# Velocity correlations and molecular chaos

## Main Question or Discussion Point

I’ve been reading up about Boltzmann transport equations, and the concept of molecular chaos has come up, in which one assumes the velocities of particles are assumed to be uncorrelated. I’m a bit confused about the concept though. In what sense do the velocities become correlated in the first place?

Does it mean that, when the particles collide with one another, their outgoing velocities become related to one another, since they exchange kinetic energy, and thus correlated?

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mfb
Mentor
Wind is an example of correlated motion.

Wind is an example of correlated motion.
By that do you mean that the velocities of air molecules at neighbouring points will be similar (in magnitude and direction), and hence are correlated?

mfb
Mentor
Yes.
With typical wind speeds the correlation is small (a few m/s over several hundred m/s random motion), but they are correlated. If one molecules moves to the left all the surrounding molecules are also a bit more likely to move to the left.

Yes.
With typical wind speeds the correlation is small (a few m/s over several hundred m/s random motion), but they are correlated. If one molecules moves to the left all the surrounding molecules are also a bit more likely to move to the left.
Ah ok. So, in the abstract case of a gas of particles, do their velocities become correlated via collisions with one another (due to transfer of momentum)?

mfb
Mentor
That can happen, but it doesn't happen if your gas is at rest, has no external forces and so on.

That can happen, but it doesn't happen if your gas is at rest, has no external forces and so on.
What if it is out of equilibrium and thermalising?

Also, am I correct in saying that the correlations in the velocities are due to collisions between particles, and momentum conservation relating their velocities?

mfb
Mentor
What if it is out of equilibrium and thermalising?
Then it depends on your initial conditions.
Also, am I correct in saying that the correlations in the velocities are due to collisions between particles, and momentum conservation relating their velocities?
I don't think that is a useful description.

I don't think that is a useful description.
How should one understand it then? How do velocities become correlated?

Chandra Prayaga
If the velocities are correlated, then the average velocity will be non-zero. That means the gas as a whole is going somewhere. If you want to make the velocities correlated, you can put a hole in the container, allowing the gas to stream out (into vacuum or lower pressure), and in the region of stream, the velocities are correlated, which means that if the velocity of one molecule is pointing out through the hole, there is a high probability that the velocity of another moloecule iss also pointing out of the hole, streaming along with the first one.
Generally, collisions between particles tend to restore equilibrium, making the velocities uncorrelated. The correlation which I described above is because of an external force (pressure difference).

If the velocities are correlated, then the average velocity will be non-zero. That means the gas as a whole is going somewhere. If you want to make the velocities correlated, you can put a hole in the container, allowing the gas to stream out (into vacuum or lower pressure), and in the region of stream, the velocities are correlated, which means that if the velocity of one molecule is pointing out through the hole, there is a high probability that the velocity of another moloecule iss also pointing out of the hole, streaming along with the first one.
Is this because they are all being acted upon by the same external force, causing a net motion in a particular direction, such that the velocities of neighbouring are likely to be similar, i.e. there is a statistical relation between their velocities as a result of them being acted upon by an external force (pointing in a particular direction)?

Chandra Prayaga
This is not an external force which you directly apply to each molecule, but is an average force transmitted through the gas, but with that understanding, I believe you are correct.

This is not an external force which you directly apply to each molecule
If this were the case then the velocities of each molecule would be perfectly correlated (at least in the sense that they will all be pointing in the same direction), right?

average force transmitted through the gas, but with that understanding, I believe you are correct.
Is it correct to say that because it is an average force, not all the particle velocities will be pointing in the same direction, however, neighbouring velocities will be correlated, such that if one particle has a given velocity, then the neighbouring particles will have velocities similar to it?

Chandra Prayaga
Well, even if you do manage to put exactly the same force on each particle, there will be other forces (collisions) which are unpredictable, destroying perfect correlation. It is only if the particles are not interacting with each other that you can have perfect correlation. But your second statement sounds right.

It is only if the particles are not interacting with each other that you can have perfect correlation.
By this do you mean that if an external force is acting on each particle, but there is no inter-particle interactions, then there will be a perfect correlation?

Why would correlations be present for a gas of particles out of thermal equilibrium? When deriving a Boltzmann equation, one assumes molecular chaos when determining the collision integral, i.e. neglects correlations between particle velocities. How do these correlations arise though, if collisions tend to destroy correlations?

Chandra Prayaga
Part 1. Yes. If these are non-interacting particles, and if there is a known force acting on each of them, the problem boils down to the motion of a single particle under a known force, the future is completely predicted for each particle, and there is complete correlation among the velocities.

Part 2. We already discussed a case of streaming in which the velocities are correlated. You can start the streaming, let the correlations develop, and then stop the streaming by putting a stopper against the stream, and then wait for thermal equilibrium to develop, and that last step happens by interparticle collisions, which destroy the velocity correlations.

that last step happens by interparticle collisions, which destroy the velocity correlations.
Is this because the collisions cause particles to scatter, resulting in neighbouring particle velocities becoming more and more randomly directed?

Chandra Prayaga