Magnetic fluctuations in the classical XY model: The origin of an exponential tail in a complex system

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 63, Issue 4 I, 2001, Pages 411061-4110622

  Bramwell, S T ^a   Fortin, J Y ^b,c   Holdsworth, P C W ^d   Peysson, S ^d   Pinton, J F ^d   Portelli, B ^d   Sellitto, M ^d  

^a UNIVERSITY COLLEGE LONDON   (United Kingdom)

^b UNIVERSITY OF WASHINGTON   (United States)

^c CNRS   (France)

^d UNIVERSITÉ DE LYON   (France)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; COMPUTER SIMULATION; EQUATIONS OF MOTION; FAST FOURIER TRANSFORMS; INTEGRATION; INTERFACES (MATERIALS); LINEARIZATION; LOW TEMPERATURE EFFECTS; MAGNETIC FIELD EFFECTS; MATHEMATICAL MODELS; MOLECULAR DYNAMICS; MONTE CARLO METHODS; PHASE EQUILIBRIA; PROBABILITY DENSITY FUNCTION;

MAGNETIC FLUCTUATIONS;

CONTINUUM MECHANICS;

EID: 0035304324     PISSN: 15393755     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (75)

1. A quote attributed to Poincaré, on the ubiquity of the normal distribution, is "everyone believes in it: experimentalists believing that it is a mathematical theorem, mathematicians believing it is an empirical fact." For a critical discussion, see B. de Finetti, Theory of Probability (Wiley, New York, 1975).
2. L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon, Oxford, 1980), Vol. 1.
3. M. Cassandro and G. Jona-Lasinio, Adv. Phys. 27, 913 (1978).
4. C. Garrod, Statistical Mechanics and Thermodynamics (Oxford University Press, Oxford, 1995).
5. K. G. Wilson and J. Kogut, Phys. Rep. 12, 75 (1974).
6. A. D. Bruce, J. Phys. C 14, 3667 (1981).
7. K. Binder, Z. Phys. B: Condens. Matter 43, 119 (1981).
8. K. Binder, in Computational Methods in Field Theory, edited by H. Gauslever and C. B. Lang (Springer-Verlag, Berlin, 1992).
9. R. Botet and M. Ploszajczak, Phys. Rev. E 62, 1825 (2000).
10. R. Labbé, J.-F. Pinton, and S. Fauve, J. Phys. II 6, 1099 (1996).
11. S. T. Bramwell, P. C. W. Holdsworth, and J.-F. Pinton, Nature (London) 396, 552 (1998).
12. J.-F. Pinton, P. C. W. Holdsworth, R. Labbé, S. T. Bramwell, and J.-Y. Fortin, Advances in Turbulence VIII, edited by C. Dopazo et al. (CIMNE, Barcelona, 2000).
13. J-F. Pinton, P. C. W. Holdsworth, and R. Labbé. Phys. Rev. E 60, R2452 (1999).
14. P. Archambault, S. T. Bramwell, and P. C. W. Holdsworth, J. Phys. A 30, 8363 (1997).
15. P. Archambault, S. T. Bramwell, J.-Y. Fortin, P. C. W. Holdsworth, S. Peysson, and J.-F. Pinton, J. Appl. Phys. 83, 7234 (1998).
16. V. L. Berezinskii, Zh. Éksp. Teor. Fiz. 59, 907 (1970) [Sov. Phys. JETP 32, 493 (1971)].
17. V. L. Berezinskii, Zh. Éksp. Teor. Fiz. 59, 907 (1970) [Sov. Phys. JETP 32, 493 (1971)].
18. J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973).
19. J. M. Kosterlitz, J. Phys. C 7, 1046 (1974).
20. J. Villain, J. Phys. (Paris) 38, 581 (1975).
21. J. V. José, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Phys. Rev. B 16, 1217 (1977).
22. S. T. Bramwell, K. Christensen, J.-Y. Fortin, P. C. W. Holdsworth, H. J. Jensen, S. Lise, J. López, M. Nicodemi, J.-F. Pinton, and M. Sellitto, Phys. Rev. Lett. 84, 3744 (2000).
23. J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman, The Theory of Critical Phenomena (Clarendon, Oxford, 1992).
24. J. L. Cardy, Scaling and Renormalization Group (Cambridge University Press, Cambridge, 1996).
25. Z. Ráz and M. Plischke, Phys. Rev. E 50, 3530 (1994).
26. N. D. Mermin, Phys. Rev. 176, 250 (1968).
27. J. Tobochnik and G. V. Chester, Phys. Rev. B 20, 3761 (1979).
28. S. F. Edwards and D. R. Wilkinson, Proc. R. Soc. London, Ser. A 381, 17 (1982).
29. G. Foltin, K. Oerding, Z. Rácz, R. L. Workman, and R. K. P. Zia, Phys. Rev. E 50, 639 (1994).
30. M. Plischke, Z. Rácz, and R. K. P. Zia, Phys. Rev. E 50, 3589 (1994).
31. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon, Oxford, 1987).
32. A. Pimpinelli and J. Villain, Physics of Crystal Growth (Cambridge University Press, Cambridge, 1999).
33. E. J. Gumbel, Statistics of Extremes (Columbia University Press, New York, 1958).
34. J.-P. Bouchaud and M. Mézard, J. Phys. A 30, 7997 (1997).
35. S. C. Chapman, G. Rowlands, and N. W. Watkins, e-print cond-mat/0007275.
36. See, for example, T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940); V. G. Vaks, A. I. Larkin, and S. A. Pikin, Zh. Éksp. Teor. Fiz. 53, 1089 (1967) [Sov. Phys. JETP 26, 647 (1968)]; J. L. Lebowitz and O. Penrose, Phys. Rev. Lett. 35, 549 (1975); G. F. Mazenko, Phys. Rev. B 14, 3933 (1976); P. W. Mitchell, R. A. Cowley, and R. Pynn, J. Phys. C 17, L875 (1984).
37. See, for example, T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940); V. G. Vaks, A. I. Larkin, and S. A. Pikin, Zh. Éksp. Teor. Fiz. 53, 1089 (1967) [Sov. Phys. JETP 26, 647 (1968)]; J. L. Lebowitz and O. Penrose, Phys. Rev. Lett. 35, 549 (1975); G. F. Mazenko, Phys. Rev. B 14, 3933 (1976); P. W. Mitchell, R. A. Cowley, and R. Pynn, J. Phys. C 17, L875 (1984).
38. See, for example, T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940); V. G. Vaks, A. I. Larkin, and S. A. Pikin, Zh. Éksp. Teor. Fiz. 53, 1089 (1967) [Sov. Phys. JETP 26, 647 (1968)]; J. L. Lebowitz and O. Penrose, Phys. Rev. Lett. 35, 549 (1975); G. F. Mazenko, Phys. Rev. B 14, 3933 (1976); P. W. Mitchell, R. A. Cowley, and R. Pynn, J. Phys. C 17, L875 (1984).
39. See, for example, T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940); V. G. Vaks, A. I. Larkin, and S. A. Pikin, Zh. Éksp. Teor. Fiz. 53, 1089 (1967) [Sov. Phys. JETP 26, 647 (1968)]; J. L. Lebowitz and O. Penrose, Phys. Rev. Lett. 35, 549 (1975); G. F. Mazenko, Phys. Rev. B 14, 3933 (1976); P. W. Mitchell, R. A. Cowley, and R. Pynn, J. Phys. C 17, L875 (1984).
40. See, for example, T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940); V. G. Vaks, A. I. Larkin, and S. A. Pikin, Zh. Éksp. Teor. Fiz. 53, 1089 (1967) [Sov. Phys. JETP 26, 647 (1968)]; J. L. Lebowitz and O. Penrose, Phys. Rev. Lett. 35, 549 (1975); G. F. Mazenko, Phys. Rev. B 14, 3933 (1976); P. W. Mitchell, R. A. Cowley, and R. Pynn, J. Phys. C 17, L875 (1984).
41. See, for example, T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940); V. G. Vaks, A. I. Larkin, and S. A. Pikin, Zh. Éksp. Teor. Fiz. 53, 1089 (1967) [Sov. Phys. JETP 26, 647 (1968)]; J. L. Lebowitz and O. Penrose, Phys. Rev. Lett. 35, 549 (1975); G. F. Mazenko, Phys. Rev. B 14, 3933 (1976); P. W. Mitchell, R. A. Cowley, and R. Pynn, J. Phys. C 17, L875 (1984).
42. W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1971).
43. The constant C is often quoted in the literature with a value C = 2 [20,26,14]. The more precise calculation here (Appendix C) leads to the value C= 1.8456.
44. In the context of intermittency in turbulence, this feature has led to analytical calculation of anomalous exponents. See K. Gawedzki and A. Kupiainen, Phys. Rev. Lett. 75, 3834 (1995); A. Pumir, Europhys. Lett. 34, 25 (1996); M. Vergassola, Phys. Rev. E 53, R3021 (1996).
45. In the context of intermittency in turbulence, this feature has led to analytical calculation of anomalous exponents. See K. Gawedzki and A. Kupiainen, Phys. Rev. Lett. 75, 3834 (1995); A. Pumir, Europhys. Lett. 34, 25 (1996); M. Vergassola, Phys. Rev. E 53, R3021 (1996).
46. In the context of intermittency in turbulence, this feature has led to analytical calculation of anomalous exponents. See K. Gawedzki and A. Kupiainen, Phys. Rev. Lett. 75, 3834 (1995); A. Pumir, Europhys. Lett. 34, 25 (1996); M. Vergassola, Phys. Rev. E 53, R3021 (1996).
47. S. Peysson (unpublished).
48. K. Bury, Statistical Distributions in Engineering (Cambridge University Press, Cambridge, 1999).
49. X. Leoncini, A. D. Verga, and S. Ruffo, Phys. Rev. E 57, 6377 (1998).
50. H. Weber and H. J. Jensen, Phys. Rev. B 44, 454 (1991).
51. M. Sellitto and P. C. W. Holdsworth (unpublished).
52. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes (Cambridge University Press, Cambridge, 1997).
53. G. Palma, T. Mayer, and R. Labbé, e-print cond-mat/0007289.
54. B. Derrida and C. Appert, J. Stat. Phys. 94, 1 (1999).
55. M. Prähofer and H. Spohn, Physica A 279, 342 (2000).
56. K. Pearson, Philos. Trans. R. Soc. London, Ser. A 186, 343 (1892); 216, 429 (1916).
57. K. Pearson, Philos. Trans. R. Soc. London, Ser. A 186, 343 (1892); 216, 429 (1916).
58. W. M. Smart, Combination of Observations (Cambridge University Press, Cambridge, 1958).
59. We note that we have made a similar change of variables for some of the data shown in Ref. [21] in order to have the exponential tail systematically on the left-hand side of the figure. The SOC data for the forest fire, Bak-Tang-Wiesenfeld and Sneppen models, and the correlated extremal data are treated this way.
60. This analysis gives the mathematical basis for "fitting an elephant" with four parameters.
61. B. Portelli and P. C. W. Holdsworth (unpublished).
62. U. Frisch, Turbulence (Cambridge University Press, Cambridge, 1995).
63. N. Goldenfeld and L. P. Kadanoff, Science 284, 87 (1999).
64. J. Harte, A. Kinzig, and J. Green, Science 284, 334 (1999).
65. S. Aumaite, S. Fauve, and J.-F. Pinton, Eur. Phys. J. B 16, 563 (2000).
66. F. Wegner, Z. Phys. 206, 465 (1967).
67. N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, Reading, MA, 1992).
68. A. I. Khinchin, Mathematical Foundations of Statistical Mechanics (Dover, New York, 1949).
69. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes (Oxford University Press, Oxford, 1992).
70. R. Von Mises, Mathematical Theory of Probability and Statistics (Academic, London, 1964).
71. G. H. Wannier, Statistical Physics (Dover, New York, 1987).
72. R. S. Ellis, Entropy, Large Deviations, and Statistical Mechanics (Springer-Verlag, Berlin, 1985).
73. J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 128 (1990).
74. F. W. Olver, Asymptotics and Special Functions (AKP Classics, New York, 1997).
75. T. Antal and Z. Ráez, Phys. Rev. E 54, 2256 (1996).