Crossover in densities of confined particles with finite range of interaction

TL;DR

This paper investigates how the equilibrium density profile of confined particles with finite-range interactions transitions from one shape to another as the interaction range varies, using analytical and numerical methods for two specific potentials.

Contribution

It provides explicit density profiles for the 1d plasma model and approximate profiles for the log-gas model across the entire range of interaction ranges, revealing diverse shape transitions.

Findings

Density profiles scale similarly to all-to-all cases for fixed interaction fraction

Explicit density profile for 1d plasma model at any interaction range

Approximate density profiles for log-gas model at extreme interaction ranges

Abstract

We study a one-dimensional classical system of $N$ particles confined within a harmonic trap. Interactions among these particles are dictated by a pairwise potential $V(x)$, where $x$ is the separation between two particles. Each particle can interact with a maximum of $d$ neighboring particles on either side (left or right), if available. By adjusting the parameter $d$, the system can be made nearest neighbour $(d=1)$ to all-to-all $(d=N-1)$ interacting. As suggested by prior studies, the equilibrium density profile of these particles is expected to undergo shape variations as $d$ is changed. In this paper, we investigate this crossover by tuning the parameter $f(=d/N)$ from $1$ to $0$ in the large $N$ limit for two distinct choices of interaction potentials, $V(x) = - |x|$ and $V(x) =- \log(|x|)$ which correspond to 1d one-component plasma and the log-gas model, respectively. For both models, the system size scaling of the density profile for fixed $f$ turns out to be the same as in their respective all-to-all cases. However, the scaling function exhibits diverse shapes as $f$ varies. We explicitly compute the average density profile for any $f \in (0,1]$ in the 1d plasma model, while for the log-gas model, we provide approximate calculations for large (close to $1$) and small (close to $0$) $f$. Additionally, we present simulation results to numerically demonstrate the crossover and compare these findings with our theoretical results.