Distribution Function of Classical Fluids of Hard Spheres. I

Measuring many-body distribution functions in fluids using test-particle insertion

We derive a hierarchy of equations, which allow a general n-body distribution function to be measured by test-particle insertion of between 1 and n particles. We apply it to measure the pair and three-body distribution functions in a simple fluid using snapshots from Monte Carlo simulations in the grand canonical ensemble. The resulting distribution functions obtained from insertion methods are compared with the conventional distance-histogram method: the insertion approach is shown to overcome the drawbacks of the histogram method, offering enhanced structural resolution and a more straightforward normalization. At high particle densities, the insertion method starts breaking down, which can be delayed by utilizing the underlying hierarchical structure of the insertion method. Our method will be especially useful in characterizing the structure of inhomogeneous fluids and investigating closure approximations in liquid state theory.

First-principles superadiabatic theory for the dynamics of inhomogeneous fluids

For classical many-body systems subject to Brownian dynamics, we develop a superadiabatic dynamical density functional theory (DDFT) for the description of inhomogeneous fluids out-of-equilibrium. By explicitly incorporating the dynamics of the inhomogeneous two-body correlation functions, we obtain superadiabatic forces directly from the microscopic interparticle interactions. We demonstrate the importance of these nonequilibrium forces for an accurate description of the one-body density by numerical implementation of our theory for three-dimensional hard-spheres in a time-dependent planar potential. The relaxation of the one-body density in superadiabatic-DDFT is found to be slower than that predicted by standard adiabatic DDFT and significantly improves the agreement with Brownian dynamics simulation data. We attribute this improved performance to the correct treatment of structural relaxation within the superadiabatic-DDFT. Our approach provides fundamental insight into the underlying structure of dynamical density functional theories and makes possible the study of situations for which standard approaches fail.

Evaluation of bridge-function diagrams via Mayer-sampling Monte Carlo simulation

We report coefficients of the h-bond expansion of the bridge function of the hard-sphere system up to order rho(4) (where rho is the density in units of the hard-sphere diameter), which in the highest-order term includes 88 cluster diagrams with bonds representing the total correlation function h(r). Calculations are performed using the recently introduced Mayer-sampling method for evaluation of cluster integrals, and an iterative scheme is applied in which the h(r) used in the cluster integrals is determined by solution of the Ornstein-Zernike equation with a closure given by the calculated clusters. Calculations are performed for reduced densities from 0.1 to 0.9 in increments of 0.1. Comparison with molecular simulation data shows that the convergence is very slow for the density expansion of the bridge function calculated this way.

Maximum entropy formulation of the Kirkwood superposition approximation

Using a variational formulation, we derive the Kirkwood superposition approximation for systems at equilibrium in the thermodynamic limit. We define the entropy of the triplet correlation function and show that the Kirkwood closure brings the entropy to its maximal value. This approach leads to a different interpretation for the Kirkwood closure relation, usually explained by probabilistic considerations of dependence and independence of particles. The Kirkwood closure is generalized to finite volume systems at equilibrium by computing the pair correlation function in finite domains. Closure relations for high order correlation functions are also found using a variational approach. In particular, maximizing the entropy of quadruplets leads to the high order closure g(1234)=g(123)g(124)g(134)g(234)/[g(12)g(13)g(14)g(23)g(24)g(34)] used in the Born-Green-Yvon 2 equations which are a pair of integral equations for the triplet and pair correlation functions.