Radial distribution function: Difference between revisions

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The total number of particles is huge, so that <math> P^{(N)}</math> in itself is not very useful. However, one can also obtain the probability of a reduced configuration, where the positions of only <math>n < N</math> particles are fixed, in <math>\textstyle \mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}</math>, with no constraints on the remaining <math>N-n</math> particles. To this end, one has to integrate ({{EquationNote|1}}) over the remaining coordinates <math>\mathbf{r}_{n+1}\, \ldots, \, \mathbf{r}_{N}</math>:

: <math> P^{(n)}(\mathbf{r}_1,\ldots,\mathbf{r}_n) =\frac{1}{Z_N} \int \cdots \int \mathrm{e}^{-\beta U_N} \, \mathrm{d}^3 \mathbf{r}_{n+1} \cdots \mathrm{d}^3 \mathbf{r}_N \, </math>.

If the particles are non-interacting, in the sense that the potential energy of each particle does not depend on any of the other particles, <math display="inline">U_N(\mathbf{r}_1,\dots,\mathbf{r}_N)=\sum_{i=1}^N U_1(\mathbf{r}_i)</math>, then the partition function factorizes, and the probability of an elementary configuration decomposes with independent arguments to a product of single particle probabilities,

The particles being identical, it is more relevant to consider the probability that ''any'' <math>n</math> of them occupy positions <math>\textstyle \mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}</math> in ''any'' permutation, thus defining the <math>n</math>-''particle density''

<math> \begin{align}

{{NumBlk|:| <math>\rho^{(n)}(\mathbf{r}_1,\ldots,\mathbf{r}_n) =\frac{N!}{(N-n)!} P^{(n)} (\mathbf{r}_1,\ldots,\mathbf{r}_n) \, </math>.|{{EquationRef|2}}}}

Z_N &=\prod_{i=1}^N \int \mathrm{d}^3 \mathbf{r}_{i}e^{-\beta U_1}=Z_1^N\\

P^{(n)}(\mathbf{r}_1,\dots,\mathbf{r}_N)&=P^{(1)}(\mathbf{r}_1)\cdots P^{(1)}(\mathbf{r}_n)

\end{align} </math>

Note how for non-interacting particles the probability is symmetric in its arguments. This is not true in general, and the order in which the positions occupy the argument slots of <math> P^{(n)}</math>matters. Given a set of positions, the way that the <math> N</math> particles can occupy those positions is <math> N!</math> The probability that those positions ARE occupied is found by summing over all configurations in which a particle is at each of those locations. This can be done by taking every [[permutation]], <math> \pi</math>, in the [[symmetric group]] on <math> N</math> objects, <math> S_N</math>, to write <math display="inline"> \sum_{\pi\in S_N} P^{(N)}(\mathbf{r}_{\pi (1)},\ldots,\mathbf{r}_{\pi (N)}) </math>. For fewer positions, we integrate over extraneous arguments, and include a correction factor to prevent overcounting,<math display="block"> \begin{align}

For <math>n=1</math>, ({{EquationNote|2}}) gives the one-particle density which, for a crystal, is a periodic function with sharp maxima at the lattice sites. For a (homogeneous) liquid, it is independent of the position <math>\textstyle \mathbf{r}_1</math> and equal to the overall density of the system:

\rho^{(n)}(\mathbf{r}_1,\ldots,\mathbf{r}_n)

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

\end{align} </math>This quantity is called the ''n-particle density'' function. For [[Indistinguishable particle|indistinguishable]] particles, one could permute all the particle positions, <math> \forall i, \mathbf{r}_i\rightarrow \mathbf{r}_{\pi(i)}</math>, without changing the probability of an elementary configuration, <math> P(\mathbf{r}_{\pi(1)},\dots,\mathbf{r}_{\pi (N)})=P(\mathbf{r}_{1},\dots,\mathbf{r}_{ N})</math>, so that the n-particle density function reduces to <math display="block"> \begin{align}

\rho^{(n)}(\mathbf{r}_1,\ldots,\mathbf{r}_n)

&=\frac{N!}{(N-n)!}P^{(N)}(\mathbf{r}_1,\ldots,\mathbf{r}_n)

\end{align} </math>Integrating the n-particle density gives the [[Permutation|permutation factor]] <math> _NP_n</math>, counting the number of ways one can sequentially pick particles to place at the <math> n</math> positions out of the total <math> N</math> particles. Now let's turn to how we interpret this functions for different values of <math> n</math>.

For <math>n=1</math>, ({{EquationNote|2}}) giveswe have the one-particle density. which, forFor a crystal, it is a periodic function with sharp maxima at the lattice sites. For a (homogeneous)non-interacting liquidgas, it is independent of the position <math>\textstyle \mathbf{r}_1</math> and equal to the overall number density, <math>\rho</math>, of the system:. To see this first note that <math>U_N = 0</math> in the volume occupied by the gas, and infinite without. The partition function in this case is

: <math> \frac{1}{V} \int \rho^{(1)}(\mathbf{r}_1) \, \mathrm{d} \mathbf{r}_1 = \rho^{(1)} = \frac{N}{V} = \rho \,</math>

:<math> Z_N = \prod_{i=1}^N\int\mathrm{d}^3\mathbf{r}_i \ 1=V^N</math>

It is now time to introduce a correlation function <math> g^{(n)}</math> by

from which the definition gives the desired result

{{NumBlk|:| <math>\rho^{(n)}(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}) = \rho^{n}g^{(n)}(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}) \, </math>.|{{EquationRef|3}}}}

: <math> \begin{align}

<math>g^{(n)}</math> is called a correlation function, since if the atoms are independent from each other <math>\rho^{(n)}</math> would simply equal <math>\rho^{n}</math> and therefore <math>g^{(n)}</math> corrects for the correlation between atoms.

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

&=\frac{N}{V} \\

&=\rho

\end{align}</math>

In fact, for this special case every n-particle density is independent of coordinates, and can be computed explicitly<math display="block"> \begin{align}

From ({{EquationNote|3}}) and ({{EquationNote|2}}) it follows that

{{NumBlk|:| <math>\rho^{(n)}(\mathbf{r}_1,\ldotsdots,\mathbf{r}_n) &= \frac{N!}{(N-n)!} P\frac{1}{V^N}\prod_{(n)i=2} (^N\mathbfint\mathrm{rd}_1,\ldots,^3\mathbf{r}_n)_i 1\\, </math>.|{{EquationRef|2}}}}

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

{{NumBlk|:| <math>g^{(n)}(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}) = \frac{V^{n}N!}{N^{n}(N-n)!} \cdot \frac{1}{Z_N} \, \int \cdots \int \mathrm{e}^{-\beta U_N} \, \mathrm{d} \mathbf{r}_{n+1} \cdots \mathrm{d} \mathbf{r}_N \, </math>.|{{EquationRef|4}}}}

\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>. With the this in hand, the ''n-point correlation'' function <math> g^{(n)}</math> is defined by factoring out the non-interacting contribution, <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}

{{NumBlk|:| <math>g^{(n)}(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}) &= \frac{V^{nN}{N!}\left(\prod_{N^{i=n+1}(^N-n)!} \cdot \frac{1}{Z_NV} \, !\int \cdots !\int \mathrm{e}^{-!\beta U_N} \,! \mathrm{d} ^3\mathbf{r}__i\right)\frac{n+1} \cdots \mathrm{dZ_N} \mathbfsum_{r}_N \,pi\in </math>.|{{EquationRef|4}}}S_N}

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

\end{align} </math>where it is clear that the n-point correlation function is dimensionless.

==Relations involving ''g''(r)==