Continuous Limit of Discrete Systems with Long-Range Interaction

TL;DR

This paper investigates the continuous limit of discrete systems with long-range interactions, deriving fractional medium equations with Riesz derivatives, and establishing a consistent framework for such limits.

Contribution

It introduces a general approach to define the continuous limit of systems with long-range interactions, including power-law cases, leading to fractional equations.

Findings

Derivation of fractional equations from long-range discrete systems

Mapping of chain equations into continuum models with Riesz derivatives

Establishment of a consistent framework for continuous limits

Abstract

Discrete systems with long-range interactions are considered. Continuous medium models as continuous limit of discrete chain system are defined. Long-range interactions of chain elements that give the fractional equations for the medium model are discussed. The chain equations of motion with long-range interaction are mapped into the continuum equation with the Riesz fractional derivative. We formulate the consistent definition of continuous limit for the systems with long-range interactions. In this paper, we consider a wide class of long-range interactions that give fractional medium equations in the continuous limit. The power-law interaction is a special case of this class.