thermodynamics - How does a gas of particles with uniform speed reach the Maxwell-Boltzmann distribution?

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.