Maxwell's Demon: would a cone work?

[krausr79]

Maxwell's demon is the little guy who opens an atomic door to a container to let atoms fly in, but shuts it before an atom flies out, thus increasing the internal pressure.

Suppose the walls of a container had several small cone-shaped holes built into it. The inside hole might be pretty small, maybe 1->several atom's widths and it would taper outward so that the outside wall's opening was larger. The slope/incline of the cone would be whatever works best for my example.

It would be more likely for an atom to enter the larger outside hole than exit through the smaller inside one. If atoms that entered the cone were likely enough to continue through the inner hole instead of eventually deflecting back out the way they came, it might be possible to build up internal pressure.

What do you think?

 

[chrisbaird]

Why would it be more likely going one direction versus the other? In the end, the atom still has to go through the same size hole.

 

[kmarinas86]

Why would it be more likely going one direction versus the other? In the end, the atom still has to go through the same size hole.

If we are talking about particles that have a finite, non-zero size, which weakly interact and rarely collide, then the size of the hole should certainly have an effect. However, if we are imagining point-sized particles, then you would be correct. But particles we actually deal with do not have a point-size. Of course, if the concentration towards one side were continued, then at some point, we do have a situation where the particles can interact and collide readily. I suppose this concentration could still be VERY small when the net transfer effect is eliminated. In any case, the holes would have to be very small, in which case Casimir forces may be involved.

 

[krausr79]

Why would it be more likely going one direction versus the other? In the end, the atom still has to go through the same size hole.

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Atoms already in the container could only exit through the small left hole but would enter through the larger outer right hole. Many more atoms will enter the outer hole based on it's size at similar pressures. Probably less than 100% of the atoms going into the outer hole would actually enter the container through the inner hole, but if that percentage were much larger than the size ratio between the holes, pressure would build up inside the container.

 

[sam_bell]

Atoms already in the container could only exit through the small left hole but would enter through the larger outer right hole. Many more atoms will enter the outer hole based on it's size at similar pressures. Probably less than 100% of the atoms going into the outer hole would actually enter the container through the inner hole, but if that percentage were much larger than the size ratio between the holes, pressure would build up inside the container.

You've got this right, you just haven't figured out what that < 100% factor is. Ultimately, the number of atoms crossing the wall coming from the outer hole is the same. If you consider a differential area element of the small hole, any particle coming from within 2pi steradians will cross the wall. The details of how it bounced around inside the cone may be complicated, but they aren't important. This should also apply if the particles have finite size, except now the effective cross-section is reduced.

You could fix up your demon by making the walls flexible (you can now collapse the inner hole to a point). That way, no particles can enter from the pointy end, and all particles pass from the outer end. Now, however, the particles do work on your cone, which (a) causes them to lose energy as they pass the wall, and (b) causes the wall to flutter from all the energy gained; allowing some particles to go the reverse direction.Sam

 

[Ken G]

Maxwell's Demon is intended to apply in a situation of thermodynamic equlibrium (there's no issue if the system starts out not in equilbrium). Here's a very important thing about thermodynamic equilibrium: every single process you can imagine is exactly balanced by its inverse process. This holds no matter what the shape of the hole is, what it is made of, and how flexible it is-- exactly the same number of particles (on average) will pass from left to right as right to left, involving exactly the same energies, and exactly the same motion, except with the sign of time reversed in the two cancelling processes. This all relates to the time reversibility of the fundamental processes of physics, and what that reversibility does in situations of thermodynamic equilibrium. Thus you can never "trick" it, no matter how hard you try-- you can never have a demon, unless something is out of thermodynamic equilibrium somewhere, and keeping track of that additional process is always the explanation for the missing entropy.

 

[Rap]

Suppose the walls of a container had several small cone-shaped holes built into it. The inside hole might be pretty small, maybe 1->several atom's widths and it would taper outward so that the outside wall's opening was larger. The slope/incline of the cone would be whatever works best for my example.

The cones will also be blocking particles that would otherwise have entered the small hole. I bet if you do the calculation, you will find that the cones won't work.