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Estimated Time: 20 minutes
Students are placed into small groups and asked to find an expression for the maximum fairness function in terms of $U$, $\beta$, and $Z$.
Before beginning the activity, write on the board:
$$F=-k_{B}\sum_{i}P_{i} \ln P_{i} \; \; \; , $$
$$U=\sum_{i}P_{i}E_{i} \; \; \; , $$
$$P_{i}=\frac{e^{-\beta E_{i}}}{Z} \; \; \; , $$
$$ Z=\sum_{i}e^{-\beta E_{i}} \; \; \; . $$
If students have not yet derived this result, the maximizing the Lagrangian activity will lead them to these expressions. Now, if we assume that the fairness function is maximized, these terms can be incorporated into the expression for fairness to derive a useful expression. Place the students into small groups and ask them to perform this operation and simplify as much as possible. Let them know that the sum over i should be gone after simplifying as much as possible.
Be sure to emphasize that the only reason we can use these expression is because we are assuming that the fairness function is maximized. After inserting the expression for probability into the maximum fairness function and simplifying, the class should find that
$$F_{max}=k_{B}\beta U + k_{B} \ln Z \; \; \; . $$
Show the class that this expression can be manipulated to get another relationship for $U$. Be sure to note that in the internal energy expression, we can see that $\beta$ is intensive since $\ln Z$ is extensive; this also provides even more evidence that $F_{max}$ is extensive.