Fractional generalization of the Ginzburg-Landau equation: An unconventional approach to critical phenomena in complex media
arXiv:cond-mat/0309577 · doi:10.1016/j.physleta.2005.01.047
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.
10 pages, improved content, submitted for publication to Phys. Lett. A
References in corpus (7)
- Electrical Resistivity Anisotropy from Self-Organized One-Dimensionality in High-Temperature Superconductors
- Some Applications of Fractional Equations
- Dimensional Crossover in Quasi One-Dimensional and High T_c Superconductors
- Current ideas on the origin of stripes
- Competing Order in the Mixed State of High Temperature Superconductors
- Fracton pairing mechanism for "strange" superconductors: Self-assembling organic polymers and copper-oxide compounds
- Experimental evidence of a fractal dissipative regime in high-T_c superconductors
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