An independent, general method for checking consistency between diffraction data and partial radial distribution functions derived from them: the example of liquid water

There are various routes for deriving partial radial distribution functions of disordered systems from experimental diffraction (and/or EXAFS) data. Due to limitations and errors of experimental data, as well as to imperfections of the evaluation procedures, it is of primary importance to confirm that the end result (partial radial distribution functions) and the primary information (diffraction data) are consistent with each other. We introduce a simple approach, based on Reverse Monte Carlo modelling, that is capable of assessing this dilemma. As a demonstration, we use the most frequently cited set of "experimental" partial radial distribution functions on liquid water and investigate whether the 3 partials (O-O, O-H and H-H) are consistent with the total structure factor of pure liquid D_2O from neutron diffraction and that of H_2O from X-ray diffraction. We find that while neutron diffraction on heavy water is in full agreement with all the 3 partials, the addition of X-ray diffraction data clearly shows problems with the O-O partial radial distribution function. We suggest that the approach introduced here may also be used to establish whether partial radial distribution functions obtained from statistical theories of the liquid state are consistent with the measured structure factors.Comment: 6 pages, 3 figure

Anisotropic distribution functions for spherical galaxies

A method is presented for finding anisotropic distribution functions for stellar systems with known, spherically symmetric, densities, which depends only on the two classical integrals of the energy and the magnitude of the angular momentum. It requires the density to be expressed as a sum of products of functions of the potential and of the radial coordinate. The solution corresponding to this type of density is in turn a sum of products of functions of the energy and of the magnitude of the angular momentum. The products of the density and its radial and transverse velocity dispersions can be also expressed as a sum of products of functions of the potential and of the radial coordinate. Several examples are given, including some of new anisotropic distribution functions. This device can be extended further to the related problem of finding two-integral distribution functions for axisymmetric galaxies.Comment: 5 figure

Effects of Bridge Functions on Radial Distribution Functions of Liquid Water

In this report the radial distribution functions (RDFs) of liquid water are calculated on the basis of the classical density functional theory combined with the reference interaction site model for molecular liquids. The bridge functions, which are neglected in the hypernetted-chain (HNC) approximation, are taken into account through the density expansion for the Helmholtz free energy functional up to the third order. A factorization approximation to the ternary direct correlation functions in terms of the site-site pair correlation functions is then employed in the expression of the bridge functions, thus leading to a closed set of integral equations for the determination of the RDFs. It is confirmed through numerical calculations that incorporation of the oxygen-oxygen bridge function substantially improves the poor descriptions by the HNC approximation at room temperature, e.g., for the second peak of the oxygen-oxygen RDF.Comment: 2 figures, Interdisciplinary Sciences: Computational Life Sciences (2014

An analysis of the fluctuation potential in the modified Poisson-Boltzmann theory for restricted primitive model electrolytes

An approximate analytical solution to the fluctuation potential problem in the modified Poisson-Boltzmann theory of electrolyte solutions in the restricted primitive model is presented. The solution is valid for all inter-ionic distances, including contact values. The fluctuation potential solution is implemented in the theory to describe the structure of the electrolyte in terms of the radial distribution functions, and to calculate some aspects of thermodynamics, viz., configurational reduced energies, and osmotic coefficients. The calculations have been made for symmetric valence 1:1 systems at the physical parameters of ionic diameter $4.25 \times 10^{-10}$ m, relative permittivity 78.5, absolute temperature 298 K, and molar concentrations 0.1038, 0.425, 1.00, and 1.968. Radial distribution functions are compared with the corresponding results from the symmetric Poisson-Boltzmann, and the conventional and modified Poisson-Boltzmann theories. Comparisons have also been done for the contact values of the radial distributions, reduced configurational energies, and osmotic coefficients as functions of electrolyte concentration. Some Monte Carlo simulation data from the literature are also included in the assessment of the thermodynamic predictions. Results show a very good agreement with the Monte Carlo results and some improvement for osmotic coefficients and radial distribution functions contact values relative to these theories. The reduced energy curve shows excellent agreement with Monte Carlo data for molarities up to 1 mol/dm$^{3}$.Comment: 16 pages, 8 figures, 3 table

Selective-pivot sampling of radial distribution functions in asymmetric liquid mixtures

We present a Monte Carlo algorithm for selectively sampling radial distribution functions and effective interaction potentials in asymmetric liquid mixtures. We demonstrate its efficiency for hard-sphere mixtures, and for model systems with more general interactions, and compare our simulations with several analytical approximations. For interaction potentials containing a hard-sphere contribution, the algorithm yields the contact value of the radial distribution function.Comment: 5 pages, 5 figure

Constructing smooth potentials of mean force, radial, distribution functions and probability densities from sampled data

In this paper a method of obtaining smooth analytical estimates of probability densities, radial distribution functions and potentials of mean force from sampled data in a statistically controlled fashion is presented. The approach is general and can be applied to any density of a single random variable. The method outlined here avoids the use of histograms, which require the specification of a physical parameter (bin size) and tend to give noisy results. The technique is an extension of the Berg-Harris method [B.A. Berg and R.C. Harris, Comp. Phys. Comm. 179, 443 (2008)], which is typically inaccurate for radial distribution functions and potentials of mean force due to a non-uniform Jacobian factor. In addition, the standard method often requires a large number of Fourier modes to represent radial distribution functions, which tends to lead to oscillatory fits. It is shown that the issues of poor sampling due to a Jacobian factor can be resolved using a biased resampling scheme, while the requirement of a large number of Fourier modes is mitigated through an automated piecewise construction approach. The method is demonstrated by analyzing the radial distribution functions in an energy-discretized water model. In addition, the fitting procedure is illustrated on three more applications for which the original Berg-Harris method is not suitable, namely, a random variable with a discontinuous probability density, a density with long tails, and the distribution of the first arrival times of a diffusing particle to a sphere, which has both long tails and short-time structure. In all cases, the resampled, piecewise analytical fit outperforms the histogram and the original Berg-Harris method.Comment: 14 pages, 15 figures. To appear in J. Chem. Phy

On the spectral distribution of kernel matrices related to\ud radial basis functions

This paper focuses on the spectral distribution of kernel matrices related to radial basis functions. The asymptotic behaviour of eigenvalues of kernel matrices related to radial basis functions with different smoothness are studied. These results are obtained by estimated the coefficients of an orthogonal expansion of the underlying kernel function. Beside many other results, we prove that there are exactly (k+d−1/d-1) eigenvalues in the same order for analytic separable kernel functions like the Gaussian in Rd. This gives theoretical support for how to choose the diagonal scaling matrix in the RBF-QR method (Fornberg et al, SIAM J. Sci. Comput. (33), 2011) which can stably compute Gaussian radial basis function interpolants