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, $V_{\text{ex}}\left(x\right)$, and repel each other via a class of pairwise interaction potentials $V_{\text{int}}\left(r\right)$ (where $r$ is distance between a pair of particles) such that $V_{\text{int}}\sim {\left|r\right|}^{-k}$ 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 behavior for $k>0$ and the entropy contribution dominates for $k<0$, and both contribute equivalently in the $k\to 0$ limit (finite-range log-gas). Given the fact that this family of systems is of broad relevance, our analytical findings are of paramount importance. These results are in excellent agreement with brute-force Monte Carlo simulations.

中文翻译：

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一类有限范围排斥相互作用将粒子限制在任意势中

在本文中，我们开发了一个 $ñ$ 系统的场论 $ñ$一维热平衡下的经典粒子。粒子被任意的外部电势限制，$V_{\text{前}}（X）$，并通过一类成对的交互电位相互排斥 $V_{\text{整型}}（\mathrm{[R}）$ （哪里 $\mathrm{[R}$ 是一对粒子之间的距离） $V_{\text{整型}}〜{\left|\mathrm{[R}\right|}^{-ķ}$ 什么时候 $\mathrm{[R}\to 0$。我们考虑每个粒子都与之相互作用的情况$d$（有限范围参数）其左侧和右侧的粒子数。由于外部约束，成对排斥和熵之间复杂的相互作用，在三种情况下密度表现出明显不同的行为$ķ>0， ķ\to 0$和 $ķ<0$。根据这个场论，我们可以解析地计算出大型油田的平均密度分布$ñ$在这些制度中。我们表明，互动的贡献支配着集体行为$ķ>0$ 熵贡献占主导地位 $ķ<0$，并且两者在 $ķ\to 0$极限（有限范围的对数气体）。鉴于该系列系统具有广泛的相关性，因此我们的分析发现至关重要。这些结果与蛮力蒙特卡洛模拟非常吻合。