Penetrable square-well fluids: Exact results in one dimension

We introduce a model of arractive penetrable spheres by adding short range attractive square well outside a penetrable core, and we provide a detailed analysis of structural and thermodynamical properties in one dimension using the exact unpenetrable counterpart as a starting point. The model is expected to describe star polymers in regimes of good and moderate solvent under dilute conditions. We derive the exact coefficients of a low density expansion up to second order for the radial distribution function and up to forth order in the virial expansion. These results are compared with approximate solutions (Percus-Yevick and hypernetted chain) which are valid for arbitrary density by finding an overall good agreement. Notwithstanding the lack of an exact solution, our results are thus expected to be rather precise within a wide range of temperatures and densities. A detailed analysis of the limiting cases is carried out. In particular we provide a complete solution of the penetrable sticky hard sphere model in one dimension up to the same order in density. The issue of Ruelle thermodynamics stability is analyzed and the region of a well defined thermodynamic limit is identified.