Radial distribution function: Difference between revisions

Revision as of 01:55, 29 August 2015

Revision as of 06:43, 7 September 2015

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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 r and r+dr 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.

Given a [[potential energy]] function, the radial distribution function can be computed either via computer simulation methods like the [[Monte Carlo method]], or via the [[Ornstein-Zernike equation]], using approximative closure relations like the [[Percus-Yevick approximation]] or the [[Hypernetted-chain equation|Hypernetted Chain Theory]]. It can also be determined experimentally, by radiation scattering techniques or by direct visualization for large enough (micrometer-sized) particles via traditional or confocal microscopy.

The radial distribution function is of fundamental importance since it can be used, using the [[Kirkwood–Buff solution theory]], to link the microscopic details to macroscopic properties. Moreover, by the reversion of the Kirkwood-Buff theory, it is possible to attain the microscopic details of the radial distribution function from the macroscopic properties.

==Definition==

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{{NumBlk|:| <math>g(\mathbf{r}) = \frac{1}{\rho} \langle \sum_{i \neq 0} \delta ( \mathbf{r} - \mathbf{r}_i) \rangle = V \frac{N-1}{N} \left \langle \delta ( \mathbf{r} - \mathbf{r}_1) \right \rangle</math>|{{EquationRef|5}}}}

where the second equality requires the equivalence of particles <math>\textstyle 1, \, \ldots, \, N-1</math>. The formula above is useful for relating <math>g(\mathbf{r})</math> to the static structure factor <math>S(\mathbf{q})</math>, defined by <math>\textstyle S(\mathbf{q}) = 1/N \langle \sum_{ij} \mathrm{e}^{-i \mathbf{q} (\mathbf{r}_i - \mathbf{r}_j)} \rangle</math>, since we have:

: <math>

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===Thermodynamic properties===

The radial distribution function is an important measure because several key thermodynamic properties, such as potential energy and pressure can be calculated from it.

For a 3-D system where particles interact via pairwise potentials, the potential energy of the system can be calculated as follows:<ref name=softmatter>{{cite book|last=Frenkel|first=Daan; Smit, Berend|title=Understanding molecular simulation from algorithms to applications|year=2002|publisher=Academic Press|___location=San Diego|isbn=0122673514|edition=2nd ed.}}</ref>

<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, u(r) is the [[pair potential]].

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