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A logarithmically deformed entropy functional

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  • Weberszpil, José

Abstract

Stretched exponential distributions appear in disordered systems, glassy dynamics, and anomalous diffusion, yet their thermodynamic origin is often phenomenological. In this work, we propose a deformed entropy functional of the form Sγ[p]=−∑ipiln1pi1/γ, which generalizes the Shannon entropy through a logarithmic deformation parameter γ. We show that, when maximized under standard constraints, this entropy leads asymptotically to stretched exponential (Weibull-type) distributions without requiring nonlinear constraints. The entropy is non-additive for γ≠1, tunably extensive, and concave in well-defined regimes. We establish its Lesche stability and derive its asymptotic variational behavior analytically. This framework offers a consistent thermodynamic foundation for modeling systems with memory, heterogeneity, or long-range correlations. The approach extends the Havrda–Charvát–Tsallis paradigm and contributes to the ongoing development of generalized thermodynamics by introducing a stretched-logarithmic entropy consistent with stretched exponential statistics.

Suggested Citation

  • Weberszpil, José, 2025. "A logarithmically deformed entropy functional," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 680(C).
    • Handle: RePEc:eee:phsmap:v:680:y:2025:i:c:s0378437125006818
    DOI: 10.1016/j.physa.2025.131029
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    References listed on IDEAS

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    1. Beck, Christian, 2006. "Stretched exponentials from superstatistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 365(1), pages 96-101.
    2. Oikonomou, Thomas & Kaloudis, Konstantinos & Bagci, G. Baris, 2021. "The q-exponentials do not maximize the Rényi entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 578(C).
    3. Kaniadakis, G., 2006. "Towards a relativistic statistical theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 365(1), pages 17-23.
    4. Weberszpil, José, 2025. "Microscopic origins of conformable dynamics: From disorder to deformation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 678(C).
    5. Okamura, Keisuke, 2020. "Affinity-based extension of non-extensive entropy and statistical mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 557(C).
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