Effects of cosmic-string framework on the thermodynamical properties of anharmonic oscillator using the ordinary statistics and the q-deformed superstatistics approaches

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 anharmonic oscillator is written as [29, 30, 31]

$$\begin{aligned} v(\rho ) = a_v \rho ^2 + b_v \rho ^4 + c_v \rho ^6, \end{aligned}$$

(4.1)

where the coefficients are real constants. Such a potential can be used in nuclear physics and the expressions of the energy rates and quadrapole transitions. In Ref. [29], the authors used the series expansion and tried to derive recurrence relation between the coefficients of the series or the Gaussian approximation to obtain the result in Ref. [30]; the authors used asymptotic iteration method to obtain the solutions. However, we used some changes of variables to derive the bi-confluent Huen differential equations [31]. Substituting (4.1) into (3.5) for \(\rho \ge \frac{a}{\alpha }\) and considering the wave function \(\Psi (t, \text{ r }) = \exp \left[ i (-\varepsilon t + l \phi + k z) \right] \Phi (\rho )\) we obtain

$$\begin{aligned}&\frac{\mathrm{d}^2 \Phi (\rho )}{\mathrm{d} \rho ^2} + \frac{1}{\rho } \frac{\mathrm{d} \Phi (\rho )}{\mathrm{d} \rho } + \left( \varepsilon - k^2 - (a_v + l^2 \alpha ^2) \rho ^2\right. \nonumber \\&\left. \quad - b_v \rho ^4 -c_v \rho ^6 \right) \Phi (\rho )=0. \end{aligned}$$

(4.2)

To solve Eq. (4.2), first derivative terms should be removed by considering \(\Phi (\rho ) = \frac{R(\rho )}{\sqrt{\rho }}\). Then we obtain

$$\begin{aligned}&\frac{\mathrm{d}^2 R (\rho )}{\mathrm{d} \rho ^2} + \Bigg ( \varepsilon + \frac{1}{4 \rho ^2} - k^2 - (a_v + l^2 \alpha ^2) \rho ^2 \nonumber \\&\quad - b_v \rho ^4 -c_v \rho ^6 \Bigg ) R (\rho )=0. \end{aligned}$$

(4.3)

The next step is using the new variable \( y = \rho ^2\). Rewriting Eq. (4.3) in terms of the new variable, results in

$$\begin{aligned}&\frac{\mathrm{d}^2 R (y)}{\mathrm{d} y^2} + \frac{1}{2 y} \frac{\mathrm{d} R (y)}{\mathrm{d} y} + \left( \frac{1}{16 y} + \frac{\frac{1}{4} (\varepsilon - k^2)}{y} - \frac{1}{4}(a_v + l^2 \alpha ^2) \right. \nonumber \\&\left. \quad - \frac{b_v}{4} y - \frac{c_v}{4} y^2 \right) R (y)=0. \end{aligned}$$

(4.4)

Once again, we need to remove the first derivative term in Eq. (4.4) to continue to get to the solutions. This time we consider \(R(y) = \frac{f(y)}{\root 4 \of {y}}\) and obtain

$$\begin{aligned}&\frac{\mathrm{d}^2 f (y)}{\mathrm{d} y^2} + \left( \frac{1}{4 y} + \frac{\frac{1}{4} (\varepsilon - k^2)}{y} - \frac{1}{4}(a_v + l^2 \alpha ^2)\right. \nonumber \\&\left. \quad - \frac{b_v}{4} y - \frac{c_v}{4} y^2 \right) f (y)=0. \end{aligned}$$

(4.5)

Considering the solution in the form of

$$\begin{aligned} f(y) = y^A_1 \exp [y( B_1 + D_1 y)]F(y), \end{aligned}$$

(4.6)

where

$$\begin{aligned} A_1&= \frac{1}{2}, \end{aligned}$$

(4.7)

$$\begin{aligned} B_1&= - \frac{b_v}{8}, \end{aligned}$$

(4.8)

$$\begin{aligned} D_1&= - \frac{\sqrt{c_v}}{2}, \end{aligned}$$

(4.9)

we have

$$\begin{aligned}&\frac{\mathrm{d}^2 F(y)}{\mathrm{d}y^2} + \left( \frac{1}{y} - \frac{b_v}{4} - 2y \right) \frac{\mathrm{d} F(y)}{\mathrm{d}y} \nonumber \\&\qquad + \left( \frac{b_v ^2}{64} - \frac{1}{4} \left( a_v ^2 + l^2 \alpha ^2 + 4 + \frac{- \frac{b_v}{8} + \frac{1}{4} (\varepsilon - k^2)}{y}\right) \right) , \nonumber \\&\quad F(y) = 0, \end{aligned}$$

(4.10)

in which we were forced to set \(c_v = 4\) (see the appendix for more details). Equation (4.10) is the bi-confluent Heun differential equation. Consequently, we have

$$\begin{aligned}&F(y) = H_b (0, \beta '_b, \gamma '_b , \delta '_b; y), \end{aligned}$$

(4.11)

$$\begin{aligned}&\beta '_b = \frac{b_v}{4}, \end{aligned}$$

(4.12)

$$\begin{aligned}&\gamma '_b = \frac{b_v ^2}{4} - \frac{1}{4} (a_v ^2 + l^2 \alpha ^2 - 4), \end{aligned}$$

(4.13)

$$\begin{aligned}&\delta '_b = \frac{1}{2} (k^2 - \varepsilon ). \end{aligned}$$

(4.14)

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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.

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

To derive thermodynamical quantities from a statistical mechanical point of view, we should construct the partition function. To construct the partition function we have \( (\hbar = 1 , m=0.5 ) \)

$$\begin{aligned} Z(\varepsilon _n)&= \sum _{n=0}^{\infty } e^{-\beta \varepsilon _n}, \nonumber \\&= \sum _{n=0}^{\infty } e^{- \beta \left( (b_v n+ \frac{5b_v}{2}) + k^2 \right) }, \nonumber \\&= e^{- \beta \left( \frac{5b_v}{2} + k^2 \right) } \sum _{n=0}^{\infty } e^{-\beta b_v n} \nonumber \\ Z(\varepsilon _n) =&\frac{e^{- \beta \left( \frac{5b_v}{2} + k^2 \right) }}{1- e^{-\beta b_v}}, \end{aligned}$$

(5.1)

where \(\beta = \frac{1}{k_{B} T}\), \(k_{B}\) is the Boltzmann constant and the temperature is denoted T. In Fig. 3, we have plotted the partition function in terms of \(\alpha \). It is seen that by increasing the value of \(\alpha \), the partition function decreases.

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As the first quantity, the Helmholtz free energy can be obtained using Eq. (5.1) as

$$\begin{aligned} A =&\frac{-1}{\beta } \ln Z(\varepsilon _n), \nonumber \\ A =&\frac{-1}{\beta } \ln \left( \frac{e^{- \beta \frac{3b_v+2k^2}{2}}}{e^{\beta b_v}-1} \right) . \end{aligned}$$

(5.2)

In Fig. 4, we have plotted Helmholtz free energy as a function of \(\alpha \). Effect of the \(\alpha \) parameter can be seen that by increasing in the \(\alpha \) parameter, the Helmholtz free energy increases too.

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We can obtain from the Helmholtz free energy the entropy. According to its definition, we have

$$\begin{aligned} S&= - \frac{\partial A}{\partial T}, \nonumber \\ S&= \frac{1}{2T} \left( 2k^2 + b \left( 5 + \frac{2}{e^{\beta b_\nu }-1} \right) \right) + k_B \ln \left( \frac{e^{-\beta \frac{3b_\nu + 2 k^2}{2}}}{e^{\beta b_\nu } -1} \right) . \end{aligned}$$

(5.3)

It is instructive to investigate the entropy behavior as a function of \(\alpha \) . In Fig. 2, we showed that by increasing \(\alpha \), the density probability of the wave function will be concentrated. Thus we expect such a treatment for the entropy. As is shown in Fig. 5, by increasing \(\alpha \), the entropy will decrease. It means that we are facing with a system that is becoming more ordered than before by increasing \(\alpha \).

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The mean energy is the next quantity that we want to derive. Using the connection between the mean energy and the partition function we can write

$$\begin{aligned} U&= A-T \left( \frac{\partial A}{\partial T} \right) , \nonumber \\ U&= k^2 + b \left( \frac{5}{2} + \frac{1}{e^{\beta b_\nu } -1} \right) . \end{aligned}$$

(5.4)

Unlike the entropy, the mean energy increases with increasing \(\alpha \). Figure 6, shows such a treatments.

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As the last quantity which we want to know as regards treatment in the cosmic-string framework, we have the specific heat capacity at constant volume. This quantity can be connected to the mean energy by

$$\begin{aligned} C_V =&\left( \frac{\partial U}{\partial T} \right) , \nonumber \\ C_V =&\frac{b^2}{4k_B T^2} \text {Csch} \left( 2 \beta b_\nu \right) . \end{aligned}$$

(5.5)

As expected from the form of the entropy, we have an ensemble with low entropy, consequently the specific heat capacity at constant volume must have a lower value than an ensemble which has higher entropy. Figure 7 shows that, by increasing \(\alpha \), the specific heat capacity at constant volume decreases.

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5.2 Thermodynamical properties; q-deformed superstatistics approach

The first step is the derivation of the partition function, which takes the form

$$\begin{aligned} Z(\varepsilon ) = \frac{{{e^{ - \frac{{3b_v + 2{k^2}}}{{2T{k_B}}}}}\left( {q \varTheta (T) + 8{T^2}k_B^2{{\left( {{e^{\frac{b_v}{{T{k_B}}}}} - 1} \right) }^2}} \right) }}{{8{T^2}k_B^2{{\left( {{e^{\frac{b_v}{{T{k_B}}}}} - 1} \right) }^3}}}, \end{aligned}$$

(5.6)

where

$$\begin{aligned} \varTheta (T)&= - 2\left( {13{b_v ^2} + 16b_v {k^2} + 4{k^4}} \right) {e^{\frac{b_v}{{T{k_B}}}}}\nonumber \\&\quad + {\left( {5b_v + 2{k^2}} \right) ^2}{e^{\frac{{2b_v}}{{T{k_B}}}}} + {\left( {3b_v + 2{k^2}} \right) ^2}. \end{aligned}$$

(5.7)

After determination of the partition function, we can obtain the Helmholtz free energy, entropy, mean energy, and specific heat at constant volume. The behaviors of these quantities are depicted in Figs. 8, 9,10, 11 and 12. We studied the presence \((q \ne 0)\) and absence \((q = 0)\) of the modification. As we stated in Sect. 2, q is a positive constant. Consequently, we can see that effects of the deformation parameter on the quantities is increasing in their values. The important effect of the deformation parameter is that in the absence of the deformation parameter, the specific heat at constant volume decreases for increasing \( \alpha \) while in the presence of deformation is increasing. It is seen easily that, for vanishing of the modification, the treatments are repeated exactly as we faced in the previous subsection.

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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.