Conformable Scaling and Critical Dynamics: A Unified Framework for Phase Transitions

TL;DR

The paper introduces a conformable-derivative framework with deformation parameter $\mu$ to model critical phenomena near continuous phase transitions. By embedding temperature-weighted scaling directly into the differential structure, it derives unified power-law expressions for key observables and expresses critical exponents as functions of $\mu$ and the critical temperature $T_c$, thereby bridging classical mean-field theory and generalized (nonextensive) scaling without full nonlocal fractional calculus. A modified Ginzburg-Landau equation incorporating conformable kinetics is developed and applied to superconducting transitions, yielding analytic predictions for the order parameter, London penetration depth, specific heat, and coherence length. The framework is validated against niobium data, showing good fits and revealing asymmetric scaling across $T_c$, which supports the model’s ability to capture finite-size, memory, and disorder effects. Overall, the work provides a tractable, thermodynamically consistent approach to generalized critical dynamics with potential applicability to a broad class of phase transitions.

Abstract

We investigate the application of conformable derivatives to model critical phenomena near continuous phase transitions. By incorporating a deformation parameter into the differential structure, we derive unified expressions for thermodynamic observables such as heat capacity, magnetization, susceptibility, and coherence length, each exhibiting power-law behavior 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 London penetration depth. Experimental data from niobium confirm the model's applicability, showing excellent fits and capturing asymmetric scaling behavior around Tc. This work offers a bridge between classical mean-field theory and generalized scaling frameworks, with implications for both theoretical modeling and experimental analysis.

Figures (3)

• Figure 1: Heat capacity of niobium fitted by the smoothed conformable piecewise model (\ref{['eq:Cv_regularized']}) with $C_{V}(T)=B_{i}(|T-T_{c}|+\epsilon)^{-\alpha_{i}}$. Fitted parameters with 95% confidence intervals: $B_{1}=0.411\pm0.023$, $\alpha_{1}=0.644\pm0.031$ ($T<T_{c}$); and $B_{2}=0.808\pm0.041$, $\alpha_{2}=0.567\pm0.028$ ($T>T_{c}$). We used the smoothing factor $\epsilon=0.293\pm0.015$ to ensure regularization near the divergence at the critical temperature $T_{c}=8.700\pm0.005$ K. Data from brown1953. The fitting employed weighted least-squares with weights proportional to $1/\sigma_{i}^{2}$ where $\sigma_{i}$ are experimental uncertainties.
• Figure 2: Fit of the normalized London penetration depth $\lambda_{L}(T)/\lambda_{0}$, where $\lambda_{0}$ denotes the London penetration depth extrapolated to zero temperature, serving as a normalization constant for comparison between theory and experiment. The data for niobium are fitted using the conformable piecewise model [Eq. (\ref{['eq:lambdaL_piecewise_alpha']})] with the regularized form $\lambda_{L}(T)=B_{i}\left(|T-T_{c}|+\epsilon\right)^{-\alpha_{i}},$ as given in Eq. (\ref{['lambdareg']}). The fitted parameters are $B_{1}=1.57$, $\alpha_{1}=0.68$ for $T<T_{c}$ and $B_{2}=5.01$, $\alpha_{2}=0.98$ for $T\geq T_{c}$, with $\epsilon=0.055$ and $T_{c}=9.78~\text{K}$. Experimental data are taken from maxfield1965.
• Figure 3: Extracted coherence length $\xi(T)$ (blue dots) and fitted model (red line) using $\xi(T)=A(T_{c}-T)^{-\nu}$ with $\nu=0.329$ and $A=40.76$. Experimental data adapted from williamson1970bulk, see text.