Radial distribution function: Difference between revisions

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==Definition==

Consider a system of <math>N</math> particles in a volume <math>V</math> (for an average number density <math>\rho =N/V</math>) and at a temperature <math>T</math> (let us also define <math>\textstyle \beta = \frac{1}{kT}</math>). The particle coordinates are <math>\mathbf{r}_{i}</math>, with <math>\textstyle i = 1, \, \ldots, \, N</math>. The [[potential energy]] due to the interaction between particles is <math>\textstyle U_{N} (\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{N})</math> and we do not consider the case of an externally applied field.

The appropriate [[Ensemble average|averages]] are taken in the [[canonical ensemble]] <math>(N,V,T)</math>, with <math>\textstyle Z_{N} = \int \cdots \int \mathrm{e}^{-\beta U_{N}} \mathrm{d} \mathbf{r}_1 \cdots \mathrm{d} \mathbf{r}_N</math> the configurational integral, taken over all possible combinations of particle positions. The probability of an elementary configuration, namely finding particle 1 in <math>\textstyle \mathrm{d} \mathbf{r}_1</math>, particle 2 in <math>\textstyle \mathrm{d} \mathbf{r}_2</math>, etc. is given by