Take an empty container and fill it with $N$ gas particles (ideally a monoatomic gas), each having the same kinetic energy $E$, then isolate the container. Since initially the speeds don't follow the Maxwell-Boltzmann distribution, such a system cannot be in thermodynamic equilibrium. On the other hand, assuming perfectly elastic collisions (and there is no reason to assume otherwise, since the only form of energy the particles can possibly have is kinetic), I see no way such a system could spontaneously evolve to equilibrium: elastic collisions among equal masses keep speeds unchanged! What gives?
I have no background in non-quasistatic processes, but I tried nonetheless to work out a solution taking into account the container, which necessarily has a certain heat capacity, a certain initial temperature, and whose walls are not necessarily perfectly elastic, etc. Knowing the number of particles and their individual speed, it's possible to compute the system's total heat content (but is this exactly $NE$, or less?) and thus derive it's equilibrium temperature. Since the system is isolated, I take it the quantity that has to change must be entropy (namely increase, as the uniform speed state seems less likely; the change can probably be arrived at from a strictly combinatorial point of view). At any rate, the process I imagined goes like this: initially, the particles bombarding the wall transfer some amount of heat to it while slowing down; in turn the wall, now heated up, will transfer back some heat to the gas; eventually, the system will reach the expected equilibrium.
Is my assumption of perfectly elastic collisions wrong, and if so, where does the dissipated energy go?
Is there an increase in temperature that accompanies the increase in entropy?
Can someone point me to the rigorous mathematical framework for analyzing the problem?
Is there direct experimental evidence that the speed of gas particles attains the Maxwell-Boltzmann distribution, or is it just a theoretical result that everyone is just happy to work with?
Thanks for any suggestion.
Best Answer
WetSavanaAnimal aka Rod Vance has given a good introduction to the issues involved.
When I originally wrote an answer, I though that you were right: if you had a perfectly ideal system of perfectly hard spheres, with perfectly elastic collisions, in a container with perfectly rigid walls, and if all the particles started out with exactly the same speed, then the velocities could not evolve into a Maxwell-Boltzmann distribution, because I thought there was no process that could make the velocities become non-equal. However, I've realised I was wrong about that. For example, consider this collision:
The total $x$-momentum is zero but the total $y$-momentum is positive. This must be the case after the collision as well, so the motion must end up looking like this
with the top particle having gained some kinetic energy while the bottom particle loses some. Through this type of mechanism the initially equal velocities will rapidly become unequal, and the system will converge to a Maxwell-Boltzmann distribution just by transferring energy between particles, with no need for energy to enter or leave the system.
However, it's still possible to imagine special initial conditions where this won't happen. For example, we can imagine that all the particles are moving in the same direction, exactly perpendicular to the chamber walls, and are positioned such that they will never collide. A system in this configuration will remain in this special state forever, and not go into a Maxwell-Boltzmann distribution.
However, such a special state is unstable, in that if you start out with even one particle moving in a slightly different direction than all the others, it will eventually collide with another particle, creating two particles out of line, which can collide with others, and so on. Soon all particles will be affected and the system will converge to the Maxwell-Boltzmann distribution.
In reality, as Rod Vance said, in practice the walls will not be perfectly rigid and will be in thermal motion, which would prevent any such precise initial state from persisting for very long.
Even so, this seems to imply that the hard sphere gas system has at least one special initial state from which it will never reach a thermal state. But this isn't necessarily a problem for statistical mechanics. In this case (if my intuition is correct) the states with this special property form such a small proportion of the overall phase space that they can essentially be ignored, since the probability of the system being in such an initial state by chance is technically zero.
There can be cases where every initial state has a property like this. This means that the system will always remain in some restricted portion of the phase space and never explore the whole thing. But these are just the cases where there is some other conservation law, in addition to the energy, and we know how to deal with that in statistical mechanics.
People used to worry a lot about proving that systems were "ergodic", which essentially means that every possible initial state will explore every other state eventually. But nowadays a lot less emphasis is put on this. As Edwin Jaynes said, the way we do physics in practice is that we use statistical mechanics to make predictions and then test them experimentally. If those predictions are broken then that's good, because we've found new physics, often in the form of a conservation law. When the new law is taken into account, the new distribution will be seen to be a thermal one after all. So we don't need to prove that systems are ergodic in order to justify statistical mechanics, we just need to assume they are "ergodic enough", until Nature tells us differently.