Fractional Dynamics of Systems with Long-Range Space Interaction and Temporal Memory

TL;DR

This paper derives fractional differential equations from variational principles for systems with long-range interactions and memory, generalizing classical models like Ginzburg-Landau and Schrödinger equations to complex media.

Contribution

It introduces a novel method to derive fractional field equations from stationary action principles for systems with power-law memory and interactions.

Findings

Derived fractional Ginzburg-Landau and Schrödinger equations

Applied method to particle chains with power-law interactions

Potential applications in modeling complex media

Abstract

Field equations with time and coordinates derivatives of noninteger order are derived from stationary action principle for the cases of power-law memory function and long-range interaction in systems. The method is applied to obtain a fractional generalization of the Ginzburg-Landau and nonlinear Schrodinger equations. As another example, dynamical equations for particles chain with power-law interaction and memory are considered in the continuous limit. The obtained fractional equations can be applied to complex media with/without random parameters or processes.