Kinetic theory of gases and velocity correlations

[Frank Castle]

I have been reading up on the kinetic theory of gases, and I'm unsure whether I have correctly understood why particle velocities become correlated after colliding. Is it because during the collision they exchange momentum and thus their velocities (and hence trajectories) are altered in a manner dictated by momentum conservation. Their velocities are then mutually related (i.e. have become correlated) post-collision, as if one were to reverse their subsequent trajectories they would return to the collision point (knowing the trajectory of one of the particles, one can predict the trajectory of the other). Would this be correct at all?

Apologies if this is a stupid question, I'm having a bit of a mental block and would like to get it sorted in my head.

 

[BvU]

Hi,

particle velocities become correlated after colliding

Their velocities are then mutually related

Are they ? Or do you mean that any initial particle velocity distribution evolves towards an equilibrium distribution (named after Boltzmann)

 

[anorlunda]

Their velocities are then mutually related

If all particles had the same mass, yes. If they have different masses, their kinetic energies become related with the most massive particles having the lowest velocities.

As @BvU said, the answer is the Botzman distribution. You can learn more about it here https://en.wikipedia.org/wiki/Boltzmann_equation

or, as a form of at-home study I recommend this video course.

 

[Frank Castle]

Hi,

Are they ? Or do you mean that any initial particle velocity distribution evolves towards an equilibrium distribution (named after Boltzmann)

Reading about molecular chaos, it is always mentioned that particle velocities become correlated after colliding.

 

[Frank Castle]

If all particles had the same mass, yes. If they have different masses, their kinetic energies become related with the most massive particles having the lowest velocities.

So in general, is it their momenta that become correlated? Is what I wrote for why they become correlated correct at all?

 

[BvU]

it is always mentioned

You have an example ?

 

[anorlunda]

So in general, is it their momenta that become correlated?

No, their kinetic energy. Momentum=mass*velocity has a sign and direction. The net momentum of a sitting collection of gas is zero. Kinetic energy=mass*velocity² is always positive; two particles moving in opposite directions both have positive K.E.

If you warm up the gas, the net momentum remains zero, but the net K.E. increases.

If we have two particles with equal K.E., then
m₁v₁²=m₂v₂².

 

[Frank Castle]

You have an example ?

See for example http://www.damtp.cam.ac.uk/user/tong/kintheory/kt.pdf at the bottom of page 22 and start ion page 23.

 

[Frank Castle]

No, their kinetic energy. Momentum=mass*velocity has a sign and direction. The net momentum of a sitting collection of gas is zero. Kinetic energy=mass*velocity² is always positive; two particles moving in opposite directions both have positive K.E.

If you warm up the gas, the net momentum remains zero, but the net K.E. increases.

If we have two particles with equal K.E., then
m₁v₁²=m₂v₂².

This confuses me, as I’ve been reading these http://www.damtp.cam.ac.uk/user/tong/kintheory/kt.pdf, in particular, at the bottom of page 22 and start of page 23 the author discusses correlations between the moments of two particles that have collided.

Surely the momenta of individual pairs of particles can become momentarily correlated after a collision, but these correlations will be lost soon after once they collide with other particles. Then it would be possible for the net momentum of the gas to be zero?

 

[anorlunda]

You're confusing two things. On a collision, both momentum and energy are conserved.

Let's consider a balloon full of gas. If the net momentum is to the right, then the whole balloon moves to the right. But if the balloon sits still, then the net momentum (meaning add them all up) of the gas particles inside is zero.

So, two particles can exchange momentum in a collision, but the net momentum change of the pair is zero.

(m₁v₁+m₂v₂)_{pre collision}=(m₁v₁+m₂v₂)_{post_collision}

 

[Frank Castle]

You're confusing two things. On a collision, both momentum and energy are conserved.

Let's consider a balloon full of gas. If the net momentum is to the right, then the whole balloon moves to the right. But if the balloon sits still, then the net momentum (meaning add them all up) of the gas particles inside is zero.

So, two particles can exchange momentum in a collision, but the net momentum change of the pair is zero.

(m₁v₁+m₂v₂)_{pre collision}=(m₁v₁+m₂v₂)_{post_collision}

View attachment 238737

Right. But the momenta of the two particles after the collision are correlated in the general sense, in that their momenta have become related by the collision - if I reverse the trajectories (i.e. I flip the signs of both of their momenta) of the two particles they will meet back at the collision point. That is, their momenta can be related even though the total net momentum is zero. (Or perhaps this is all only the case in non-equilibrium scenarios?)

 

[anorlunda]

But the momenta of the two particles after the collision are correlated in the general sense, in that their momenta have become related by the collision

Yes that correlation is true, but it does not get you to the kinetic theory of gasses.

Maybe I'm missing the point of your question.

 

[Frank Castle]

Yes that correlation is true, but it does not get you to the kinetic theory of gasses.

Maybe I'm missing the point of your question.

Sorry, I think my title was a bit misleading. I'm trying to understand this in the context of constructing Boltzmann transport equations and the molecular chaos assumption. This being said, is what I written about why the momenta of the particles become correlated after a collision correct at all?

 

[BvU]

mutually related (i.e. have become correlated) post-collision

still searching ...
during $\ne$ post

 

[Frank Castle]

still searching ...
during $\ne$ post

Right, good point. They become correlated during the collision and remain so afterwards until they collide with another particle. As far as I understand, the molecular chaos assumption is that the momenta of two particles that mutually interact are uncorrelated before they interact (i.e. any previous correlations between them have been lost due to many collisions with other particles since their last interaction). They then become correlated during the interaction, such that after the interaction, their momenta are correlated for the time interval associated with their mean free path.

 

[anorlunda]

I'm trying to understand this in the context of constructing Boltzmann transport equations and the molecular chaos assumption.

That video course I recommended in #3 is very enjoyable. After watching that, you'll understand completely.

 

[Frank Castle]

That video course I recommended in #3 is very enjoyable. After watching that, you'll understand completely.

Ok, thank you. I’ll take a look.

 

[BvU]

remain so afterwards

Nah, not really. On top of p 32 he needs

Moreover, we assume that the two particles are once again uncorrelated by the time they leave the collision, albeit now with their new momenta"

 

[Frank Castle]

Nah, not really. On top of p 32 he needs

How is this the case though? How can they become correlated during the collision but not afterwards? Is some amount of coarse graining assumed, such that the timescale over which the correlation is lost is much smaller than the timescale over which the system evolves?

Edit: Having read about it more, it seems that one assumes that correlations after the collision are lost precisely because one assumes that the gas of particles is sufficiently diffuse that a given particle will interact with many different particles before it interacts with the same particle again. As such, any correlation between their momenta will have long been lost due to multiple different interactions inbetween.

 

[BvU]

Tell us what you think that it means if "two particle velocities are correlated"

 

[Lord Jestocost]

A statement like “particle velocities become correlated after colliding” should be read in the context of the molecular chaos assumption. When speaking of molecular chaos, one assumes that the pre-collision velocities of particles are uncorrelated before the collision. This assumed lack of correlations is often referred to as a mean-field behavior and allows to write the two-particle distribution function in phase space as the product of two single-particle distribution functions. This assumption works (Boltzmann’s Stoßzahlenansatz), but is in principle only justified if the velocities of the two colliding particles have nothing to do with each other, i.e., if the two colliding particles are not somehow or other correlated owing to an intersection of their collisional histories. They would “share” a common history and thus exhibit some correlation in case they would have, for example, collided some time before or would have collided with particles that have collided before.

 

[Frank Castle]

Tell us what you think that it means if "two particle velocities are correlated"

My understanding is that the velocities of two particles that participate in a binary collision become correlated during the collision in the sense that if one reverses the time evolution (i.e. flip the signs of their velocities), then the two particles will return to the collision point.

 

[Frank Castle]

A statement like “particle velocities become correlated after colliding” should be read in the context of the molecular chaos assumption. When speaking of molecular chaos, one assumes that the pre-collision velocities of particles are uncorrelated before the collision. This assumed lack of correlations is often referred to as a mean-field behavior and allows to write the two-particle distribution function in phase space as the product of two single-particle distribution functions. This assumption works (Boltzmann’s Stoßzahlenansatz), but is in principle only justified if the velocities of the two colliding particles have nothing to do with each other, i.e., if the two colliding particles are not somehow or other correlated owing to an intersection of their collisional histories. They would “share” a common history and thus exhibit some correlation in case they would have, for example, collided some time before or would have collided with particles that have collided before.

Thanks for the clarification. When it is said that particle velocities become correlated during a collision is it meant that the velocities become related in such a way that, if one reverses their subsequent trajectories (i.e. flips the sign of their velocities), they will both return to the collision point?

 

[BvU]

Thanks for the clarification. When it is said that particle velocities become correlated during a collision is it meant that the velocities become related in such a way that, if one reverses their subsequent trajectories (i.e. flips the sign of their velocities), they will both return to the collision point?

I think Tong explains it quite well: when he writes the combined distribution function as a product of two individual distribution functions he needs the uncorrelated assumption. That's all there is to it. Time reversal symmetry of the governing collision equations ensures your 'returning to collision' point, completely, and much stronger than the 'correlated' concept.

 

[Lord Jestocost]

When it is said that particle velocities become correlated during a collision is it meant that the velocities become related in such a way that, if one reverses their subsequent trajectories (i.e. flips the sign of their velocities), they will both return to the collision point?

Maybe, one should understand this in the following way:
In case you assume a time-symmetric dynamics, you will never “get rid” of correlations between the velocities and positions of all those particles which have collided with each other in the past. These correlations might in principle be encoded in an extremely complex manner.

 

[Frank Castle]

In case you assume a time-symmetric dynamics, you will never “get rid” of correlations between the velocities and positions of all those particles which have collided with each other in the past. These correlations might in principle be encoded in an extremely complex manner.

But what is intuitively meant by their velocities being correlated? Is what I wrote in post #23 a correct intuitive understanding?

 

[Lord Jestocost]

Maybe, the following might be of help:
Thermodynamics And Statistical Physics: Teaching Modern Physics - ...

 

[Frank Castle]

Maybe, the following might be of help:
Thermodynamics And Statistical Physics: Teaching Modern Physics - ...

Thanks for the reference. So do particle velocities become correlated due to a common collisional history? And by correlated it is meant that their velocity components become related as a result of the collision between them. Is this right?