Abstract
The q-generalized quantum distributions, arising within the context of q-nonextensive thermostatistics, have recently found interesting applications to cosmology. These applications rest upon an approximated form of the q-distributions whose properties are not yet completely known. In order to shed some new light on this subject we consider here a maximum entropy principle leading to the q-generalized Fermi–Dirac distribution. This variational principle is formulated entirely in terms of the mean occupation numbers. It constitutes a natural generalization to the nonextensive regime of the well-known prescription leading, in the standard q= 1 case, to the usual distribution for fermions. We analyze some important properties of this variational approach. In particular, we discuss 1) the invariance of the associated solutions under uniform shifts of the energy spectrum; 2) the role played by different kinds of constraints (i.e. linear or q-generalized constraints); and 3) the probabilistic interpretation of the variational procedure. The possibility of extending the present approach to bosons is also considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Buyukklilic, F. and Demirhan, D.: 1993, Phys. Lett. A 181, 24.
Buyukklilic, F., Demirhan, D. and Gulec, A.: 1995, Phys. Lett. A 197, 209.
Pennini, F., Plastino, A. and Plastino, A.R.: 1995, Phys. Lett. A 208, 309.
Tirnakli, U., Buyukklilic, F. and Demirhan, D.: 1998, Phys. Lett. A 245, 62.
Buyukklilic, F. and Demirhan, D.: 2000, Eur. Phys. J. B 14, 705.
Wang, Q. and Le Mehaute, A.: 1997, Phys. Lett. A 235, 222.
Uys, H., Miller, H.G. and Khanna, F.C.: 2001, Phys. Lett. A 289, 264–272.
Torres, D.F., Vucetich, H. and Plastino, A.: 1997, Phys. Rev. Lett. 79, 1588 [Erratum: 80, 3889 (1998)].
Torres, D.F. and Vucetich, H.: 1998, Physica A 259, 397.
Tirnakli, U. and Torres, D.F.: 1999, Physica A 268, 225.
Pessah, M.E., Torres, D.F. and Vucetich, H.: 2001, Physica A 297, 164.
Tsallis, C.: 1988, J. Stat. Phys. 52, 479–482.
Curado, E.M.F. and Tsallis, C.: 1991, J. Phys. A 24, L69–L72.
Tsallis, C.: 1999, Braz. J. Phys. 29, 1–35.
Tsallis, C.: 2001, in: S. Abe and Y. Okamoto (eds.), Nonextensive Statistical Mechanics and Its Applications, Springer, Berlin
Tsallis, C., Mendes, R.S. and Plastino, A.R.: 1998, Physica A 261, 534.
Plastino, A.R. and Plastino, A.: 1993, Phys. Lett. A 177, 177.
Beck, C.: 2001, Phys. Rev. Lett. 87, 180601.
Lavagno, A., Kaniadakis, G., Rego-Monteiro, M., Quarati, P. and Tsallis, C.: 1998, Astrophys. Lett. Commun. 35, 449.
Lima, J.A.S., Silva Jr., R. and Janilo Santos: 2000, Phys. Rev. E 61, 3260–3263.
Tsallis, C.: 1997, Phys. World 10, 42.
Boghosian, B.M.R.: 1995, Phys. Rev. E 53, 4754–4763.
Beck, C.: 2000, Physica A 277, 115–123.
Plastino, A.R. and Plastino, A.: 1995, Physica A 222, 347.
Frank, T.D. and Daffertshofer, A.: 2001, Physica A 295, 455–474.
Tirnakli, U.: 2000, Phys. Rev. E 62, 7857–7860.
Tirnakli, U., Tsallis, C. and Lyra, M.L.: 2002, Phys. Rev. E 65, 036207.
Uys, H. and Miller, H.G.: submitted.
Kapur, J.N. and Kesavan, H.K.: 1992, Entropy Optimization Principles with Applications, Academic Press. San Diego.
Sommerfeld, A.: 1993, Thermodynamics and Statistical Mechanics, Academic Press, New York.
Beck, C. and Schlogl, F.: 1993, Thermodynamics of Chaotic Systems, Cambridge University Press, Cambridge, U.K.
Ehrenfest, P. and Ehrenfest, T.: 1990, The Conceptual Foundations of the Statistical Approach in Mechanics,Dover, New York.
Zeh, H.D.: 1992, The Physical Basis of the Direction of Time, Springer-Verlag, Berlin.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Plastino, A., Plastino, A., Plastino, A. et al. Foundations of Nonextensive Statistical Mechanics and Its Cosmological Applications. Astrophysics and Space Science 290, 275–286 (2004). https://doi.org/10.1023/B:ASTR.0000032529.67037.21
Issue Date:
DOI: https://doi.org/10.1023/B:ASTR.0000032529.67037.21