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Thermal properties of Klein-Gordon Oscillator in the Context of Amelino-Camelia and Magueijo-Smolin Doubly Special Relativity (DSR) frameworks

Abdelmalek Boumali, Nosratollah Jafari, Bekdaulet Shukirgaliyev, Fadila Serdouk

TL;DR

This work analyzes how Amelino-Camelia and Magueijo–Smolin Doubly Special Relativity (DSR) deformations affect the thermal properties of the one-dimensional Klein–Gordon oscillator. By deriving the modified spectra $E_n^{AC}$ and $E_n^{MS}$ and isolating the positive-energy sector via Foldy–Wouthuysen, the authors construct the partition function $Z(\beta)$ through Euler–Maclaurin summation and extract thermodynamic quantities such as the specific heat $C_v$ and entropy $S$. Planck-scale corrections controlled by $E_p$ produce model-dependent shifts in the $C_v$ peak, while $S(\beta)$ remains analytic and monotonic, indicating the absence of phase transitions; the peaks are Schottky-type anomalies due to the finite spectrum. The results demonstrate that thermodynamic observables can serve as sensitive probes of Planck-scale kinematics and offer a diagnostic means to distinguish between DSR prescriptions, with potential relevance for experimental analogue platforms and higher-dimensional generalizations.

Abstract

We examine the thermal and statistical properties of the one dimensional Klein-Gordon oscillator within two prominent Doubly Special Relativity (DSR) frameworks: Amelino-Camelia and Magueijo-Smolin. Using the modified dispersion relations specific to each formulation, we derive the positive energy spectra, construct the partition function via the Euler-Maclaurin method, and compute key thermodynamic quantities, including the specific heat $C_v$, as functions of temperature and the deformation scale. Planck-scale corrections produce distinct, theoretically resolvable shifts in both the position and magnitude of the $C_v$ peak in the two models. An accompanying entropy analysis reveals that these peaks correspond to smooth Schottky-type anomalies: the specific heat curves remain analytic and positive across the explored temperature range, and thus do not indicate latent or continuous thermodynamic phase transitions. These comparative results provide a robust diagnostic framework for differentiating DSR prescriptions in relativistic quantum systems and reinforce the transition-free character of their thermal response.

Thermal properties of Klein-Gordon Oscillator in the Context of Amelino-Camelia and Magueijo-Smolin Doubly Special Relativity (DSR) frameworks

TL;DR

This work analyzes how Amelino-Camelia and Magueijo–Smolin Doubly Special Relativity (DSR) deformations affect the thermal properties of the one-dimensional Klein–Gordon oscillator. By deriving the modified spectra and and isolating the positive-energy sector via Foldy–Wouthuysen, the authors construct the partition function through Euler–Maclaurin summation and extract thermodynamic quantities such as the specific heat and entropy . Planck-scale corrections controlled by produce model-dependent shifts in the peak, while remains analytic and monotonic, indicating the absence of phase transitions; the peaks are Schottky-type anomalies due to the finite spectrum. The results demonstrate that thermodynamic observables can serve as sensitive probes of Planck-scale kinematics and offer a diagnostic means to distinguish between DSR prescriptions, with potential relevance for experimental analogue platforms and higher-dimensional generalizations.

Abstract

We examine the thermal and statistical properties of the one dimensional Klein-Gordon oscillator within two prominent Doubly Special Relativity (DSR) frameworks: Amelino-Camelia and Magueijo-Smolin. Using the modified dispersion relations specific to each formulation, we derive the positive energy spectra, construct the partition function via the Euler-Maclaurin method, and compute key thermodynamic quantities, including the specific heat , as functions of temperature and the deformation scale. Planck-scale corrections produce distinct, theoretically resolvable shifts in both the position and magnitude of the peak in the two models. An accompanying entropy analysis reveals that these peaks correspond to smooth Schottky-type anomalies: the specific heat curves remain analytic and positive across the explored temperature range, and thus do not indicate latent or continuous thermodynamic phase transitions. These comparative results provide a robust diagnostic framework for differentiating DSR prescriptions in relativistic quantum systems and reinforce the transition-free character of their thermal response.

Paper Structure

This paper contains 5 sections, 35 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Klein--Gordon Oscillator under DSR Frameworks
  3. Statistical Properties
  4. Conclusion
  5. Acknowledgment

Figures (4)

  • Figure 1: The behavior of the partition function and specific heat of the one-dimensional Klein-Gordon oscillator (1D KGO) as a function of $1/\beta$ in the framework of Amelino-Camelia DSR, and its comparison with the standard 1D KGO
  • Figure 2: The behavior of the partition function and specific heat of the one-dimensional Klein-Gordon oscillator (1D KGO) as a function of $1/\beta$ in the framework of Magueijo-Smolin Doubly DSR, and its comparison with the standard 1D KGO
  • Figure 3: Comparison of the specific heat as a function of $1/\beta$ for the one-dimensional Klein-Gordon oscillator (1D KGO) in the frameworks of Amelino-Camelia and Magueijo-Smolin Doubly Special Relativity, with that of the standard 1D KGO.
  • Figure 4: The behavior of the entropy function of the one-dimensional Klein-Gordon oscillator (1D KGO) as a function of $1/\beta$ in the framework of both Amelino-Camelia and Magueijo-Smolin Doubly DSR, and its comparison with the standard 1D KGO