Microscopic origins of conformable dynamics: From disorder to deformation

Abstract

Conformable derivatives provide a mathematically tractable approach to modeling anomalous relaxation and scaling in complex systems, yet their physical origin remains poorly understood. We address this gap by deriving conformable relaxation dynamics from first principles. Our approach is based on a spatially-resolved Ginzburg–Landau model incorporating quenched disorder and temperature-dependent kinetic coefficients. Applying statistical averaging and transport-theoretic arguments, we demonstrate that spatial heterogeneity and energy barrier distributions generate power-law memory kernels of the form $K\left(\tau \right)\sim {\tau }^{\mu -1}$. In the adiabatic limit, these memory effects reduce to a local conformable evolution law $T^{1-\mu }d\psi /dT$. We show that the deformation parameter $\mu$ is directly linked to measurable quantities such as transport coefficients, susceptibility, energy barrier distributions, and the underlying disorder exponent. Furthermore, $\mu$ is related to Tsallis nonextensive entropy via the relation $\mu =1/(q-1)$. These results establish a microscopic foundation for conformable dynamics in disordered media, provide a physical interpretation of the deformation parameter, ensure thermodynamic consistency with entropy production, and yield experimentally testable predictions. Observable consequences include specific relaxation spectra and susceptibility decay patterns. Overall, the framework unifies memory effects, nonextensive thermodynamics, and critical phenomena within a coherent and physically grounded description.