Put two systems side by side. Say the left has W_1 states and the right has W_2. Any of the left may go with any of the right, so the pair has W = W_1 W_2 in all. The grid in the lower part of the figure lays out every one of those combinations.
Thermodynamic entropy, however, is a quantity that adds when two systems are put together. Like volume or amount of substance, doubling the system doubles it. That is a different manner of growth from W, which multiplies.
So W cannot simply be called the entropy. Some turn is needed to change a multiplication into an addition.
There is only one tool that turns multiplication into addition, and that is the logarithm. \log W = \log W_1 + \log W_2, and now it adds.
The bars in the figure show it. Stack the right system's \log W_2 on top of the left system's \log W_1 and you have the \log W of the whole. With W itself there is no such stacking.
In the previous piece we used \log W because the number of digits was beyond handling. Here a second reason joins it. Only the logarithm grows in the same manner as a thermodynamic quantity.
So we settle on this: S = k \log W. Boltzmann's relation. k is Boltzmann's constant, and a later scene shows what work it is doing.
This was not derived. It was decided. The previous scene took us as far as wanting a quantity that adds, and past that point we are simply declaring what to write down.
What follows the declaration is another matter. From this one line come temperature, pressure and free energy, everything thermodynamics had built up from experience, without exception. The figure sets out the argument so far one line at a time. The last line, S = S_1 + S_2, is the form we wanted at the start.
\log W is a bare number with no units. Thermodynamic entropy, on the other hand, has the dimensions of \mathrm{J/K}. k is the factor that converts between them.
The left scale in the figure is \log W and the right one is S. They point at the same place and yet the numbers marked on them differ. What closes that difference is k.
The conversion is needed because people fixed the temperature scale first. The decision to cut the span between freezing and boiling water into 100 came first, and statistical mechanics came afterwards. Measure temperature in units of energy and k = 1, and the factor disappears.
Shut the gas into one side of the box and then take the partition out of the middle. The particles spread, and never once go back to the one side.
Count it. Twice the room is available, so each particle has twice the places to be, and N of them have 2^{N} times as many. \log W rises by N\log 2 and the entropy by Nk\log 2.
They do not go back, and not because the way back is blocked. The state with everything on one side is fully counted among the rest. It is simply one part in 2^{N}. As the previous piece showed, with N at 10^{23} that is the same as never.