Pair correlation function, statistical analysis

The adsorbed colloid systems also can be used for mimicking fluctuation phenomena occurring on molecular level. This is so because the adsorbed particle monolayer resembles a frozen fluctuation whose statistical analysis can furnish interesting structural information of general validity. In order to express quantitatively fluctuation phenomena, one usually determines the variance of the number of particles absorbed over equal-sized surface areas S [181-183]. This quantity is connected with the pair-correlation function by the Omstein-Zemicke relationship [132]... 

Analysis of the correlation functions demonstrates also impressive general agreement between the superposition approximation and computer simulations. Note, however certain overestimate of the similar particle correlations, X r,t), at small r, especially for d = 1. In its turn the correlation function of dissimilar particles, Y(r,t), demonstrates complete agreement with the statistical simulations. Since the time development of concentrations is defined entirely by Y(r, t), Figs 5.2 and 5.3 serve as an additional evidence for the reliability of the superposition approximation. An estimate of the small distances here at which the function Y (r, t) is no longer zero corresponds quite well to the earlier introduced correlation length o, equation (5.1.47) as one can see in fact that at moment t there are no AB pairs separated by r < o-... 

Crystal structures can be determined exactly by means of x-ray diffraction and the periodicity of the lattice introduces some simplicity into the mathematical analysis. There is no such simplicity for amorphous materials. Only a statistical description is possible. In particular, a one-dimensional correlation function is often presented in the form of a radial distribution function, which is a pair-distribution function averaged over all atomic pairs. It is compatible with a large number of possible structures. The challenge is to separate out one of these realistic and compatible structures from the even greater number of random networks that are poor representations of the structure. 

To estimate errors in structural or other properties, expressed not just as numbers but as functions of either distance (e.g., g(r)), or time (e.g., time correlation functions), a similar analysis should be carried out for different values of the argument. For instance, when the time correlation function is calculated, much better statistics is obtained for short times, as there are much more data available. The time correlation function values for times comparable to the length of the simulation run are obtained with larger errors, as the average is taken over only a few pairs of values. 

In this section, we discuss the correlations and effective interactions between ions in aqueous solution. For this purpose we typically look at simulation boxes with about 2000 water molecules and add between 1 and 200 salt pairs in the solution. The main output from the simulation is the radial distribution function jij (r) between the ions. In order to obtain good statistics for further analysis, long simulation runs of about 200 ns are needed. The potential of mean force (PMF) follows by Boltzmann inversion from the radial distribution function according to... 

---