Microscopic Origins of Conformable Dynamics: From Disorder to Deformation

TL;DR

This paper derives conformable relaxation dynamics from microscopic principles, linking them to physical properties and providing a solid foundation for their use in modeling complex systems.

Contribution

It offers a systematic derivation of conformable derivatives from a microscopic Ginzburg-Landau model with disorder, connecting the deformation parameter to measurable physical quantities.

Findings

Emergent power-law memory kernels from disorder and heterogeneity

Connection between conformable parameter  and physical properties

Unified framework for memory effects and anomalous relaxation

Abstract

Conformable derivatives have attracted increasing interest for bridging classical and fractional calculus while retaining analytical tractability. However, their physical foundations remain underexplored. In this work, we provide a systematic derivation of conformable relaxation dynamics from microscopic principles. Starting from a spatially-resolved Ginzburg-Landau framework with quenched disorder and temperature-dependent kinetic coefficients, we demonstrate how spatial heterogeneity and energy barrier distributions give rise to emergent power-law memory kernels. In the adiabatic limit, these kernels reduce to a conformable temporal structure of the form T^{1-\mu}\,d\psi/dT. The deformation parameter \mu is shown to be connected to experimentally measurable properties such as transport coefficients, disorder statistics, and relaxation time spectra. This formulation also reveals a natural link with nonextensive thermodynamics and Tsallis entropy. By unifying memory effects, anomalous relaxation, and spatial correlations under a coherent physical mechanism, our framework transforms conformable derivatives from heuristic tools into physically grounded operators suitable for modeling complex critical dynamics.