The particles of a gas strike one another. When they do, two velocities are exchanged for two others. Momentum and total energy do not change, but the combination of velocities does.
The figure is one such collision. Before and after, the arrows have swapped direction and length. It is these exchanges, piled up, that move the distribution of velocities.
Run a single collision backwards in time and it is still a collision. Nothing in mechanics objects. And yet gather enough of them and there is no going back. This piece is about where that gap comes from.
Boltzmann built a single number out of the velocity distribution f(v): H = \int f\log f\,dv. It gathers the shape of the distribution into one number.
The upper curve in the figure is the distribution, the lower one f\log f. The more sharply the distribution peaks in one place the larger H is, and the flatter and broader it spreads the smaller.
Change the sign and this is the entropy: S = -kH. It is the -k\sum p\log p of the piece on turning spread into entropy, written again for a continuous velocity.
Here we put in one assumption: the two velocities just before a collision have nothing to do with each other. This is called the assumption of molecular chaos. That single line is in fact the most important thing in this piece.
Counting collisions under that assumption, H never increases with time. The curve in the figure is that, coming to rest once it has fallen as far as it will. This is the H theorem.
Since S = -kH, this says the entropy never stops rising. The second law of thermodynamics has come out of counting collisions. So far, this is going very well indeed.
Once H has fallen as far as it will, the distribution stops changing. Its shape then is the Maxwell distribution. The same shape found by counting in the piece on the ideal gas turns up here as a destination.
The figure is that change. It begins split into two peaks, and as the collisions accumulate it gathers into one. The number of particles and the total energy stay as they were; only the shape moves.
There is only one destination because, holding those two fixed, only one shape makes H as small as it can be. Impose what is known and leave the rest as flat as possible. That is exactly the argument of the maximum entropy principle.
But there is trouble. Reverse every velocity at some moment and the system retraces the path it came by, and over that stretch H must rise. Mechanics runs backwards perfectly well, while the H theorem allows one direction only.
The way out lies in the assumption of molecular chaos. The velocities just after the reversal are not unrelated to one another: the traces of the previous collisions are still there, as correlations between them. The assumption holds only where those traces may be ignored.
The left of the figure is what a fine view shows, the right what averaging over cells shows. Finely, the area occupied is exactly what it was, only drawn out into threads. Averaged, it appears to have spread over everything. What increased was the entropy of the averaged view. The second law is a law one can state only after settling how finely to look.