Derivative of log(x!)
This simple Demonstration illustrates the important limit as . For noninteger values of , we take to be .
When optimizing the energy repartition in a statistical ensemble by maximizing the total number of states with respect to each state occupation number , it is convenient to take the log of this expression to transform the product into a sum and simplify the differentiation with respect to . The formula then becomes .
When the system is in contact with an energy reservoir that keeps the average energy per state constant, one can use a Lagrange multiplier to account for the constraint that is constant.
One then has to differentiate the function with respect to , which gives .
The only difficult step in this computation is the left-hand side of this equation. The trick here is to rewrite for large . This amounts to neglecting next to : for large . Hence the formula becomes for large , which leads straightforwardly to the Boltzmann distribution.
This elementary Demonstration illustrates the fast accuracy of this important formula by plotting both functions side by side for a varying range up to .