An Extended Model of Non-Integer-Dimensional Space for Anisotropic Solids with q-Deformed Derivatives

TL;DR

This paper introduces a novel non-integer-dimensional model for anisotropic solids using q-deformed derivatives, linking nonextensive statistics to physical heterogeneity and accurately fitting experimental data.

Contribution

It develops a unified analytical framework incorporating q-deformed derivatives to model anisotropic thermodynamic properties in solids, connecting microscopic disorder to macroscopic behavior.

Findings

Derived explicit expressions for phonon density of states and specific heat.

Achieved excellent agreement with experimental data across temperature ranges.

Linked the deformation parameter q to microscopic disorder and memory effects.

Abstract

We propose a non-integer-dimensional spatial model for anisotropic solids by incorporating a q-deformed derivative operator, inspired by the Tsallis nonadditive entropy framework. This generalization provides an analytical framework to explore anisotropic thermal properties, within a unified and flexible mathematical formalism. We derive explicit expressions for the phonon density of states and specific heat capacity, highlighting the impact of the deformation parameter q on the thermodynamic behavior. We apply the model to various solid-state materials, achieving excellent agreement with experimental data across a wide temperature range, and demonstrating its effectiveness in capturing anisotropic and subextensive effects in real systems. Beyond providing accurate fits, we anchor the q-deformation in a microscopic disorder/kinetics exponent \mu emerging from conformable dynamics, thereby linking nonextensive statistics to measurable heterogeneity and memory effects.