system
Given the following post, respond as if you're commenting on Reddit:
human
If an isolated system reach a microstate where it has a macroscopic velocity, doesn't it violate the law of conservation of energy?

I know this is very, very, very unlikely edge case, but let's say that an isolated system, with an incredible luck, reach a microstate where the direction of their microscopic speed synchronise, in such a way that the system has a macroscopic speed. This mean we have a ΔE=ΔEc (ΔU=0 and ΔEp=0 too), yet the system is isolated so it should have a ΔE=W+Q=0. I was wondering where the mistake was in my reasonning.
gpt
The mistake in your reasoning is in the process from going from a stationary system to your edge case. That macrostate can never happen, for fundamental (not probabilistic) reasons. Consider the macroscopic system as simply a gas of particles. For simplicity, we can just think of the case where all the particles have the same mass (but this argument can easily be extended for the different masses case by considering momentum conservation). Initially, there is 0 average velocity in this mess of randomly-moving particles. Now say your desired incredibly lucky state has all the particles moving in the +x direction. Do you see why there cannot be any series of collisions that takes you to this state? At the beginning, half the particles will have a negative x-velocity. To make one of these particles get a positive x-velocity, another particle has to collide with it and lose some of its own positive x-velocity. So we can never make all of the particles have positive x-velocity! The same is true in the other dimensions.

Here's a cute and more intuitive proof if we just considered elastic collisions: in this case, all collisions are time-symmetric. So, we can simply start with your end case of all the particles moving in the direction with the same speed. How on earth can a series of collisions go from this to a stationary system? Firstly, this idea already shows you that there must be some small variations in the speed or we could never leave the synchronized state. But even then, any time you try to make a particle go in the backward direction, you have to make another particle speed up in this forward direction.

tl;dr: It can't be done because momentum must be conserved and so the total net speed cannot change unless an external force influences the system, in which case ΔE≠W+Q anyway.