Radial distribution function: Difference between revisions

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In [[computational mechanics]] and [[statistical mechanics]], a '''radial distribution function''' (RDF), ''g''(''r''), describes how the density of surrounding matter varies as a function of the distance from a distinguished point. This is normalized by the average density such that the function goes to 1 far from the distinguished point when the medium is even slightly disordered. If, for example, we choose to locate our distinguished point at the center of a hard-core particle with radius '''&sigma;''', ''g''(''r'') will be 0 for ''r'' < '''&sigma;'''.

{{Cleanup|March 2006}}

Suppose, for example, that we choose a molecule at some point O in the volume. What is then the average density at some point P at a distance r away from O? If <math>\rho=N/V</math> is the average density, then the mean density at P ''given'' that there is a molecule at O would differ from &rho; by some factor g(r). One could say that the radial distribution function takes into account the correlations in the distribution of molecules arising from the forces they exert on each other:

In [[computational mechanics]] and [[statistical mechanics]], a '''radial distribution function''' (RDF), ''g''(''r''), describes how the density of surrounding matter varies as a function of the distance from a distinguished point. This is normalized by the average density such that the function goes to 1 far from the distinguished point when the medium is even slightly disordered. If, for example, we choose to locate our distinguished point at the center of a hard-core particle with radius '''&sigma;''', ''g''(''r'') will be 0 for ''r'' < '''&sigma;'''.

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Given a [[potential energy]] function, the radial distribution function can be found via sampling -- see [[statistical mechanics]].

(mean local density at distance r from O) = <math>\rho</math>g(r) (1)

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As long as the gas is '''dilute''' the correlations in the positions of the molecules that g(r) takes into account are due to the potential <math>\phi</math>(r) that a molecule at P feels owing to the presence of a molecule at O. Using the Boltzmann distribution law:

What makes the RDF important is that for certain systems it can be used to calculate most thermodynamically interesting quantities, such as the average energy U or entropy S.

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<math>g(r) = e^{-\phi(r)/kT} </math> (2)

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If <math>\phi(r)</math> would be zero for all r - i.e. if the molecules did not exert any influence on each other g(r) = 1 for all r. Then from (1) the mean local density would be equal to the mean density <math>\rho</math>: the presence of a molecule at O would not influence on the presence or absence of any other molecule and the gas would be ideal. As long as there is a <math>\phi(r)</math> the mean local density will always be different from the mean density <math>\rho</math> due to the interactions.

When the density of the gas gets higher the so called low-density limit (2) is not applicable anymore due to the fact that the molecules attratced to and reppeled by the molecule at O also reppel and attract eac other. The correctuion terms needed to correctly describe g(r) resemble the [[virial equation]], it is an expansion in the density:

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<math>g(r)=e^{-\phi(r)/kT}+\rho g_{1}(r)+\rho^{2}g_{2}(r)+\ldots</math> (3)

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in which additional functions <math>g_{1}(r), g_{2}(r)</math> appear which may depend on temperature <math>T</math> and distance <math>r</math> but not on <math>\rho</math>.

Given a [[potential energy]] function, the radial distribution function can be found via computer simulation methods like the [[Mote Carlo method]]. It could also be calculated numerically using rigourous methods obtained from [[statistical mechanics]] like the [[Perckus-Yevick approximation]].

==Importance of g(r)==

g(r) is of fundamental importance in thermodynamics for macroscopic thermodynamic quantities can be calculated using g(r). A few examples:

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''The virial equation for the pressure:''

<math>p=\rho kT-\frac{2\pi}{3}\rho^{2}\int d r r^{3} u^{\prime}(r)/kT g(r, \rho, T) </math>

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''The energy equation:''

<math>\frac{E}{NkT}=\frac{3}{2}+\frac{\rho}{2kT}\int d r \,4\pi r^{2} u(r)g(r, \rho, T) </math>

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''The compressibility equation:''

<math>kT\left(\frac{\partial \rho}{\partial p}\right)=1+\rho \int d r [g(r)-1] </math>

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===References===

#D.A. McQuarrie, Statistical Mechanics (Harper Collins Publishers) 1976

==External links==