Chemical potential and entropy in monodisperse and polydisperse hard-sphere fluids using Widom’s particle insertion method and a pore size distribution-based insertion probability

Rejection- and importance-sampling-based perfect simulation for Gibbs hard-sphere models

Coupling-from-the-past (CFTP) methods have been used to generate perfect samples from finite Gibbs hard-sphere models, an important class of spatial point processes consisting of a set of spheres with the centers on a bounded region that are distributed as a homogeneous Poisson point process (PPP) conditioned so that spheres do not overlap with each other. We propose an alternative importance-sampling-based rejection methodology for the perfect sampling of these models. We analyze the asymptotic expected running time complexity of the proposed method when the intensity of the reference PPP increases to infinity while the (expected) sphere radius decreases to zero at varying rates. We further compare the performance of the proposed method analytically and numerically with that of a naive rejection algorithm and of popular dominated CFTP algorithms. Our analysis relies upon identifying large deviations decay rates of the non-overlapping probability of spheres whose centers are distributed as a homogeneous PPP.

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

A detailed comparison between the Boublík-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state of hard-sphere mixtures is made with Molecular Dynamics (MD) simulations of the same compositions. The Labík and Smith simulation technique [S. Labík and W. R. Smith, Mol. Simul. 12, 23-31 (1994)] was used to implement the Widom particle insertion method to calculate the excess chemical potential, βμ₀^ex, of a test particle of variable diameter, σ₀, immersed in a hard-sphere fluid mixture with different compositions and values of the packing fraction, η. Use is made of the fact that the only polynomial representation of βμ₀^ex which is consistent with the limits σ₀ → 0 and σ₀ → ∞ has to be of the cubic form, i.e., c₀(η)+c¯₁(η)σ₀/M₁+c¯₂(η)(σ₀/M₁)²+c¯₃(η)(σ₀/M₁)³, where M₁ is the first moment of the distribution. The first two coefficients, c₀(η) and c¯₁(η), are known analytically, while c¯₂(η) and c¯₃(η) were obtained by fitting the MD data to this expression. This in turn provides a method to determine the excess free energy per particle, βa^ex, in terms of c¯₂, c¯₃, and the compressibility factor, Z. Very good agreement between the BMCSL formulas and the MD data is found for βμ₀^ex, Z, and βa^ex for binary mixtures and continuous particle size distributions with the top-hat analytic form. However, the BMCSL theory typically slightly underestimates the simulation values, especially for Z, differences which the Boublík-Carnahan-Starling-Kolafa formulas and an interpolation between two Percus-Yevick routes capture well in different ranges of the system parameter space.

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

The Labík 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, P₀(η,σ₀), of tracer hard-sphere (HS) particles of different diameters, σ₀, in a host HS fluid of diameter σ and packing fraction, η, up to 0.5. It is shown analytically that the only polynomial representation of -ln⁡P₀(η,σ₀) consistent with the limits σ₀→0 and σ₀→∞ has necessarily a cubic form, c₀(η)+c₁(η)σ₀/σ+c₂(η)(σ₀/σ)²+c₃(η)(σ₀/σ)³. Our MD data for -ln⁡P₀(η,σ₀) are fitted to such a cubic polynomial and the functions c₀(η) and c₁(η) are found to be statistically indistinguishable from their exact solution forms. Similarly, c₂(η) and c₃(η) agree very well with the Boublík-Mansoori-Carnahan-Starling-Leland and Boublík-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 σ₀→σ, as βμ^ex=c₀+c₁+c₂+c₃. Excellent agreement between the Carnahan-Starling and Carnahan-Starling-Kolafa theories with our MD data is evident.