  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.
