eur. Phys. J. B 10, 739-760 (1999).
68. W. Krauth, "Quantum Monte Carlo calculations for a large number of bosons in a harmonic trap," Phys. Rev. Lett. 77, 3695-3698 (1996).
69. S. Heinrichs and W. J. Mullin, "Quantum Monte Carlo calculations for bosons in a two-dimensional harmonic trap," J. Low Temp. Phys. 113, 231-236 (1998); erratum 114, 571 (1999).
70. W. J. Mullin, S. Heinrichs, and ). P. Fernández, "The condensate number in PIMC treatments of trapped bosons," Physica B 284-288, 9-10 (2000).
71. An exception to the statement that recurrence relations hold only for ideal systems is the harmonically interacting system studied in Refs. 35, 49, and 50.