 Derivative of log(x!)

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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

Snapshots   Details

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 .

Permanent Citation

Cedric Voisin

 Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send