Moving Walls and 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: 14

Section 1.3 follows the derivation laid out in 1.2, but with a variable volume to arrive at:



As in 1.2, a parameter is defined that must be equalized:



The above subscript notation indicates that N and E are held constant and that V is equal to its equilibrium value.

The basic formula of thermodynamics is then stated as:



Immediately after which, it is stated that:



But how?

First, remembering that N and E are constant, rewrite the basic formula above as:



Now, using the expression from section 1.2 relating the micro and macro states:

We can write:

However,

so,

Cool Math Tricks: Pulling Linear Factors out of Binomial Sums

While doing my statistical mechanics homework today I arrived at a sum that looked like


Sums like this come up frequently when you're working with random walks, or flipping coins, or counting states of quantum mechanical systems where only two energies are allowed, or in any other number of contrived situations.

It's almost a good looking sum because everything to the right of the factor of m looks like the sum for a binomial expansion:


which simply evaluates to:


It turns out that there's an easy way to get the factor of m out of the sum and get on with your life! First notice that:

so that the sum


can be re-written as:


So, we rather handily got rid of the factor of m. The extra factor of p can be taken outside of the sum since it has nothing to do with the summation index m. Furthermore, the order of summation and differentiation can be interchanged to arrive at:


Now the sum actually is a binomial expansion and after simplifying and performing the derivative that was introduce we arrive at:

I didn't come up with this trick, I'm merely passing it along. I found it in a Statistical Mechanics text that I'm very impressed with:

Reif, Statistical and Thermal Physics, 1965

NOTE: I hope these little pointers are helping folks out, because they're definitely helping me. While preparing this post, I initially put in a summation index of i and found myself wondering what m had to do with anything. It turns out that I didn't have the concept firmly in my head yet, and upon re-investigating I corrected the summation index and got a much better understanding of what's going on. It just goes to show that what my childhood piano teacher said is true:

"The best way to learn something is to prepare yourself to be able to teach it to someone else."

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: