assumption of molecular chaos and the Loschmidt paradox

Question

The assumption of molecular chaos says the velocities of two colliding particles are uncorrelated and also independent of time. Boltzmann actually used this assumption in his formulation of the H-theorem. Loschmidt objected Boltzmann's argument by arguing that the time-reversibility of microscopic dynamics will in no way leads to a macroscopically irreversible process. The wikipage claimed that it is the molecular chaos assumption, which ignores the correlation of particle velocities, that introduced asymmetry in the microscopic dynamics which leads to the time arrow in H-theorem.

so my question is: why the velocities of the colliding particles are still correlated?

Answer 1

The molecular chaos assumption is not that all particle velocities are always uncorrelated. If you make two particles collide, of course their final velocities will be correlated. Rather, molecular chaos assumes that between collision events, each particle has "forgotten" that its velocity came from a prior collision. Since the particles have no way to "remember" their past velocities, the molecular chaos assumption actually inserts the time asymmetry necessary to get irreversibility, which resolves the Loschmidt paradox.

If you're not satisfied by this, that's ok! Assuming molecular chaos is actually begging the question, using the result (irreversibility) in the initial assumptions.