Nonextensive Thermodynamics of the Morse Oscillator: Signature and Solid State Application

TL;DR

This work analyzes the Morse oscillator within the Tsallis nonextensive framework to illuminate thermodynamics of bound quantum systems with finite spectra. By deriving analytical expressions for the generalized partition function $Z_q(T)$, internal energy $U_q$, entropy $S_q$, and the effective state count $n_{ ext{eff}}$, the authors reveal how nonextensivity ($q<1$) suppresses high-energy occupations and reduces accessible states, producing a dip in $U_q/U_{ ext{BG}}$ at intermediate temperatures and a Schottky-like peak in $C_q(T)$. They extend the approach to solids by incorporating Morse anharmonicity into Debye-like thermodynamics, showing that $q$-deformation modulates vibrational contributions and could inform experimental vibrational and thermal properties of diatomic molecules and molecular solids. Across high temperatures the results converge to Boltzmann-Gibbs behavior, while at moderate to low temperatures the bounded spectrum and $q$-exponential truncation yield distinct thermodynamic signatures, establishing the Morse oscillator as a valuable testing ground for nonextensive quantum thermodynamics with potential experimental relevance in molecular thermodynamics and non-equilibrium solid-state systems.

Abstract

In this work, we present a detailed thermodynamic analysis of a bound quantum system: the Morse oscillator within the framework of Tsallis nonextensive statistics. Using the property of the bound spectrum (upper bound) of the Morse potential, limited by the bond dissociation energy, we analytically derive the generalized partition function. We present results for both the high- and low-temperature limits. We propose the effective number of accessible states as a measure of nonextensivity. The calculation shows that the nonextensive framework further restricts the number of accessible states. We also derive the generalized internal energy and entropy and examine their dependence on temperature and the nonextensivity parameter \( q \). Numerical results confirm the strong effect of nonextensive behavior in the low-temperature regime (precisely low to moderate temperature), where the ratio of generalized internal energy and internal energy calculated from the Boltzmann Gibbs (BG) formula develops a nontrivial dip structure for \( q < 1 \). Moreover, the generalized specific heat shows the Schottky-type anomaly. We extend our study by deriving the specific heat of solids with BG and Tsallis statistics using the anharmonic energy levels of the Morse oscillator. This study suggests that the Morse oscillator is a solvable and physically meaningful testing ground for exploring the thermodynamics of quantum systems driven by nonextensive statistics, with implications for the vibrational properties of the non-equilibrium molecular thermodynamics (especially diatomic molecules).

Figures (6)

• Figure 1: Surface plots of internal energy $U_q(T, q)$ for the Morse oscillator at increasing anharmonicity values: $x_e = 0.00, 0.02, 0.04, 0.06$ arranged from left to right and top to bottom. As $x_e$ increases, the number of bound states decreases, leading to earlier saturation of internal energy. For $q < 1$, nonextensivity further suppresses contributions from high-energy levels due to the compact support of the $q$-exponential. The combined effect results in significant thermodynamic suppression, especially at intermediate to high temperatures. All panels $n_{\mathrm{max}}$ is adjusted as a function of $x_e$. $k_BT$ and $U_q$ are in eV.
• Figure 2: (a) Generalized partition function $Z_q(T)$ for different values of $q < 1$ as a function of temperature ($T$). (b) Probability distribution $p_q(E_n)$ over energy levels $n$ at fixed temperature. The parameters are as follows: $\hbar \omega_0 =1~\mathrm{eV}$, $D_e=25~\mathrm{eV}$ and $x_e=0.01,$ with $n_{\mathrm{max}}=49$. The plots illustrate that smaller q values narrow the distribution and modify the $Z_q(T)$ growth.
• Figure 3: (a) Temperature dependence of the fraction of effectively contributing states $\frac{n_{\mathrm{eff}}}{n_{\mathrm{max}}}$. (b) Ratio of internal energy in Tsallis and BG frameworks $\frac{U_q}{U_{\mathrm{BG}}}$, showing a dip due to nonextensive suppression of higher-energy states. $k_B T$ is in eV. The parameters are as follows: $\hbar \omega_0 =1~\mathrm{eV}$, $D_e=25~\mathrm{eV}$ and $x_e=0.01,$ with $n_{\mathrm{max}}=49$.
• Figure 4: Plot for generalized internal energy $U_q(T)$ as a function of temperature ($T$) for different $q$ values. $U_q$ and $k_BT$ are in eV. The parameters are as follows: $\hbar \omega_0 =1~\mathrm{eV}$, $D_e=25~\mathrm{eV}$ and $x_e=0.01,$ with $n_{\mathrm{max}}=49$. The difference in $U_q(T)$ occurs in the intermediate-temperature region and vanishes at high temperature, and collapses with the BG prediction.
• Figure 5: (a) Tsallis entropy $S_q(T)$, (b) Specific heat $C_q(T)$ for different values of $q < 1$ has been plotted as a function of temperature (T). The parameters are as follows: $\hbar \omega_0 =1~\mathrm{eV}$, $D_e=25~\mathrm{eV}$ and $x_e=0.01,$ with $n_{\mathrm{max}}=49$. Entropy shows $q$ dependent saturation, and Specific heat shows $q$ dependent peak position as a signature of nonextensivity. $k_B T$ is in eV.