The previous chapter treated the two systems as equals. Here we make one of them very large. The large one is called a heat bath. It is the wide outer box in the figure, and the small box inside it is the system we care about.
Saying the bath is large means that taking a little energy from it does not move its temperature. A cupful drawn from the sea does not lower the sea.
The small system's energy is now no longer fixed. It changes from instant to instant as it trades with the bath. What is fixed is the temperature. Fixing the temperature rather than the energy is what we call the canonical ensemble.
The whole, system and bath together, is isolated. So the previous chapter's equal weighting applies as it stands. Every microstate of the whole carries the same weight.
Suppose the system is in one particular state, of energy E. The bath is then left with E_{\mathrm{t}} - E, and the count of bath states answering to that is W_{\mathrm{R}}(E_{\mathrm{t}} - E). That is how many cases the whole has, so the probability goes as it too.
The curve in the figure is that W_{\mathrm{R}}. The more the system takes, the less the bath keeps, and the bath's state count drops sharply. As the previous piece showed, the state count grows violently with energy, so it falls just as steeply.
A curve that falls that fast is awkward to handle, so we take its logarithm. It was S_{\mathrm{R}} = k \log W_{\mathrm{R}}. This one has a plain shape and looks, in the figure, very nearly straight.
The share E that the system takes is a very small thing beside all the bath holds. So the tangent at E_{\mathrm{t}} will do in its place. The slope of that tangent is \dfrac{\partial S_{\mathrm{R}}}{\partial E}, which is \dfrac{1}{T}. That is how the previous chapter defined temperature.
Here we set \beta = \dfrac{1}{kT}. From now on temperature turns up almost always in this form. Coming back out of the logarithm, the probability goes as e^{-\beta E}.
e^{-\beta E} is called the Boltzmann factor. The higher a state's energy the less often it happens, and the substance of the factor is that the falling off is exponential.
The figure shows the fall at various temperatures. A high temperature makes \beta small, and the curve lies flatter and trails a tail out to high energies. A low temperature stands it up and pins it near the bottom.
\dfrac{1}{\beta} = kT is the measure of how far that tail reaches. States up to about kT happen readily, and anything far beyond it is cut off exponentially.
Look at a system whose energy levels are arranged like a staircase. The horizontal bars in the figure are the levels and the bars on the right are the odds of being on each.
Raise the temperature and kT grows, and the bars reach the upper levels. Lower it and they gather on the bottom one or two. The spacing of the levels against the size of kT, and nothing else, decides how far up gets used.
That gives us the whole of how to handle a system at fixed temperature. All that remains is to add this factor over every state. What that sum is called is the partition function, and it is the next piece.