  As is well known, one-dimensional systems with interactions restricted to
first nearest neighbors admit a full analytically exact statistical-mechanical
solution. This is essentially due to the fact that the knowledge of the first
nearest-neighbor probability distribution function, $p_1(r)$, is enough to
determine the structural and thermodynamic properties of the system. On the
other hand, if the interaction between second nearest-neighbor particles is
turned on, the analytically exact solution is lost. Not only the knowledge of
$p_1(r)$ is not sufficient anymore, but even its determination becomes a
complex many-body problem. In this work we systematically explore different
approximate solutions for one-dimensional second nearest-neighbor fluid models.
We apply those approximations to the square-well and the attractive two-step
pair potentials and compare them with Monte Carlo simulations, finding an
excellent agreement.
