The 'all air in a room collecting on one side' adage

[DaveC426913]

TL;DR Summary

Isn't it actually impossible for all air to collect on one side of a room - even given infinite time - because mechanics?

Speaking of the the oft-referenced adage 'Given sufficient (time akin to orders of mag greater than the age of the universe ) the probability of all air molecules randomly collecting on one side of a room, leaving the other in vacuum - while very remote - is not zero, .'

Is this not actually impossible? The molecules are not free-moving. They collide with each other after an arbitrarily short mean free path. How is it possible for every single molecule to be imparted with motion sending it to the left side of the room without any other molecules being sent the other way?

I guess they would have to all bounce off the same (right) wall (or at least, have zero bouncing off the left wall), causing the box itself to shift in equally and oppositely. For the duration of the improbable event, the box would be physically shifted to the right (by an amount inversely proportionate to its mass compared to the air volume).

Yes? Did I just answer my own question?

 

[Bystander]

Summary:: Isn't it actually impossible for all air to collect on one side of a room - even given infinite time - because mechanics?

Did I just answer my own question?

Yes.

 

[DaveC426913]

Yes.

Wait.

Yes it's impossible?
or
Yes I answered my own question (that it's possible)?

 

[Bystander]

Yes I answered my own question (that it's possible)?

 

[hutchphd]

Just consider a very dilute gas (nearly a vacuum). Clearly with 10 molecules it is possible (although not likely) for them to segregate. Unless you can argue that there is a critical density where the physics changes there will be a nonzero probability for any density..

 

[DaveC426913]

Just consider a very dilute gas (nearly a vacuum). Clearly with 10 molecules it is possible (although not likely) for them to segregate.

Yeah. I was wondering about that. I took it further and imagined a single bowling ball. I pictured a bowling ball richocheting around inside the room, and obviously the room would recoil.

Unless you can argue that there is a critical density where the physics changes there will be a nonzero probability for any density..

I guess the tipping point would be when the MFP becomes shorter than the width of the room.
Initially I was imagining incoming air molecules being repelled by an ever-increasing density of air, and so there's a negative feedback loop where they can't all be heading into the higher density volume -not with a finite MFP.

But I guess that density gradient just a nuance of the general problem - just part of the mechanism that makes it merely "really unlikely" and not "impossible".

 

[hutchphd]

Noisy, too...

 

[vanhees71]

Within classical mechanics it's not impossible, because you can prepare a gas in such a way initially, just by filling one half of a box and then taking out the separating wall (at least as a gedanken experiment ;-)).

It's of course extremely unlikely that spontaneously a gas moves exclusively to one half of your room. You can calculate the probability for that by considering the entropy of this situtation in comparison to the entropy of the equilibrium state where the gas is spread homogeneously over the entire room.

 

[A.T.]

Is this not actually impossible?

Yes.

Yes it's impossible?
or
Yes I answered my own question (that it's possible)?

That's what you get for using double negatives.

 

[stevendaryl]

Summary:: Isn't it actually impossible for all air to collect on one side of a room - even given infinite time - because mechanics?

I guess you already have your answer, but another way to look at it is to think backwards. Suppose that you put an airtight divider in the middle of the room, and pumped all the air out of one side into the other side. Now, remove the divider.

Immediately after the divider, the air molecules are in some state $$A$$, with all the molecules on one side of the room. Wait 15 minutes, and the air molecules will be in some state $$B$$, with the molecules evenly distributed throughout the room.

Now, let $$B'$$ be the state that is exactly like state $$B$$ except that all momenta of the molecules are reversed. If the molecules happened to start in state $$B'$$ then after 15 minutes, they will be in state $$A'$$, which is the momentum-reversed version of $$A$$, and all the molecules will be on one side of the room.

 

[256bits]

Within classical mechanics it's not impossible, because you can prepare a gas in such a way initially, just by filling one half of a box and then taking out the separating wall (at least as a gedanken experiment ;-)).

It's of course extremely unlikely that spontaneously a gas moves exclusively to one half of your room. You can calculate the probability for that by considering the entropy of this situtation in comparison to the entropy of the equilibrium state where the gas is spread homogeneously over the entire room.

I tend to think that the gedanken experiment describing the situation, although generally expressing the role of entropy, does leave some aspects out.

Kinetic energy and potential energy are part of classical mechanics AFAIK.

Water molecules in the air do congregrate into droplets that fall as precipitation ( or humidity forming on a glass window ). Both of these examples do not take to the end of the universe to occur.

A problem such as in the OP does not give the existing state of the gas or conditions.
Most likely it is assumed that the gas is idealized of 'non-interacting air particles' at "room temperature " and incorrectly given a higher ranking for what must always occur for any situation, along with other non-stated assumptions.

Temperature - at low temperature the kinetic energy of the gas molecules is also low
Intermolecular forces - resistive or attractive can make a difference
Elastic / inelastic collisions - is energy dissipated during a collision
Isolated, closed, or open system - interaction with the environment would make a time reversal of only the particles impossible
Size of room

 

[DaveC426913]

A problem such as in the OP does not give the existing state of the gas or conditions.
Most likely it is assumed that the gas is idealized of 'non-interacting air particles' at "room temperature "

Don't blame OP. Those details are not included in the standard adage - as widely promulgated - to which I refer.

You may provide your own states ad project as you see fit.But frankly, I'm not sure how to interpret the bulk of your post. Are you claiming that there are certain initial conditions that make the stated outcome impossible?

 

[jbriggs444]

Don't blame OP. Those details are not included in the standard adage - as widely promulgated - to which I refer.

You may provide your own states ad project as you see fit.But frankly, I'm not sure how to interpret the bulk of your post. Are you claiming that there are certain initial conditions that make the stated outcome impossible?

Clearly, there are such conditions. Take the classical model. Point-like non-interacting particles in a rigid, un-moving cubical container where the walls result in perfectly elastic collisions. Align the cube east-west, north-south, up-down.

Set up the particles in some spatial distribution within the box so that no two particles are on a direct east-west line from one another. Give them all zero north-south and up-down velocities. Give all of the particles an identical east-west speed. But point half of them east and half of them west.

This is clearly isomorphic to a cylindrical arrangement where the particles all orbit a central axis in lock step. If not all were on one side of the cylinder at the beginning, not all will be on one side ever.

 

[DaveC426913]

Clearly, there are such conditions.

Er. I'm not sure it's valid to posit conditions that are disqualified by the physical world:

Take the classical model. Point-like non-interacting particles in a rigid, un-moving cubical container where the walls result in perfectly elastic collisions.

(As you say, that's a model, one not intended to accurately describe the physical world.)

However, I take your point that it is still theoretically possible to set up such conditions without ignoring known physics.

 

[jbriggs444]

Er. I'm not sure it's valid to posit conditions that are disqualified by the physical world:

(As you say, that's a model, one not intended to accurately describe the physical world.)

However, I take your point that it is still theoretically possible to set up such conditions without ignoring known physics.

In the real world, the box would evaporate before the air molecules migrate...

... with probability so close to 1 that it is ludicrous.

 

[DaveC426913]

In the real world, the box would evaporate before the air molecules migrate...
... with probability so close to 1 that it is ludicrous.

Fair enough, but there's still a difference between a thought experiment that's practically impossible to complete and one that can't be started because it specifies fictional conditions just to set up.

 

[bob012345]

I guess you already have your answer, but another way to look at it is to think backwards. Suppose that you put an airtight divider in the middle of the room, and pumped all the air out of one side into the other side. Now, remove the divider.

Immediately after the divider, the air molecules are in some state $A$, with all the molecules on one side of the room. Wait 15 minutes, and the air molecules will be in some state $B$, with the molecules evenly distributed throughout the room.

Now, let $B'$ be the state that is exactly like state $B$ except that all momenta of the molecules are reversed. If the molecules happened to start in state $B'$ then after 15 minutes, they will be in state $A'$, which is the momentum-reversed version of $A$, and all the molecules will be on one side of the room.

If, once the gas equalized in the room, Maxwell's demon comes along and reverses all the momenta, I think you would just see the gas continue statistically the same. $B'$ would look and act like $B$. You would not notice a thing. In the transition from states $A →B$, the collisions tended to increase entropy. In the transition from states $B' → A'$ you are requiring them to statistically prefer to reduce entropy. That won't happen. Information has been lost. Reversing the momenta is not like reversing a video. $A'$ will look a lot like $B$ and not like $A$.

 

[nsaspook]

Fair enough, but there's still a difference between a thought experiment that's practically impossible to complete and one that can't be started because it specifies fictional conditions just to set up.

A recent PBS video on the subject.

 

[256bits]

Don't blame OP. Those details are not included in the standard adage - as widely promulgated - to which I refer.

You may provide your own states ad project as you see fit.But frankly, I'm not sure how to interpret the bulk of your post. Are you claiming that there are certain initial conditions that make the stated outcome impossible?

Yes . I agree. It is a standard description, albeit simplified thus making the concepts more understandable.
It appears an isolated equilibrium system is the one in mind, with constant entropy.

For the system, it is just a description of the 2nd law of thermodynamics.
That is all it is.

For the real world, the such a room would not last to the end of the universe.
And, it would not be isolated, which would change the dynamics of the system continuously with energy flux across the boundary destroying any possible sense of mirror dynamics part way through to the end of the universe.
I think, in fact, with an external reservoir of of same temperature, one would have to take into account the microstates of incoming/outgoing radiation and heat flow, even if miniscule in amount, and localized here and there on the boundary, ever changing, which implies that "to the end of the universe" having to be very long indeed.

No. My post was really in reference to the fact that we see gases condensing all the time and thing nothing of it - no questions asked. That's the world we actually live in.

 

[Office_Shredder]

No. My post was really in reference to the fact that we see gases condensing all the time and thing nothing of it - no questions asked. That's the world we actually live in.

My mind is blown.

 

[256bits]

My mind is blown.

You have just increased the entropy of the universe a little bit
( and so have those condensing gases )

 

[stevendaryl]

If, once the gas equalized in the room, Maxwell's demon comes along and reverses all the momenta, I think you would just see the gas continue statistically the same. $B'$ would look and act like $B$. You would not notice a thing. In the transition from states $A →B$, the collisions tended to increase entropy. In the transition from states $B' → A'$ you are requiring them to statistically prefer to reduce entropy. That won't happen. Information has been lost. Reversing the momenta is not like reversing a video. $A'$ will look a lot like $B$ and not like $A$.

I’m talking from the point of view of Newtonian mechanics. Whether time evolution increases or decreases macroscopic entropy depends on the initial conditions.

The prediction of Newtonian physics is that state B’ will lead to state A’, which has all the molecules on one side of the room.

This can never be done in practice, because there is no way to reverse all the individual momenta.

 

[sophiecentaur]

Water molecules in the air do congregrate into droplets that fall as precipitation ( or humidity forming on a glass window ). Both of these examples do not take to the end of the universe to occur.

In circumstances where the particles are nothing like Ideal Gas particles then all bets are off. The collisions are not describable with Newtonian Mechanics once the intra molecular forces are significant.

But condensation on the objective lens of my telescope does actually constitute the 'end of my world', for at least one evening.