Conformable scaling and critical phenomena: a unified framework for phase transitions

Abstract We investigate the application of conformable derivatives to model critical phenomena near continuous phase transition points. By incorporating a deformation parameter into the differential structure, we derive unified expressions for thermodynamic observables such as heat capacity, magnetisation, susceptibility, and coherence length, each exhibiting a power-law behaviour near the critical temperature. The conformable derivative framework naturally embeds scale invariance and critical slowing down into the dynamics without resorting to fully nonlocal fractional calculus. Modified Ginzburg–Landau equations are constructed to model superconducting transitions, leading to analytical expressions for the order parameter and the London penetration depth. Experimental data from niobium confirm the model’s applicability, showing excellent fits and capturing asymmetric scaling behaviour around the critical point. This work offers a bridge between classical mean-field theory and generalised scaling frameworks, with implications for both theoretical modelling and experimental analysis.