Interpretation of the nonextensivity parameter q in some applications of Tsallis statistics and Levy distributions

Physical Review Letters

Volumn 84, Issue 13, 2000, Pages

  Wilk, G ^a   Wlodarczyk Z ^b  

^a SOLTAN INSTITUTE FOR NUCLEAR STUDIES   (Poland)

^b JAN KOCHANOWSKI UNIVERSITY   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

BOLTZMANN-GIBBS EXPONENTIAL FORMULEA; LÉVY DISTRIBUTIONS; TRANSVERSE MOMENTA; TSALLIS STATISTICS;

APPROXIMATION THEORY; COSMIC RAYS; DATA REDUCTION; FUNCTIONS; GAMMA RAYS; GAUSSIAN NOISE (ELECTRONIC); PARAMETER ESTIMATION; PROBABILITY DISTRIBUTIONS; THERMODYNAMICS;

STATISTICAL METHODS;

EID: 15544364321     PISSN: 00319007     EISSN: 10797114     Source Type: Journal    
DOI: 10.1103/PhysRevLett.84.2770     Document Type: Article

References (32)

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2. C. Tsallis, J. Stat. Phys. 52, 479 (1988); for an updated bibliography on this subject, cf. http://tsallis.cat.cbpf.br/biblio.htm. Recent summary is provided in the special issue of Braz. J. Phys. 29, No. 1 (1999) (available also at http://sbf.if.usp.br/WWW/pages/Journals/BJP/Vol29/Num1/index.htm).
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9. Actually, the recently proposed use of quantum groups in studying Bose-Einstein correlations (cf. D. V. Anchishkin, A. M. Gavrilik, and N. Z. logorov, CERN Report No. CERN-TH/99-177 and nucl-th/9906034) belong also to that category because [as demonstrated in C. Tsallis, Phys. Lett. A 195, 329 (1994)] there is close correspondence between deformation parameter of quantum groups and nonextensivity parameter of Tsallis statistics. The same can be said on the works on intermittency using the so called Levy stable distributions [cf., for example, Ph. Brax and R. Peschanski, Nucl. Phys. B253, 225 (1991); S. Hegyi, Phys. Lett. B 387, 642 (1996)].
10. Actually, the recently proposed use of quantum groups in studying Bose-Einstein correlations (cf. D. V. Anchishkin, A. M. Gavrilik, and N. Z. logorov, CERN Report No. CERN-TH/99-177 and nucl-th/9906034) belong also to that category because [as demonstrated in C. Tsallis, Phys. Lett. A 195, 329 (1994)] there is close correspondence between deformation parameter of quantum groups and nonextensivity parameter of Tsallis statistics. The same can be said on the works on intermittency using the so called Levy stable distributions [cf., for example, Ph. Brax and R. Peschanski, Nucl. Phys. B253, 225 (1991); S. Hegyi, Phys. Lett. B 387, 642 (1996)].
11. Actually, the recently proposed use of quantum groups in studying Bose-Einstein correlations (cf. D. V. Anchishkin, A. M. Gavrilik, and N. Z. logorov, CERN Report No. CERN-TH/99-177 and nucl-th/9906034) belong also to that category because [as demonstrated in C. Tsallis, Phys. Lett. A 195, 329 (1994)] there is close correspondence between deformation parameter of quantum groups and nonextensivity parameter of Tsallis statistics. The same can be said on the works on intermittency using the so called Levy stable distributions [cf., for example, Ph. Brax and R. Peschanski, Nucl. Phys. B253, 225 (1991); S. Hegyi, Phys. Lett. B 387, 642 (1996)].
12. Actually, the recently proposed use of quantum groups in studying Bose-Einstein correlations (cf. D. V. Anchishkin, A. M. Gavrilik, and N. Z. logorov, CERN Report No. CERN-TH/99-177 and nucl-th/9906034) belong also to that category because [as demonstrated in C. Tsallis, Phys. Lett. A 195, 329 (1994)] there is close correspondence between deformation parameter of quantum groups and nonextensivity parameter of Tsallis statistics. The same can be said on the works on intermittency using the so called Levy stable distributions [cf., for example, Ph. Brax and R. Peschanski, Nucl. Phys. B253, 225 (1991); S. Hegyi, Phys. Lett. B 387, 642 (1996)].
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16. T. C. P. Chui, D. R. Swanson, M. J. Adriaans, J. A. Nissen, and J. A. Lipa, Phys. Rev. Lett. 69, 3005 (1992); C. Kittel, Phys. Today 41, No. 5, 93 (1988); B. B. Mandelbrot, Phys. Today 42, No. 1, 71 (1989); cf. also E. V. Shuryak, Phys. Lett. B 423, 9 (1998); S. Mró wczyń ski, Phys. Lett. B 430, 9 (1998).
17. T. C. P. Chui, D. R. Swanson, M. J. Adriaans, J. A. Nissen, and J. A. Lipa, Phys. Rev. Lett. 69, 3005 (1992); C. Kittel, Phys. Today 41, No. 5, 93 (1988); B. B. Mandelbrot, Phys. Today 42, No. 1, 71 (1989); cf. also E. V. Shuryak, Phys. Lett. B 423, 9 (1998); S. Mró wczyń ski, Phys. Lett. B 430, 9 (1998).
18. T. C. P. Chui, D. R. Swanson, M. J. Adriaans, J. A. Nissen, and J. A. Lipa, Phys. Rev. Lett. 69, 3005 (1992); C. Kittel, Phys. Today 41, No. 5, 93 (1988); B. B. Mandelbrot, Phys. Today 42, No. 1, 71 (1989); cf. also E. V. Shuryak, Phys. Lett. B 423, 9 (1998); S. Mró wczyń ski, Phys. Lett. B 430, 9 (1998).
19. T. C. P. Chui, D. R. Swanson, M. J. Adriaans, J. A. Nissen, and J. A. Lipa, Phys. Rev. Lett. 69, 3005 (1992); C. Kittel, Phys. Today 41, No. 5, 93 (1988); B. B. Mandelbrot, Phys. Today 42, No. 1, 71 (1989); cf. also E. V. Shuryak, Phys. Lett. B 423, 9 (1998); S. Mró wczyń ski, Phys. Lett. B 430, 9 (1998).
20. L. D. Landau and I. M. Lifschitz, Course of Theoretical Physics: Statistical Physics (Pergamon Press, New York, 1958).
21. This use of what is essentially the Mellin transformation has been discussed in different contexts of Tsallis statistics in a number of places; cf., for example, D. Prato, Phys. Lett. A 203, 165 (1995); P. A. Alemany, Phys. Lett. A 235, 452 (1997); C. Tsallis et al., Phys. Rev. E 56, R4922 (1997).
22. This use of what is essentially the Mellin transformation has been discussed in different contexts of Tsallis statistics in a number of places; cf., for example, D. Prato, Phys. Lett. A 203, 165 (1995); P. A. Alemany, Phys. Lett. A 235, 452 (1997); C. Tsallis et al., Phys. Rev. E 56, R4922 (1997).
23. This use of what is essentially the Mellin transformation has been discussed in different contexts of Tsallis statistics in a number of places; cf., for example, D. Prato, Phys. Lett. A 203, 165 (1995); P. A. Alemany, Phys. Lett. A 235, 452 (1997); C. Tsallis et al., Phys. Rev. E 56, R4922 (1997).
24. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier Science Publishers B. V., North-Holland, Amsterdam, 1987), Chap. VIII.
25. Our discussion (and results that follow) resembles to some extent approaches where Tsallis-type microscopic distributions were derived as an exact solution of the standard Fokker-Planck equation; see, for example, L. Borland, Phys. Lett. A 245, 67 (1998); Phys. Rev. E 57, 6634 (1998).
26. Our discussion (and results that follow) resembles to some extent approaches where Tsallis-type microscopic distributions were derived as an exact solution of the standard Fokker-Planck equation; see, for example, L. Borland, Phys. Lett. A 245, 67 (1998); Phys. Rev. E 57, 6634 (1998).
27. C. A. Ahmatov, Y. E. Diakov, and A. Tchirkin, Introduction to Statistical Radio-physics and Optics (Nauka, Moscow, 1981) (in Russian).
28. L. D. Landau and I. M. Lifschitz, Course of Theoretical Physics: Hydrodynamics (Pergamon Press, New York, 1958); Course of Theoretical Physics: Mechanics of Continuous Media (Pergamon Press, Oxford, 1981).
29. L. D. Landau and I. M. Lifschitz, Course of Theoretical Physics: Hydrodynamics (Pergamon Press, New York, 1958); Course of Theoretical Physics: Mechanics of Continuous Media (Pergamon Press, Oxford, 1981).
30. Compare, for example, S. Curilef, Z. Phys. B 100, 433 (1996), and references therein.
31. See, for example, M. L. Lyra and C. Tsallis, Phys. Rev. Lett. 80, 53 (1998); C. Anteneodo and C. Tsallis, Phys. Rev. Lett. 80, 5313 (1998), and references therein.
32. See, for example, M. L. Lyra and C. Tsallis, Phys. Rev. Lett. 80, 53 (1998); C. Anteneodo and C. Tsallis, Phys. Rev. Lett. 80, 5313 (1998), and references therein.