Map of Discrete System into Continuous

TL;DR

This paper explores how discrete systems with long-range interactions can be mapped into continuous models, resulting in fractional differential equations, specifically using Riesz derivatives for one-dimensional coupled oscillators.

Contribution

It introduces a general method to map discrete long-range interacting systems into continuous fractional models, expanding the understanding of such systems.

Findings

Discrete models are mapped into continuous equations with Riesz fractional derivatives.

Long-range interactions lead to fractional differential equations in the continuum limit.

The approach applies to one-dimensional coupled oscillator systems.

Abstract

Continuous limits of discrete systems with long-range interactions are considered. The map of discrete models into continuous medium models is defined. A wide class of long-range interactions that give the fractional equations in the continuous limit is discussed. The one-dimensional systems of coupled oscillators for this type of long-range interactions are considered. The discrete equations of motion are mapped into the continuum equation with the Riesz fractional derivative.