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Partition function for position-dependent mass systems from superestatistics

Ignacio S. Gomez, Matheus Gabriel Alves Santos, Daniela de Almeida dos Santos, Ronaldo Thibes

TL;DR

This work addresses the thermodynamics of position-dependent mass (PDM) systems by linking canonical PDM partition functions to superstatistics through a fluctuating inverse temperature. The authors show that choosing a delta-like inverse-temperature distribution f(β)=δ(β−γ(x)) with γ(x)=β0 m0/m(x) and an appropriate transformed potential yields a biunivocal mapping between PDM configurations and superstatistical descriptions, enabling a closed-form partition function Z for 1D PDM systems. They apply the method to a PDM ideal gas with Tsallis and Kaniadakis statistics and to two harmonic-oscillator–type PDMs (quadratic and exponential masses), obtaining explicit Z1 and thermodynamic expressions; notably, a quadratic PDM oscillator is thermodynamically equivalent to a 3D ideal gas, while an exponential PDM yields linear specific heat and potential Nernst-law compatibility. The framework provides a unified, tractable approach to study inhomogeneous mass effects in nonextensive thermodynamics, with potential relevance for heterogeneous materials and complex systems.

Abstract

In this work, we show a connection between superstatistics and position-dependent mass (PDM) systems in the context of the canonical ensemble. The key point is to set the fluctuation distribution of the inverse temperature in terms od the system PDM. For PDMs associated to Tsallis and Kaniadakis nonextensive statistics, the pressure and entropy of the ideal gas result lower than the standard case but maintaining monotonic behavior. Gas of non-interacting harmonic oscillators provided with quadratic and exponential PDMs exhibit a behavior of standard ED harmonic oscillator gas and a linear specific heat respectively, the latter being consistent with Nernst's third law of thermodynamics. Thus, a combined PDM-superstatistics scenario offers an alternative way to study the effects of the inhomogeneities of PDM systems in their thermodynamics.

Partition function for position-dependent mass systems from superestatistics

TL;DR

This work addresses the thermodynamics of position-dependent mass (PDM) systems by linking canonical PDM partition functions to superstatistics through a fluctuating inverse temperature. The authors show that choosing a delta-like inverse-temperature distribution f(β)=δ(β−γ(x)) with γ(x)=β0 m0/m(x) and an appropriate transformed potential yields a biunivocal mapping between PDM configurations and superstatistical descriptions, enabling a closed-form partition function Z for 1D PDM systems. They apply the method to a PDM ideal gas with Tsallis and Kaniadakis statistics and to two harmonic-oscillator–type PDMs (quadratic and exponential masses), obtaining explicit Z1 and thermodynamic expressions; notably, a quadratic PDM oscillator is thermodynamically equivalent to a 3D ideal gas, while an exponential PDM yields linear specific heat and potential Nernst-law compatibility. The framework provides a unified, tractable approach to study inhomogeneous mass effects in nonextensive thermodynamics, with potential relevance for heterogeneous materials and complex systems.

Abstract

In this work, we show a connection between superstatistics and position-dependent mass (PDM) systems in the context of the canonical ensemble. The key point is to set the fluctuation distribution of the inverse temperature in terms od the system PDM. For PDMs associated to Tsallis and Kaniadakis nonextensive statistics, the pressure and entropy of the ideal gas result lower than the standard case but maintaining monotonic behavior. Gas of non-interacting harmonic oscillators provided with quadratic and exponential PDMs exhibit a behavior of standard ED harmonic oscillator gas and a linear specific heat respectively, the latter being consistent with Nernst's third law of thermodynamics. Thus, a combined PDM-superstatistics scenario offers an alternative way to study the effects of the inhomogeneities of PDM systems in their thermodynamics.

Paper Structure

This paper contains 12 sections, 24 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Position-dependent mass systems
  4. Superestatistical partition function and thermodynamics
  5. PDM canonical partition function from superestatistical partition function
  6. Applications: thermodynamics from PDM partition functions
  7. Case 1: PDM ideal gas
  8. Tsallis PDM ideal gas
  9. Kaniadakis PDM ideal gas
  10. Case 2: quadratic PDM $m(x)=m_0ax^2$ and harmonic potential $V(x)=\alpha x^2/2$
  11. Case 3: exponential PDM $m(x)=m_0e^{-bx}$ and harmonic potential $V(x)=\alpha x^2/2$
  12. Conclusions

Figures (2)

  • Figure 1: Pressure (uper panel) and entropy (bottom panel) per particle in function of the volume at the temperature $T=T_0$ of the PDM ideal gas for the Tsallis, Kaniadakis and constant mass cases with $1-q=\kappa=0.5$.
  • Figure 2: Entropy (uper panel), internal energy (center panel) and specific heat (bottom panel) per particle in function of the temperature of a gas of non-interacting harmonic oscillators for the quadratic, exponential and constant mass cases with $c=0.5$.