Particles confined in arbitrary potentials with a class of finite-ranged interactions

In this paper, we develop a large-$N$ field theory for a system of $N$ classical particles in one dimension at thermal equilibrium. The particles are confined by an arbitrary external potential, $V_\text{ex} (x)$, and repel each other via a class of pairwise interaction potentials $V_\text{int}(r)$ (where $r$ is distance between a pair of particles) such that $V_\text{int} \sim |r|^{-k}$ when $r \to 0$. We consider the case where every particle is interacting with $d$ (finite range parameter) number of particles to its left and right. Due to the intricate interplay between external confinement, pairwise repulsion and entropy, the density exhibits markedly distinct behavior in three regimes $k>0$, $k \to 0$ and $k<0$. From this field theory, we compute analytically the average density profile for large $N$ in these regimes. We show that the contribution from interaction dominates the collective behaviour for $k > 0$ and the entropy contribution dominates for $k<0$, and both contributes equivalently in the $k\to 0$ limit (finite range log-gas). Given the fact that these family of systems are of broad relevance, our analytical findings are of paramount importance. These results are in excellent agreement with brute-force Monte-Carlo simulations.