Tsallis Statistics Restrict Accessible States in the Morse Oscillator System

The behaviour of systems far from equilibrium often defies explanation using traditional thermodynamic models, prompting researchers to explore alternative statistical frameworks. Arpita Goswami, from the Department of Physics at the Indian Institute of Technology Tirupati, and colleagues demonstrate how Tsallis nonextensive statistics can successfully describe the Morse oscillator, a fundamental model for molecular vibrations. Their work analytically derives a generalized partition function, revealing how nonextensivity restricts the number of accessible states and significantly alters energy and entropy calculations, particularly at lower temperatures. This approach, which predicts a distinctive dip in the ratio of generalized to standard internal energy and a Schottky-type anomaly in specific heat, extends to solid-state systems, offering a new way to understand the vibrational properties of molecules in non-equilibrium conditions and potentially advancing our understanding of complex materials.

Researchers analytically derive the generalized partition function to investigate thermodynamic properties within a nonextensive framework, proposing the effective number of accessible states as a quantifiable measure of nonextensivity. Calculations demonstrate that the nonextensive framework restricts the number of accessible states available to the system. The team also derives expressions for the generalized internal energy and entropy, examining their dependence on both temperature and the nonextensivity parameter, q. Numerical results confirm a strong effect of nonextensive behaviour, particularly in the low-to-moderate temperature regime, where the ratio of generalized internal energy to standard Boltzmann internal energy is significantly altered.

Tsallis Statistics Applied to Morse Oscillators

This research explores the thermodynamic properties of a Morse oscillator, a realistic model of molecular vibrations, using Tsallis non-extensive statistics. The goal is to understand how deviations from standard statistical mechanics affect the system’s behaviour, particularly in confined or non-standard scenarios. A key concept is the parameter ‘q’, which quantifies the degree of non-extensivity; a value of q=1 recovers standard statistical mechanics. The Morse oscillator, unlike simpler models, accounts for anharmonicity and has a finite number of bound energy levels.

Researchers used the Morse potential to describe the system’s energy levels and applied Tsallis statistics to calculate thermodynamic properties like the partition function, internal energy, entropy, and specific heat. They derived analytical expressions for these properties, simplifying calculations at low and high temperatures, and verified these results with numerical simulations. The team also extended the model to study the specific heat of solids, incorporating the anharmonicity of the Morse potential.

The results demonstrate that the effective number of contributing states decreases with smaller values of ‘q’ at low temperatures, meaning fewer energy levels are effectively occupied. They also observed a depletion of internal energy at intermediate temperatures for non-extensive systems. The model successfully incorporates anharmonicity into the calculation of specific heat, providing a more realistic description of solid-state thermodynamics. At high temperatures, all thermodynamic quantities converged to standard statistical mechanics results, confirming the approach’s consistency.

This study demonstrates that Tsallis non-extensive statistics provides a useful framework for understanding the thermodynamic properties of confined or anharmonic systems. The Morse oscillator serves as a minimal model for exploring the effects of non-extensivity, with potential implications for condensed matter physics, molecular thermodynamics, and vibrational spectroscopy. Future research could extend the model to more complex molecules, investigate dynamical observables, explore quantum information-theoretic measures, and apply the framework to experimental systems.

Nonextensivity Quantifies Complex System Behaviour

Researchers have extended statistical mechanics to better describe complex systems exhibiting behaviours beyond traditional methods. Standard statistical mechanics struggles with systems displaying long-range interactions, memory effects, or those existing far from equilibrium. To address these limitations, scientists are exploring generalized frameworks, with the nonextensive formalism proposed by Tsallis gaining attention. This approach introduces a parameter, ‘q’, which quantifies the degree of non-extensivity; q approaching 1 aligns with standard statistical mechanics, while deviations indicate nonextensive behaviour.

This research focuses on applying Tsallis statistics to the Morse oscillator, a more realistic model of molecular vibrations than the simple harmonic oscillator. The Morse oscillator possesses a finite number of bound states, limited by the energy required to break the chemical bond, and provides a more accurate representation of real molecular systems. By combining analytical calculations with numerical simulations, the team investigated the thermodynamic properties of the Morse oscillator within the Tsallis framework.

The results demonstrate that the nonextensive framework restricts the number of accessible states within the system, effectively reducing the available energy levels. Furthermore, the calculated internal energy exhibits a distinctive dip structure at low to moderate temperatures for values of q less than 1, a feature not observed in standard statistical mechanics. The generalized specific heat displays a Schottky-type anomaly, indicating a sharp peak in heat capacity at specific temperatures. Researchers extended their analysis to estimate the specific heat of solids, utilizing the Morse oscillator potential within both standard and Tsallis statistics, providing a pathway to understanding the vibrational properties of materials under non-equilibrium conditions.

The study establishes the Morse oscillator as a valuable, solvable model for exploring the thermodynamics of quantum systems governed by nonextensive statistics, with potential implications for understanding the behaviour of molecules in diverse and complex environments.

Nonextensivity Limits Molecular Vibration States

This study presents a detailed analysis of the Morse oscillator, a model more representative of real molecular vibrations than the simple harmonic oscillator, within the framework of Tsallis nonextensive statistics. Researchers successfully derived an analytical expression for the generalized partition function, which describes the distribution of energy states, and examined its behaviour at both high and low temperatures. The results demonstrate that incorporating nonextensivity further limits the number of accessible states within the system, influencing its thermodynamic properties.

The investigation reveals a notable impact of nonextensive behaviour, particularly at low to moderate temperatures, where the ratio of generalized to standard internal energy exhibits a distinct dip structure. Furthermore, the generalized specific heat displays a Schottky-type anomaly, indicating a change in heat capacity due to quantum effects.

By extending the analysis to solids, the study suggests the Morse oscillator provides a valuable, solvable model for exploring the implications of nonextensive statistics on vibrational properties, especially in systems far from equilibrium, such as diatomic molecules. The authors acknowledge that while the model offers insights, further work is needed to connect these findings with experimental observations. Future research could focus on applying this framework to more complex molecular systems and exploring its relevance to recent advances in trapping and cooling atomic gases.