Derivative of log(x!)

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This simple Demonstration illustrates the important limit as
. For noninteger values of
, we take
to be
.
Contributed by: Cedric Voisin (June 2012)
Open content licensed under CC BY-NC-SA
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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 .
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