Noro-Frenkel scaling in short-range square well: A Potential Energy Landscape study

We study the statistical properties of the potential energy landscape of a system of particles interacting via a very short-range square-well potential (of depth $-u_0$), as a function of the range of attraction $\mathrm{\Delta }$ to provide thermodynamic insights of the Noro and Frenkel [ M.G. Noro and D. Frenkel, J.Chem.Phys. {\bf 113}, 2941 (2000)] scaling. We exactly evaluate the basin free energy and show that it can be separated into a {\it vibrational} ($\mathrm{\Delta }$-dependent) and a {\it floppy} ($\mathrm{\Delta }$-independent) component. We also show that the partition function is a function of $\mathrm{\Delta }{e}^{\beta u_o}$, explaining the equivalence of the thermodynamics for systems characterized by the same second virial coefficient. An outcome of our approach is the possibility of counting the number of floppy modes (and their entropy).