Boltzmann made a key assumption known as the Stosszahlansatz (molecular chaos assumption), that during any collision event in the gas, the two particles participating in the collision have 1) independently chosen kinetic energies from the distribution, 2) independent velocity directions, 3) independent starting points. Under these assumptions, and given the mechanics of energy transfer, the energies of the particles after the collision will obey a certain new random distribution that can be computed.