Correlation functions - radial distribution function

[SchroedingersLion]

Greetings,

I am about to start my master thesis in computational physics and I need to make myself familiar with correlation functions, in particular with the radial distribution function of a system of N identical particles.

At Wiki, there is a short explanation of the definition of the radial distribution function $g^{n}(\mathbf r_{1},...,\mathbf r_{n})$
https://en.wikipedia.org/wiki/Radial_distribution_function#Definition
I think I understood that part.

However, in the part after this (https://en.wikipedia.org/wiki/Radial_distribution_function#The_structure_factor):
They want to consider $g^{2}(\mathbf r_{1},\mathbf r_{2})$ and the say that, in the case of spherical symmetrical particles, it only depends on the distance between the particles, $\mathbf r_{12}=\mathbf r_{1}-\mathbf r_{2}$.
Because of this, they can write $g(\mathbf r):=g^{2}(\mathbf r_{12}):=g^{2}(\mathbf r_{1},\mathbf r_{2})$

From eq. (2) and (3) of the former Wiki section, it should follow that (with $\rho=\frac {N}{V}$, the average / ideal gas number density) $\rho^{2} g(\mathbf r)d\mathbf r_{1}d\mathbf r_{2}$ (*) is the probability to find one particle at $\mathbf r_{1}$ and another one at $\mathbf r_{2}$.

However, they write that ($\mathbf r_{1}$ being set to the origin)
$\rho g(\mathbf r)d^{3}r=dn(\mathbf r)$ is the average number density dn at $\mathbf r$

I don't understand this conclusion. They seemed to have divided my term (*) by $\rho d\mathbf r_{1}$, but why would that give me the average number density?
On top of that, their quote "...is the average number of particles [...] to be found in the volume $d^{3}\mathbf r$ around the position $\mathbf r$." does say nothing about the particle at 0.
It is not the average number density at $\mathbf r$, but, if anything, it should be the average number density at $\mathbf r$ under the condition that I have a particle at 0, right?

Would appreciate the help!

Regards,
SL.

 

[eljose79]

Dear SL,

First of all, congratulations on starting your master thesis in computational physics! The radial distribution function is an important concept in this field, so it's great that you are already familiarizing yourself with it.

To address your question, the key to understanding this conclusion lies in the definition of the radial distribution function. As you mentioned, the radial distribution function is defined as the probability of finding two particles at a distance r from each other, divided by the ideal gas probability. This means that the radial distribution function takes into account the number of particles in a given volume, as well as the probability of finding those particles at a specific distance from each other.

Now, let's look at the equation in question, $\rho g(\mathbf r)d^{3}r=dn(\mathbf r)$. This equation is basically saying that the product of the average number density, $\rho$, and the radial distribution function, g, gives you the average number of particles, dn, in a small volume, d^3r, around a specific position, r. In other words, it is the average number of particles that you would expect to find in a small volume around a specific position, taking into account both the number of particles and the probability of finding them at a specific distance from each other.

To your point about the particle at the origin, keep in mind that the radial distribution function is a function of the distance between two particles, not just one specific particle. So, when we say "the average number of particles to be found in the volume d^3r around the position r," it is implied that one of those particles is at the origin, while the other is at a distance r from it. This is why we can divide your term (*) by $\rho d\mathbf r_{1}$ and still get the average number density at r.

I hope this helps to clarify the conclusion for you. If you have any further questions, please don't hesitate to ask. Best of luck with your master thesis!