- #1

binbagsss

- 1,265

- 11

## Homework Statement

Hi,

I am trying to follow the working attached which is showing that the average energy is equal to the most probable energy, denoted by ##E*##,

where ##E*## is given by the ##E=E*## such that:

##\frac{\partial}{\partial E} (\Omega (E) e^{-\beta E}) = 0 ##

MY QUESTION: the third equality, i.e. the second line

I have it explained the first term is taking care of the explicit dependence and the second term is taking care of the implicit dependence.

I'm pretty confused, I have never seen an example like this before. The only thing I can see is that if there is implicted and explicit dependene you do the chain rule, getting a product of terms, not a sum. I.e. letting ##f(E(\beta))## denote the function we are taking the deriviate of, I would conclude :

##\frac{\partial}{\partial \beta} f(E(\beta)) = \frac{\partial E}{\partial \beta}\frac{\partial}{\partial E}##...

I have never seen a sum of terms obtained from differentiation of explicit and implicit dependence of some variable.

Can some please expalin and tell me why the chain rule is not correct here? or (Links to any material on this also appreciated, thanks )

*context is canonical ensemble, statistical mechanics.*

Many thanks in advance.

## Homework Equations

see above

## The Attempt at a Solution

see above [/B]