Structure of penetrable-rod fluids: Exact properties and comparison   between Monte Carlo simulations and two analytic theories
  Bounded potentials are good models to represent the effective two-body interaction in some colloidal systems, such as dilute solutions of polymer chains in good solvents. The simplest bounded potential is that of penetrable spheres, which takes a positive finite value if the two spheres are overlapped, being 0 otherwise. Even in the one-dimensional case, the penetrable-rod model is far from trivial, since interactions are not restricted to nearest neighbors and so its exact solution is not known. In this paper we first derive the exact correlation functions of penetrable-rod fluids to second order in density at any temperature, as well as in the high-temperature and zero-temperature limits at any density. Next, two simple analytic theories are constructed: a high-temperature approximation based on the exact asymptotic behavior in the limit $T\to\infty$ and a low-temperature approximation inspired by the exact result in the opposite limit $T\to 0$. Finally, we perform Monte Carlo simulations for a wide range of temperatures and densities to assess the validity of both theories. It is found that they complement each other quite well, exhibiting a good agreement with the simulation data within their respective domains of applicability and becoming practically equivalent on the borderline of those domains. A perspective on the extension of both approaches to the more realistic three-dimensional case is provided. 
Alexandr Malijevsky and Andres Santos