Fractional generalization of the Ginzburg-Landau equation: An unconventional approach to critical phenomena in complex media

TL;DR

This paper introduces a fractional derivative approach to modify the Ginzburg-Landau equation, capturing nonlocal effects and fractal influences in critical phenomena, leading to a renormalized transition temperature.

Contribution

It extends the classical Ginzburg-Landau theory by incorporating fractional derivatives to model complex media with nonlocal interactions.

Findings

Fractional modifications affect the free energy functional at criticality.

The fractional Ginzburg-Landau equation predicts a shift in transition temperature.

Nonlocal effects are effectively modeled using fractional calculus.

Abstract

Equations built on fractional derivatives prove to be a powerful tool in the description of complex systems when the effects of singularity, fractal supports, and long-range dependence play a role. In this paper, we advocate an application of the fractional derivative formalism to a fairly general class of critical phenomena when the organization of the system near the phase transition point is influenced by a competing nonlocal ordering. Fractional modifications of the free energy functional at criticality and of the widely known Ginzburg-Landau equation central to the classical Landau theory of second-type phase transitions are discussed in some detail. An implication of the fractional Ginzburg-Landau equation is a renormalization of the transition temperature owing to the nonlocality present.