A Solvable Model for Decoupling of Interacting Clusters

TL;DR

This paper introduces a model that simplifies the analysis of interacting particle clusters by decoupling in-group and inter-group dynamics, enabling analytical solutions for complex many-body problems.

Contribution

It provides explicit conditions for decoupling in a system of interacting clusters, reducing the problem to independent in-group and solvable inter-group dynamics.

Findings

Decoupling conditions are derived for arbitrary in-group interactions.

The model applies to systems like ions and impurities, enabling analytical solutions.

Impurity can probe the collective dynamics of ion clusters.

Abstract

We consider M clusters of interacting particles, whose in-group interactions are arbitrary, and inter-group interactions are approximated by oscillator potentials. We show that there are masses and frequencies that decouple the in-group and inter-group degrees of freedom, which reduces the initial problem to M independent problems that describe each of the relative in-group systems. The dynamics of the M center-of-mass coordinates is described by the analytically solvable problem of M coupled harmonic oscillators. This paper derives and discusses these decoupling conditions. Furthermore, to illustrate our findings, we consider a charged impurity interacting with a ring of ions. We argue that the impurity can be used to probe the center-of-mass dynamics of the ions.