Evolutionary optimization of the Verlet closure relation for the hard-sphere and square-well fluids

The Ornstein-Zernike equation is solved for the hard-sphere and square-well fluids using a diverse selection of closure relations; the attraction range of the square-well is chosen to be $λ=1.5.$ In particular, for both fluids we mainly focus on the solution based on a three-parameter version of the Verlet closure relation [Mol. Phys. 42, 1291-1302 (1981)]. To find the free parameters of the latter, an unconstrained optimization problem is defined as a condition of thermodynamic consistency based on the compressibility and solved using Evolutionary Algorithms. For the hard-sphere fluid, the results show good agreement when compared with mean-field equations of state and accurate computer simulation results; at high densities, i.e., close to the freezing transition, expected (small) deviations are seen. In the case of the square-well fluid, a good agreement is observed at low and high densities when compared with event-driven molecular dynamics computer simulations. For intermediate densities, the explored closure relations vary in terms of accuracy. Our findings suggest that a modification of the optimization problem to include, for example, additional thermodynamic consistency criteria could improve the results for the type of fluids here explored.