# One way door violate the 2nd law of thermodynamics

I have designed this simple thought experiment that seems to contradict 2nd law of thermodynamics. Could you please find a mistake in my reasoning?

Setup:

• small door that only opens to the right (w.l.o.g), between two chambers of gas.
• The chambers are full with small fast particles and big and slow particles
• The small particles fit through the doorway with epsilon degrees wiggle room. This means that the probability that a particle that approaches the (center of the) doorway crosses it is (without the door itself), is epsilon/180
• The door opens up to 90 degrees only to the right.
• The door’s mass is negligible (does not change the course of the particle).

In this setup, the probability that a small particle that approaches the doorway from the left crosses it is, is epsilon/180. Because the door itself is not an obstacle (low mass, and opens freely to the right).

But the probability that a particle that approaches the doorway from the right crosses it, is the probability that the particle is heading the right direction (epsilon/180) times the probability that the door is sufficiently open (epsilon/90).

In total there’s 90/epsilon times more chance that a particle will cross with the direction of the door than against it.

This seems to result in most fast particles ending on the right side, causing a heat difference, violating the 2nd law.

Note: I'm pretty much ignoring the big slow particles in the text below, since your argument works without them.

Suppose the door is initially closed, and a particle hits it from the left. The door's mass is negligible compared to that of the particle, so the particle's trajectory is not appreciably changed. However, in order to swing the door open, the particle must have transferred some energy to the door. So the door is not only open but moving, probably at a very high speed given its low mass.

You said the door opens only to 90 degrees, so there must be some object in place that prevents it from opening further than that. Since we're on the molecular scale the collision between the door and this object will be an elastic one, and it will swing back the other way at the same velocity, colliding again with its doorframe and bouncing back, and so on. So after the first particle passes through, the door is not just open but oscillating wildly, and that means it's open most of the time, so it's easy for particles coming from the right to get past it.

Further collisions between particles and the door will tend to increase the door's kinetic energy even more, until the average kinetic energy in the door is equal to the average kinetic energy in each of the particles. (We know this because of the equipartition theorem.) Since the door has such low mass, this means it will be oscillating very fast indeed. So the equilibrium state is one in which there are an equal number of small particles on the right-hand side as on the left, and in which the door is oscillating very rapidly.

You might object to my claim that the door swings back elastically --- you probably intended there to be some damping that makes it eventually return to its closed position. This is possible, but to implement that the door must lose energy over time, which means it has to be connected to a heat bath. This heat bath has to be at a much lower temperature than that of the moving particles. If it wasn't, then energy would flow into the door from the heat bath, causing it to start swinging back and forth even without any collisions from the particles.

If the door is connected to a very low temperature heat bath in this way then there's no reason your setup wouldn't work. The door will preferentially let particles through from the left to the right but not the other way around, and this will result in a pressure gradient that could be used to extract work. But we're not violating the second law here, because adding heat to the low-temperature heat bath is an irreversible process that produces entropy. Essentially, the system is a kind of heat engine that uses the temperature differential between the (hot) moving particles and the (cold) heat bath to do work on the gas.

• Plus 1 for the last sentence "Essentially ......". – Alchimista Jan 25 '18 at 8:14
• I agree that the door will be oscillating. That does mean that the door will be open most of the time. But it has to be fully open for a particle coming from the right to pass it. And this happens only in a very small portion of the time. This is what creates the different crossing probability. – Jordan Valansi Jan 26 '18 at 2:01
• I think I figured out why this doesn't break the 2nd law. The particles hitting the door, double it's speed each time, increasing it's energy, taking a lot of the particle's energy. so it's a lossy reaction. Particles from the other side don't affect the door's speed, since it moves faster then them. Causing the door to increase in speed, and behaving as a barrier for particles to switch chambers. – Jordan Valansi Jul 8 '18 at 19:48