Perturbation theory for very long-range potentials

TL;DR

This paper introduces two innovative methods to analyze systems with very long-range interactions, such as gravitational and charged systems, by approximating the original potential with more manageable forms for statistical mechanics analysis.

Contribution

Proposes two novel approaches—finite sum of Yukawa potentials and damped potentials—to facilitate the study of long-range interacting systems.

Findings

Potential expressed as finite sum of Yukawa potentials

Potential rewritten as a damped interaction with a controllable range

Methods enable conventional statistical mechanics analysis of long-range systems

Abstract

Systems with very long-range interactions (that decay at large distances like $U(r)\sim r^{-l}$ with $l\le d$ where $d$ is the space dimensionality) are difficult to study by conventional statistical mechanics perturbation methods. Examples of these systems are gravitational and charged (non-electroneutral). In this work we propose two alternative methodologies to avoid these difficulties and capture some of the properties of the original potential. The first one consists in expressing the original potential in terms of a finite sum of hard-core Yukawa potentials. In the second one, the potential is rewritten as a damped potential, using a damping function with a parameter that controls the range of the interaction. These new potentials, which mimic the original one, can now be treated by conventional statistical mechanics methods.