In this episode we study the amazing properties of 1-dimensional classical gas.
There are N 'molecules', each with the same mass m, in a narrow cylinder of length L,
moving back and forth, colliding with each other and we assume that the collisions are
perfectly elastic. The cylinder is closed on both ends, but at the right end the rightmost
molecule can escape if its energy exceeds the threshold Eo, providing a 'window' for a
physicist to observe what is happening inside.
How many 'molecules' do we expect to
leak out from the 'window' and what can we learn from observing them (like peeking into a 1-dimensional oven)?
It turns out that the statistical mechanics of this system is remarkably simple,
once we consider what happens when two molecules collide.
Assume that 'molecule' A moves with velocity v1 and collides with B moving with velocity v2.
Using momentum and energy conservation (and the fact that both 'molecules' have exactly the same
mass) we find immediately that after the collision A moves with velocity v2 and B moves with velocity v1.
But this is equivalent to assuming that we are actually dealing with two (quasi)particles 1 and 2, which move with constant velocities v1 and v2 and do not interact with each other.
I guess a string theorist might call this a duality; We have two equivalent pictures of the same
situation. In one picture we are dealing with interacting molecules, bouncing off each other, and in
the other picture we are dealing with non-interacting molecules moving with constant velocities. As long as we cannot distinguish the molecules (they have the same mass) both pictures describe the same situation.
Of course, statistical mechanics is quite simple in the non-interaction picture and we can immediately answer the question from above: If initially there are n molecules with kinetic energy E > Eo,
we will observe n molecules leaking out and the time this takes will be less than 2*L/v, with v = sqrt(2*Eo/m). We only need to consider a freely moving (quasi)particle with E = Eo (+ epsilon), traveling
from the right end to the left, getting reflected and moving back to the right end where it leaves
the cylinder; All other (quasi)particles with E > Eo will leave even earlier.
And what do we learn about the gas (remaining) in the cylinder from observing the escaping molecules? The answer is exactly nothing, because of the independence of the (quasi)particles. We won't even know the temperature of the remaining gas, but we would know for sure that the kinetic energy of each remaining molecule is less than Eo.
Notice that the molecules of the 1-dimensional gas will in general not follow a
Boltzmann distribution, instead the probability distribution for energy and momentum remains whatever it was
initially - the 1-d system does not 'thermalize' as one would expect from experience with real 3-d gas.
It is left as an exercise for the interested reader to determine if (or in which sense) the 0th and 2nd law still hold.
I assume somebody must have studied the properties of 1-dimensional classical gas already, but I am just not aware of it. Please let me know if you have a reference.
6 comments:
I think its called a "Knudsen Gas". It also reminds me of the Flory "phantom" chain in polymer theory.
RZ,
very good point. I think one way to put it is that the one-dimensional classical gas (with hard-point interaction) is indeed equivalent to a Knudsen gas.
What I find interesting is the equivalence of the 'interaction picture' (hardpoint-molecules) with the 'non-interacting molecules picture'.
E.g. it should be possible to repeat Boltzmann's Stosszahl Ansatz in the 'interaction picture' and see how it compares with the interaction free result.
after some more searching it seems that (very) similar models have been investigated under the name 'Jepsen gas'.
Yes, the "duality" is strange. Probably related to conservation of energy and momentum in 1-d.
What is the interpolating parameter between fully non interacting and hard core? As long as the particles have zero size, the still behave like phantoms (note that for d>1 dimensions zero-size particles never collide). In fact, even with finite size, 2-particle collisions are phantom like. Only 3 particle collisions may give non trivial behaviour.
>> In fact, even with finite size, 2-particle collisions are phantom like.
Yes, as long as the collisions are fully elastic the argument still applies.
But I wonder if one additional molecule with different mass would already be enough to thermalize the 1-d gas.
Shouldn't be too hard to analyze ( or at least simulate). I am sure it has been done. A simple limit is an infinite mass particle. It would act as a moving boundary or piston.
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