Such analyses are given the name Statistical Mechanics . Boltzmann showed that given a collection of molecules with any velocity distribution, after enough time has passed the velocities of the molecules will eventually acquire a continuous normal-like distribution described by James Clerk Maxwell and that the gas will eventually acquire a uniform temperature. That is, Boltzmann showed that the arrow of time was stastical in origin. A smashed and spilled coffee cup will never be seen to reassemble into a cup of coffee, not because it can't happen, but because it's extremely unlikely that it ever could happen. The Second Law of Thermodynamics is just a way of telling us not to worry about such things.
Time runs forward because that is the most likely way for it to run.