A logarithmically deformed entropy functional

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 $S_{\gamma }\left[p\right]=-{\sum }_i{p}_i{\left(ln\frac{1}{p_i}\right)}^{1/\gamma }$, which generalizes the Shannon entropy through a logarithmic deformation parameter $\gamma$. 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 $\gamma \ne 1$, 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.