Abstract
We investigate the thermodynamic properties of some quantum statistical systems with a fractional Hamiltonian in D-dimensional space. We calculate the partition function of the system of N fractional quantum oscillators and the thermodynamic quantities associated with it. We consider the thermal and critical properties of both Bose and Fermi gases in the context of the fractional energy and described by a fractional derivative.
Similar content being viewed by others
References
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York (1993).
I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution, and Some of Their Applications (Math. Sci. Engin., vol. 198), Acad. Press, San Diego, Calif. (1999).
R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000).
K. S. Fa, Phys. A, 350, 199–206 (2005).
F. Riewe, Phys. Rev. E, 53, 1890–1899 (1996).
F. Riewe, Phys. Rev. E, 55, 3581–3592 (1997).
G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford Univ. Press, Oxford (2005).
A. Hussain, S. Ishfaq, and Q. A. Naqvi, Prog. Electromagn. Res., 63, 319–335 (2006).
V. E. Tarasov, Phys. Lett. A, 336, 167–174 (2005).
V. E. Tarasov, Chaos, 15, 023102 (2005); arXiv:nlin/0602029v1 (2006).
B. A. Carreras, V. E. Lynch, and G. M. Zaslavsky, Phys. Plasmas, 8, 5096–5103 (2001).
V. E. Tarasov, Phys. Plasmas, 12, 082106 (2005).
G. M. Zaslavsky, Phys. D, 76, 110–122 (1994).
R. R. Nigmatullin, Phys. A, 363, 282–298 (2006).
G. M. Zaslavsky and M. A. Edelman, Phys. D, 193, 128–147 (2004).
N. Laskin and G. M. Zaslavsky, Phys. A, 368, 38–54 (2006); arXiv:nlin/0512010v2 (2005).
V. E. Tarasov and G. M. Zaslavsky, Chaos, 16, 023110 (2006).
V. V. Uchaikin, Phys. Usp., 46, 821–849 (2003).
N. Laskin, Phys. Lett. A, 268, 298–305 (2000); arXiv:hep-ph/9910419v2 (1999); Phys. Rev. E, 62, 3135–3145 (2000); arXiv:0811.1769v1 [math-ph] (2008).
X. Guo and M. Xu, J. Math. Phys., 47, 082104 (2006).
J. Dong and M. Xu, J. Math. Phys., 48, 072105 (2007).
R. Herrmann, Gam. Ori. Chron. Phys., 1, 1–12 (2013); arXiv:1210.4410v2 [math-ph] (2012).
R. Herrmann, Cent. Eur. J. Phys., 11, 1212–1220 (2013); arXiv:1308.4587v2 [physicsgen-ph] (2013).
Z. Korichi and M. T. Meftah, J. Math. Phys., 55, 033302 (2014).
Z. Z. Alisultanov and R. P. Meylanov, Theor. Math. Phys., 171, 769–779 (2012).
N. Laskin, Phys. Rev. E, 66, 056108 (2002); arXiv:quant-ph/0206098v1 (2002).
N. Laskin, “Principles of fractional quantum mechanics,” arXiv:1009.5533v1 [math-ph] (2010).
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Natl. Bur. Stds. Appl. Math. Ser., vol. 55), Dover, New York (1972).
K. Huang, Statistical Mechanics, Wiley, New York (1963).
Z. Yan, Eur. J. Phys., 21, 625–631 (2000).
G. Su, Y. Cai, and J. Chen, J. Phys. A: Math. Theor., 42, 125003 (2009).
S. Luca, J. Math. Phys., 41, 8016–8024 (2000).
Z. Yan, M. Li, L. Chen, C. Chen, and J. Chen, J. Phys. A: Math. Gen., 32, 4069–4078 (1999).
Author information
Authors and Affiliations
Corresponding author
Additional information
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 186, No. 3, pp. 433–442, March, 2016.
Rights and permissions
About this article
Cite this article
Korichi, Z., Meftah, M.T. Quantum statistical systems in D-dimensional space using a fractional derivative. Theor Math Phys 186, 374–382 (2016). https://doi.org/10.1134/S0040577916030065
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577916030065