On the Free Energy of the Hard Sphere Fluid

The equilibrium properties of the hard sphere fluid are investigated in the low/medium density regime by finding upper and lower bounds for the free energy of the system. It is shown that the coincidence of the bounds in the low density limit justifies the Van der Waal's approximation at very low densities. A better approximation is obtained valid for moderately higher densities. Similarities and properties of the equilibrium packing and parking processes are employed, allowing a simple expression to be obtained for the partition function of the hard sphere fluid.