# Statistical mechanics: energy variance of ensemble

1. Jan 18, 2010

### Dawei

I'm really lost on where to get started here. It's a two state system, one with energy 0 and the other with energy ε. I already have ensemble average, <E>, found to be:

ε / (e^βε + 1) , where β is thermodynamic beta, 1/KbT.

How do I convert this to an expression for the variance of the energy?

A more complete description of the problem can be found http://books.google.com/books?id=z6...in+thermal+physics&cd=1#v=onepage&q=&f=false", on the top half of page 41:

I've already found http://en.wikipedia.org/wiki/Partit...chanics)#Relation_to_thermodynamic_variables" explanation, but I'm not connecting how it could be applied to this example:

Please, can anyone point me in the right direction?

Last edited by a moderator: Apr 24, 2017
2. Jan 19, 2010

### mathman

In general: variance = 2nd moment (m2) - mean square.

With a two level system, m2=a2P(a)+b2P(b), where a and b are the 2 levels. In your case a=0 and b=ε.