Disorder and critical phenomena

TL;DR

This paper explores how fractal defect distributions influence critical phenomena, showing that combined random field and temperature defects can alter the system's statistical distribution and critical behavior.

Contribution

It introduces a fractional-order extension of the Landau-Khalatnikov equation and derives a renormalization group framework for systems with fractal defects.

Findings

Distribution shifts from Gibbs to Tsallis or q-type statistics.

Critical indices are calculated for the modified system.

Fractional differential equations describe the order parameter dynamics.

Abstract

In the present paper we are discussing the influence of the fractal distribution of defects on the critical behavior of the system. We consider a case when the equation of motion for the order parameter contains influences of defects of random field and random temperature types at the same time and we show that it can lead to a change of distribution function from Gibbs distribution to Tsallis distribution or more general statistics of q-type. We are able to extend the Landau-Khalatnikov equation for the order parameter and represent it in the form of a differential equation of the fractional order. We deduce and we solve the corresponding renormalization group equation with the nonlinear dispersion law and we calculate the critical indices of the system.