Microscopic Origins of Conformable Dynamics: From Disorder to Deformation

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.