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: