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Free energy is a fundamental property of a thermodynamic system, from which pressure, entropy, and other interesting properties can be derived. It is useful, then, to be able to accurately compute the free energy at various densities and temperatures in a way that can serve as the basis for further thermodynamic predictions.
We use Monte Carlo simulations to compute the free energy of a homogeneous hard sphere fluid, as a function of the filling fraction. We find the free energy by shrinking a valid configuration in which the component spheres are non-overlapping and checking for overlaps in the smaller volume. We develop techniques to optimize the free energy simulation for speed and accuracy, such as neighbor tables and run length tracking. Finding the absolute free energy is also discussed. This model will serve as the foundation for future simulations of more complex fluids, as free energy is an essential quantity to understand and hard spheres are the standard basis model for fluids.
We find that our free energy simulation agrees with the Carnahan-Starling equation of state, and is able to accurately predict free energy up into relatively dense states. This simulation can serve as the basis for a more complete fluid simulation that includes attractive square well interactions to model phenomena such as surface tension.