Effects of cosmic-string framework on the thermodynamical properties of anharmonic oscillator using the ordinary statistics and the q-deformed superstatistics approaches
- 178 Downloads
Abstract
In this article, we determine the thermodynamical properties of the anharmonic canonical ensemble within the cosmic-string framework. We use the ordinary statistics and the q-deformed superstatistics for this study. The q-deformed superstatistics is derived by modifying the probability density in the original superstatistics. The Schrödinger equation is rewritten in the cosmic-string framework. Next, the anharmonic oscillator is investigated in detail. The wave function and the energy spectrum of the considered system are derived using the bi-confluent Heun functions. In the next step, we first determine the thermodynamical properties for the canonical ensemble of the anharmonic oscillator in the cosmic-string framework using the ordinary statistics approach. Also, these quantities have been obtained in the q-deformed superstatistics. For vanishing deformation parameter, the ordinary results are obtained.
1 Introduction
Topological defects were first theorized by Kibble [1]. These defects are physical structures produced in symmetry-breaking phase transitions in the early universe. Among all the possible types of defects, the one-dimensional cosmic strings are the focus of most of the studies in this area [2]. This is because of the compatibility with the current cosmological models and their association with several brane inflation scenarios [3, 4] and super-symmetric grand unified theories [5]. These are constrained by the cosmic microwave background [6, 7, 8, 9, 10] and gravitational wave facilities [11, 12, 13, 14, 15]. In recent developments, cosmic strings have been considered to study solutions of black holes [16], to investigate the average rate of change of energy for a static atom immersed in a thermal bath of electromagnetic radiation [17], to study Hawking radiation of massless and massive charged particles [18], to study the non-Abelian Higgs model coupled with gravity [19], in the quantum dynamics of scalar bosons [20], in hydrodynamics [21], to study the non-relativistic motion of a quantum particle subjected to magnetic field [22], to investigate dynamical solutions in the context of super-critical tensions [23], describing scattering states of the Dirac equation in the presence of cosmic string for Coulomb interaction [24] and to study the spin-zero Duffin–Kemmer–Petiau equation in a cosmic-string space-time with the Cornell interaction [25].
- The probability density should be non-negative and$$\begin{aligned} \int _{0}^{\infty } f(\beta '_0, \beta ) \mathrm{d}\beta ' = 1. \end{aligned}$$(1.2)
- The average of \(\beta '_0\) should be \(\beta = \frac{1}{k_{B} T}\)$$\begin{aligned} \left\langle \beta '_0 \right\rangle = \int _{0}^{\infty } \beta '_0 f(\beta '_0,\beta ) \mathrm{d} \beta '_0 = \beta . \end{aligned}$$(1.3)
-
The superstatistics must be normalizable. For instance the integral \(\int _{0}^{\infty } B(E) \mathrm{d}E\) should exist and, in the general case, the integral \(\int _{0}^{\infty } \rho _d(E) B(E) \mathrm{d}E\) has to exist in which \( \rho _d(E) \) is the density of states.
-
The superstatistics, if there are no fluctuations of intensive quantities, should reduce to the Boltzmann–Gibbs statistic.
2 Modified of probability density
3 Schrödinger equation within cosmic-string framework
4 The anharmonic oscillator
Treatments of the anharmonic potential. The first plot is for the minus sign of b and the second plots is for the positive sign of b
Now the wave function and energy spectrum relation are determined. Thus we are ready to investigate some thermodynamical properties from a statistical mechanics point of view.
5 Thermodynamical properties within cosmic-string framework
In this section, the thermodynamical properties of the considered system are investigated in two manners. In the first subsection of this section, we want to study thermodynamical properties such as the entropy, the Helmholtz free energy, the mean energy and the entropy. In the other subsection, these properties are investigated in the q-deformed superstatistics manner.
5.1 Thermodynamical properties; ordinary statistics approach
Effects of \(\alpha \) on \(|\Psi (\rho )|^2\) considering \(n = 3, l = 1, \alpha = 0.2, a = 1;\) and \( k = 1\)
Partition function vs. \(\alpha \) considering \( n = 1, l = 1, k = 1, k_{B} = 1, a = 1 \) and \( T = 100\)
Helmholtz free energy vs. \(\alpha \) considering \( n = 1, l = 1, k = 1, k_{B} = 1, a = 1 \) and \( T = 100\)
Entropy vs. \(\alpha \) considering \( n = 1, l = 1, k = 1, k_B = 1, a = 1\) and \( T = 100\)
Mean energy vs. \(\alpha \) considering \( n = 3, l = 1, k = 1, k_B = 1, a = 1\) and \( T = 100\)
Specific heat energy at constant volume vs. \(\alpha \) considering \( n = 1, l = 1, k = 1, k_B = 1, a = 1\) and \( T = 100\)
Partition functions vs. \(\alpha \) in the q-deformed superstatistics case considering \( n = 1, l = 1, k = 1, k_B = 1, a = 1 \) and \( T = 100\) for different values of the deformation parameter
5.2 Thermodynamical properties; q-deformed superstatistics approach
Helmholtz free energy vs. \(\alpha \) in the q-deformed superstatistics case considering \( n = 1, l = 1, k = 1, k_B = 1, a = 1 \) and \( T = 100\) for different values of the deformation parameter
Entropy free energy vs. \(\alpha \) in the q-deformed superstatistics case considering \( n = 1, l = 1, k = 1, k_B = 1, a = 1\) and \( T = 100\) for different values of the deformation parameter
Mean energy free energy vs. \(\alpha \) in the q-deformed superstatistics case considering \( n = 1, l = 1, k = 1, k_B = 1, a = 1\) and \( T = 100\) for different values of the deformation parameter
Specific heat at constant volume vs. \(\alpha \) in modified Dirac delta distribution case considering \( n = 3, l = 1, k = 1, k_B = 1, a = 1 \) and \( T = 100 \) for different values of the deformation parameter
6 Conclusions
In this article, we have studied the thermodynamical properties of the anharmonic oscillator within cosmic-string framework using ordinary and the q-deformed superstatistics. After an introduction of the superstatistics, we presented a new probability density using the Dirac delta function and its derivatives and then the effective Boltzmann factor was derived according to the new probability density. Next, we derived the Schrödinger equation within the cosmic-string framework. The wave function of the system considered was derived using the bi-confluent Heun function and the series form of the bi-confluent Heun functions. Having determined the spectrum, using the ordinary statistics approach, some of the thermodynamical properties for the ensemble of the considered system were derived. In the next step, we obtained once again these quantities according to the effective Boltzmann factor derived by the q-deformed superstatistics. Treatments of these quantities were illustrated graphically. It was shown that, for the q-deformed superstatistics case, by removing the modification parameter, the results of the ordinary statistics approach were derived.
Notes
Acknowledgements
It is a great pleasure for the authors to thank the referee for the helpful comments.
References
- 1.T.W.B. Kibble, J. Phys. A 9, 1387 (1976)ADSCrossRefGoogle Scholar
- 2.A. Vilenkin, E.P.S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge, 1994)MATHGoogle Scholar
- 3.S. Sarangi, S.-H.H. Tye, Phys. Lett. B 536, 185 (2002)ADSCrossRefGoogle Scholar
- 4.E.J. Copeland, R.C. Myers, J. Polchinski, JHEP 2004, 013 (2004)CrossRefGoogle Scholar
- 5.R. Jeannerot, J. Rocher, M. Sakellariadou, Phys. Rev. D 68, 103514 (2003)ADSCrossRefGoogle Scholar
- 6.M. Landriau, E.P.S. Shellard, Phys. Rev. D 69, 023003 (2004)ADSCrossRefGoogle Scholar
- 7.P.A.R. Ade et al. (Planck), Astron. Astrophys. 571, A25 (2014)Google Scholar
- 8.A. Lazanu, P. Shellard, JCAP 1502, 024 (2015)ADSCrossRefGoogle Scholar
- 9.C. Ringeval, D. Yamauchi, J. Yokoyama, F.R. Bouchet, JCAP 1602, 033 (2016)ADSCrossRefGoogle Scholar
- 10.T. Charnock, A. Avgoustidis, E.J. Copeland, A. Moss, Phys. Rev. D 93, 123503 (2016)ADSCrossRefGoogle Scholar
- 11.T. Damour, A. Vilenkin, Phys. Rev. Lett. 85, 3761 (2000)ADSCrossRefGoogle Scholar
- 12.T. Damour, A. Vilenkin, Phys. Rev. D 71, 063510 (2005)ADSCrossRefGoogle Scholar
- 13.X. Siemens, V. Mandic, J. Creighton, Phys. Rev. Lett. 98, 111101 (2007)ADSCrossRefGoogle Scholar
- 14.S. Olmez, V. Mandic, X. Siemens, Phys. Rev. D 81, 104028 (2010)ADSCrossRefGoogle Scholar
- 15.P. Binetruy, A. Bohe, C. Caprini, J.-F. Dufaux, JCAP 1206, 027 (2012)ADSCrossRefGoogle Scholar
- 16.H. Vieira, V. Bezerra, G. Silva, Ann. Phys. 362, 576 (2015)ADSCrossRefGoogle Scholar
- 17.H. Cai, H. Yu, W. Zhou, Phys. Rev. D 92, 084062 (2015)ADSCrossRefGoogle Scholar
- 18.K. Jusufi, Gen. Relativ. Gravit. 47, 1 (2015)MathSciNetCrossRefGoogle Scholar
- 19.A. de Pádua Santos, E.R.B. de Mello, Class. Quantum Gravity 32, 155001 (2015)ADSCrossRefGoogle Scholar
- 20.L.B. Castro, Eur. Phys. J. C 75, 287 (2015)ADSCrossRefGoogle Scholar
- 21.A. Beresnyak, Astrophys. J. 804, 121 (2015)ADSCrossRefGoogle Scholar
- 22.H. Hassanabadi, A. Afshardoost, S. Zarrinkamar, Ann. Phys. 356, 346 (2015)ADSCrossRefGoogle Scholar
- 23.F. Niedermann, R. Schneider, Phys. Rev. D 91, 064010 (2015)ADSCrossRefGoogle Scholar
- 24.M. Hosseinpour, H. Hassanabadi, Int. J. Mod. Phys. A 30, 1550124 (2015)ADSCrossRefGoogle Scholar
- 25.M. de Montigny, M. Hosseinpour, H. Hassanabadi, Int. J. Mod. Phys. A 31, 1650191 (2016)CrossRefGoogle Scholar
- 26.C. Becka, E.G.D. Cohen, Phys. A 322, 267 (2003)MathSciNetCrossRefGoogle Scholar
- 27.S. Sargolzaeipor, H. Hassanabadi, W.S. Chung, Eur. Phys. J. Plus 133, 5 (2018)Google Scholar
- 28.C.R. Muniz, V.B. Bezerra, M.S. Cunha, Ann. Phys. 350, 105 (2014)ADSCrossRefGoogle Scholar
- 29.A.K. Dutta, R.S. Willey, J. Math. Phys. 29, 892 (1988)ADSMathSciNetCrossRefGoogle Scholar
- 30.N. Saad, R.L. Hall, H. Ciftci, J. Phys. A Math. Gen. 39, 8477 (2006)ADSCrossRefGoogle Scholar
- 31.H. Sobhani, A.N. Ikot, H. Hassanabadi, Eur. Phys. J. Plus 132, 240 (2017)CrossRefGoogle Scholar
- 32.A. Ishkhanyan, K.A. Suominen, J. Phys. A Math. Gen. 34, 6301 (2001)ADSCrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP3