Radial distribution function: Difference between revisions

Revision as of 21:26, 14 December 2015

Revision as of 20:21, 21 March 2016

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If a given particle is taken to be at the origin O, and if <math>\rho =N/V</math> is the average number density of particles, then the local time-averaged density at a distance <math>r</math> from O is <math>\rho g(r)</math>. This simplified definition holds for a [[homogeneous]] and [[isotropic]] system. A more general case will be considered below.

In simplest terms it is a measure of the probability of finding a particle at a distance of <math>r</math> away from a given reference particle, relative to that for an ideal gas. The general algorithm involves determining how many particles are within a distance of <math>r</math> and <math>r+dr</math> away from a particle. This general theme is depicted to the right, where the red particle is our reference particle, and blue particles are those which are within the circular shell, dotted in orange.

The RDF is usually determined by calculating the distance between all particle pairs and binning them into a histogram. The histogram is then normalized with respect to an ideal gas, where particle histograms are completely uncorrelated. For three dimensions, this normalization is the number density of the system multiplied by the volume of the spherical shell, which mathematically can be expressed as <math>g(r)_I = 4\pi r^2\rho dr</math>, where <math>\rho</math> is the number density.

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<math>PE=\frac{N}{2}4\pi\rho\int^{\infty}_0r^2u(r)g(r)dr </math>

Where N is the number of particles in the system, <math> \rho </math> is the number density, <math> u(r)</math> is the [[pair potential]].

The pressure of the system can also be calculated by relating the 2nd [[virial coefficient]] to <math> g(r)</math> . The pressure can be calculated as follows:<ref name=softmatter/>

<math>P = \rho k_BT-\frac{2}{3}\pi\rho^2\int_{0}^{\infty}dr\frac{du(r)}{dr}r^3g(r)</math>

Where <math>T</math> is the temperature and k<submath>Bk_B</submath> is Boltzmann's constant. Note that the results of potential and pressure will not be as accurate as directly calculating these properties because of the averaging involved with the calculation of <math>g(r)</math>.

==Approximations==

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{{NumBlk|:| <math>g(r) = \exp \left [ -\frac{u(r)}{kT} \right ] </math>.|{{EquationRef|11}}}}

If <math>u(r)</math> were zero for all <math>r</math> &ndash; i.e., if the particles did not exert any influence on each other, then '<math>g(r) = 1 </math> for all ''<math>\bold{r''}</math> and the mean local density would be equal to the mean density <math>\rho</math>: the presence of a particle at O would not influence the particle distribution around it and the gas would be ideal. For distances <math>r</math> such that <math>u(r)</math> is significant, the mean local density will differ from the mean density <math>\rho</math>, depending on the sign of <math>u(r)</math> (higher for negative interaction energy and lower for positive <math>u(r)</math>).

As the density of the gas increases, the low-density limit becomes less and less accurate since a particle situated in <math>\mathbf{r}</math> experiences not only the interaction with the particle in O but also with the other neighbours, themselves influenced by the reference particle. This mediated interaction increases with the density, since there are more neighbours to interact with: it makes physical sense to write a density expansion of <math>g(r)</math>, which resembles the [[virial equation]]: