Radial distribution function: Difference between revisions

Revision as of 14:59, 27 September 2022

Revision as of 16:59, 27 September 2022

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\rho^{(n)}(\mathbf{r}_1,\dots,\mathbf{r}_n) &= \frac{N!}{(N-n)!}\frac{1}{V^N}\prod_{i=2}^N\int\mathrm{d}^3\mathbf{r}_i 1\\

&=\frac{N!}{(N-n)!}\frac{1}{V^n}

\end{align}</math>For <math>N\gg n</math>, the non-interacting n-particle density is approximately <math>\rho^{(n)}_\text{non-interacting}(\mathbf{r}_1,\dots,\mathbf{r}_N)= \left(1-n(n-1)/2N+\cdots \right)\rho^n\approx \rho^n</math><ref>{{cite journal |last1=Tricomi |first1=F. |last2=Erdélyi |first2=A. |title=The asymptotic expansion of a ratio of gamma functions |journal=Pacific Journal of Mathematics |date=1 March 1951 |volume=1 |issue=1 |pages=133–142 |doi=http://dx.doi.org/10.2140/pjm.1951.1.133}}</ref>. With this in hand, the ''n-point correlation'' function <math> g^{(n)}</math> is defined by factoring out the non-interacting contribution{{Citation needed|date=September 2022}}, <math display="block">\rho^{(n)}(\mathbf{r}_{1}, \ldots, \, \mathbf{r}_{n}) = \rho^{(n)}_\text{non-interacting}g^{(n)}(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}) </math>Explicitly, this definition reads <math display="block">\begin{align}

g^{(n)}(\mathbf{r}_{1}, \ldots, \, \mathbf{r}_{n}) &=\frac{V^N}{N!}\left(\prod_{i=n+1}^N\frac{1}{V}\!\!\int \!\! \mathrm{d}^3\mathbf{r}_i\right)\frac{1}{Z_N}\sum_{\pi\in S_N}

e^{-\beta U(\mathbf{r}_{\pi(1)}, \ldots, \, \mathbf{r}_{\pi(N)})}