Maximizing Microstates: It's Obvious! Not!

This installment of “It’s Obvious. Not!” looks at:

Book: “Statistical Mechanics”

Edition: second

Authors: R.K. Pathria

Publisher: Elsevier Butterworth Heinemann

Page: 12

The idea in Pathria is to start with a function for the number of microstates as a function of energy and then maximize it to study the implications of a system in equilibrium, (the maximum number of microstates). Pathria skips a few steps in the differentiation and simplification. There shown below to help me and others along. Have fun!

Starting with

Maximize

with respect to

Keep in mind that

is a function of

Use the chain rule of differentiation to expand the second partial derivative:

The last derivative term simplifies to -1:

So, to maximize we have:

Now, consider the differentiation of a function . The chain rule gives:

Applying this to our result above, we get:

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