Effects of cosmicstring framework on the thermodynamical properties of anharmonic oscillator using the ordinary statistics and the qdeformed superstatistics approaches
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Abstract
In this article, we determine the thermodynamical properties of the anharmonic canonical ensemble within the cosmicstring framework. We use the ordinary statistics and the qdeformed superstatistics for this study. The qdeformed superstatistics is derived by modifying the probability density in the original superstatistics. The Schrödinger equation is rewritten in the cosmicstring framework. Next, the anharmonic oscillator is investigated in detail. The wave function and the energy spectrum of the considered system are derived using the biconfluent Heun functions. In the next step, we first determine the thermodynamical properties for the canonical ensemble of the anharmonic oscillator in the cosmicstring framework using the ordinary statistics approach. Also, these quantities have been obtained in the qdeformed 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 symmetrybreaking phase transitions in the early universe. Among all the possible types of defects, the onedimensional 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 supersymmetric 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 nonAbelian Higgs model coupled with gravity [19], in the quantum dynamics of scalar bosons [20], in hydrodynamics [21], to study the nonrelativistic motion of a quantum particle subjected to magnetic field [22], to investigate dynamical solutions in the context of supercritical tensions [23], describing scattering states of the Dirac equation in the presence of cosmic string for Coulomb interaction [24] and to study the spinzero Duffin–Kemmer–Petiau equation in a cosmicstring spacetime with the Cornell interaction [25].
 The probability density should be nonnegative 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 cosmicstring framework
4 The anharmonic oscillator
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 cosmicstring 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 qdeformed superstatistics manner.
5.1 Thermodynamical properties; ordinary statistics approach
5.2 Thermodynamical properties; qdeformed superstatistics approach
6 Conclusions
In this article, we have studied the thermodynamical properties of the anharmonic oscillator within cosmicstring framework using ordinary and the qdeformed 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 cosmicstring framework. The wave function of the system considered was derived using the biconfluent Heun function and the series form of the biconfluent 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 qdeformed superstatistics. Treatments of these quantities were illustrated graphically. It was shown that, for the qdeformed 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.
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