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Radial distribution of particle separation in a liquid at small distances

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Draw schematically the radial distribution function $g(r)$ for a Lennard-Jones fluid at low and high particle densities. Discuss in both cases the behavior at small $r$. At high densities $g(r)$ has some damped oscillations. Explain their origin and what would happen at long distances.

What I know:

The radial distribution function $g(r)$ describes how density varies as a function of distance from a reference particle (Wikipedia).

Specifically in Lennard-Jones case, we have (Pasteboard still does not work for me):


Here we can observe that if repulsion outweighs attraction, the curve will die out on the x axis. On the contrary, if it is the other way around, the curve will grow exponentially. For the sake of clarity, I would remark y = 0 as the Ideal Gas behaviour.

I have read that $g(r)$ vanishes at short distances, because the probability of finding two particles close to each other vanishes due to the repulsive part of the potential. At high densities $g(r)$ can show some damped oscillations: https://imgur.com/a/i32D39I.

These oscillations express the preference of the particles to be found at specific distances from a reference particle at the origin. For instance in the LJ case, a first layer of particles will be localized closed to the minimum of LJ's potential, which is the origin of the first peak in $g(r)$.

This layer prevents other particles from getting close to it, which is what causes the first
minimum in $g(r)$.

What I do not know.

1)The explanation: 'These oscillations express the preference of the particles to be found at specific distances from a reference particle at the origin'. How can particles have 'preference' to be at different distances? I think this is not a good physical explanation. I have been thinking about this and I would say that as there is a high density of particles at small distances, the repulsion force is exerted on these, which triggers such oscillations. Finally there is a point where they are so close to each other that the repulsion force provides them with an initial kinetic energy which will be equal to a final electrostatic potential energy of zero (selecting our zero of electrostatic potential energy at y=0) , and that is the moment when the curve dies out on the x axis. Do you agree with this explanation?

2)Are we dealing with underdamped oscillations at high densities? Could you give an insight into these kind of oscillations (if what I have just said is not enough or incorrect)?

3)When we are dealing with long distances, would $g(r)$ tend to be one? What would we observe, the Ideal Gas behaviour? Why? Is it because there is no interaction between the particles?

As always, I am interested in explaining such phenomena from a physical point of view.

asked Nov 1 in Physics Problems by JD_PM (490 points)
edited Nov 2 by sammy gerbil
0) There seems to have been a software change to the Pasteboard site. It is not accepting input from me, and no contact details are given to report bugs.

1) The word "preference" suggests conscious choice. Of course the particles do not possess consciousness. What is meant is that the spatial distribution of particles settles into a configuration which minimises potential energy, and this is achieved when there is some structure. For example, **hexagonal close packing**.

Gravitational energy is far too small on the scale of microscopic particles. The balance is between kinetic energy and electrostatic potential energy as defined by the Lennard-Jones potential.

Ideal gas particles are assumed to have no attraction at all, and only "hard sphere" repulsion. The potential function for ideal gas interactions is therefore $U(r)=0$ for $r>2r_0$ and $U(r)=\infty$ for $r\le 2r_0$, where $r_0$ is the radius of particles.
2) I think you have misunderstood the oscillations. These are not **dynamic** oscillations in the motion of particles, like a mass on a  spring. So the question about over/under-damping is not relevant - although it is possible some analogy could be made with dynamic oscillations. These are **static** oscillations in the distribution function, as in the ripples of a wave function in quantum mechanics, or the ripples in the distribution of sand on a beach. (For oscillations in radial wave function, see Figure 11 on page 19 here : http://www.umich.edu/~chem461/QMChap7.pdf).

$g(r)$ is a probability distribution function, so the "oscillations" indicate that there are periodic values of particle separation which are more likely than average (peaks) or less likely than average (troughs). These "oscillations" are not necessarily sinusoidal (harmonic).

High density causes the "oscillations" by restricting the space which particles can move about in. When particles are forced close together they slide into relative positions where they can keep as much motion as possible - ie they form structures such as hexagonal close packing. At low densities particles have plenty of freedom to move about and can occupy all relative positions equally. All values of separation $r$ are equally likely - the probability distribution function is uniform.

3) Yes for long distances (= low densities) $g(r)$ becomes a uniform probability density function. That is right, this describes ideal gas particles in which there is no interaction  - except at very short distances.
1) Why $2r_0$? (I mean the factor of 2 and no other number)?

2) Yes, after seeing carefully the graphs it is highly likely that we are not dealing with harmonic oscillations. I have a problem here, as I have just seen harmonic ones. How could we confront non-sinusoidal oscillations? If we say $m \frac{d^2 x}{d t^2} = -kx$ we get the harmonic oscillations. Is there any other force law which using Newtonian mechanics gives as another kind of oscillations? If there is not, do you recommend me using quantum mechanics? I am not sure if it would be feasible, but I thought about using the quantum harmonic oscillator to make an approximation by perturbation theory.
1) $2r_0$ (twice particle radius) is the minimum distance between the centres of particles which are hard spheres.

2) I am not sure you have understood the nature of the "oscillations" in the probability density function. Try reading my previous comment again. These are **not** mechanical oscillations. So trying to use Newtonian Mechanics (or Quantum Mechanics) to find their period is not appropriate.

The graph you linked does **look** like a damped harmonic oscillation. However, it is not an oscillation in time and space, it is an oscillation in the probability density function $g(r)$. It is not even an oscillation in space (like ripples in the sand on the beach), because $r$ is not a position vector.

The oscillations are only periodic variations in the probability of a particle being found at a distance $r$ from an arbitrary particle.

The explanation you quoted suggests that the graph is statistical. It has been constructed by taking a snapshot of the jiggling particles, and measuring the distance of every particle from one aribitrary particle, which is chosen as the origin. (Perhaps this is repeated with every particle in turn being used as the origin. This is the same as measuring the distance between every pair of particles.) The distances of every particle from the origin are measured, and a histogram is plotted of the **density** of particles at a distance in the range $r-\Delta r$ to $r+\Delta r$ (vertical axis) vs $r$ (horizontal axis).

For a gas you would expect a uniform density distribution. For a 'cold' 3D cubic crystalline solid you find sharp peaks at separations $r$ which are multiples of $1, \sqrt2, \sqrt3, \sqrt5, \sqrt6, ...$ units. (In fact for all $r^2=l^2+m^2+n^2$ with all possible combinations of integers $l, m, n$.) In between are ranges in $r$ which have zero frequency - eg $1.25$ units. These peaks and troughs extend over a large range in $r$ - ie there is **long range order**. For a 'hot' crystal these peaks and the troughs in between are broader and more rounded, like rolling hills and valleys, giving the impression of oscillations. (The unit of spacing also increases as the solid expands.)

For a liquid or amorphous solid there are peaks (and troughs) only at small values of $r$ - ie there is **short range order**. At large values of $r$ the distribution is more uniform like a gas.
OK thank you Professor, I think now I understood the oscillations issue.

The following comment has nothing to do with the problem but I wanted to ask:

I am interested in learning more about classical mechanics and mechanics in general. I have been using books like Tipler, Physics for scientists and Engineers and so on, but these most of the time do not derive physics laws. So I have been looking for such a kind of book (https://physics.stackexchange.com/questions/12175/book-recommendations) and I came across Classical Mechanics by Taylor and Classical Mechanics by Goldstein.

I have been looking for a book related to mathematical methods as well. I started reading Mathematical Methods for Physicists by Arfken.

Any personal recommendations? I would appreciate it, thank you.
Sorry, I do not know many textbooks, and do not like making recommendations anyway. It is very easy to find what textbooks are available for any topic, and get recommendations, reviews and previews, from sites such as Amazon, GoogleBooks, etc. Which one suits your requirements is a very subjective decision.
I have posted some instructions for embedding images using Imgur : see http://physics.qandaexchange.com/?qa=253/how-to-upload-an-image

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