Chemical potential of a test hard sphere of variable size in a hard-sphere fluid

The Lab\'ik and Smith Monte Carlo simulation technique to implement the Widom particle insertion method is applied using Molecular Dynamics (MD) instead to calculate numerically the insertion probability, $P_0(\eta,\sigma_0)$, of tracer hard-sphere (HS) particles of different diameters, $\sigma_0$, in a host HS fluid of diameter $\sigma$ and packing fraction, $\eta$, up to $0.5$. It is shown analytically that the only polynomial representation of $-\ln P_0(\eta,\sigma_0)$ consistent with the limits $\sigma_0\to 0$ and $\sigma_0\to\infty$ has necessarily a cubic form, $c_0(\eta)+c_1(\eta)\sigma_0/\sigma+c_2(\eta)(\sigma_0/\sigma)^2+c_3(\eta)(\sigma_0/\sigma)^3$. Our MD data for $-\ln P_0(\eta,\sigma_0)$ are fitted to such a cubic polynomial and the functions $c_0(\eta)$ and $c_1(\eta)$ are found to be statistically indistinguishable from their exact solution forms. Similarly, $c_2(\eta)$ and $c_3(\eta)$ agree very well with the Boubl\'ik-Mansoori-Carnahan-Starling-Leland and Boubl\'ik-Carnahan-Starling-Kolafa formulas. The cubic polynomial is extrapolated (high density) or interpolated (low density) to obtain the chemical potential of the host fluid, or $\sigma_{0}\to\sigma$, as $\beta\mu^{\text{ex}}=c_0+c_1+c_2+c_3$. Excellent agreement between the Carnahan-Starling and Carnahan-Starling-Kolafa theories with our MD data is evident.