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# implicit explicit dependence derivaitve, context statistical mechanics canonical ense watch

1. 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 ,

where is given by the such that:

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 denote the function we are taking the deriviate of, I would conclude : ..

. 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.

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2. (Original post by xfootiecrazeesarax)
..
Consider differentitating , You get (product rule).

What you've got is similar but a little more compicated. But the extra term is coming because in the process of diffing the whole thing you need to differentiate the product
in the term.

Edit: and explictly,the chain rule doesn't work in the way you seem to expect here, because we can't write the whole expression as a function of because of that single in the product that doesn't depend on E.
3. (Original post by DFranklin)
Consider differentitating , You get (product rule).

What you've got is similar but a little more compicated. But the extra term is coming because in the process of diffing the whole thing you need to differentiate the product
in the term.

Edit: and explictly,the chain rule doesn't work in the way you seem to expect here, because we can't write the whole expression as a function of because of that single in the product that doesn't depend on E.

I get (using log additive properties) , So I have the second term, can't see how I am going to get the first term.
4. (Original post by xfootiecrazeesarax)
I get (using log additive properties) , So I have the second term, can't see how I am going to get the first term.
I can't read your latex, but I explained how you get the first term in the previous post.

From what I can make out from your LaTeX, your response doesn't seem to have any attempt to use what I said in the previous post. Until you do so, I don't think I can help.
5. (Original post by DFranklin)
I can't read your latex, but I explained how you get the first term in the previous post.

From what I can make out from your LaTeX, your response doesn't seem to have any attempt to use what I said in the previous post. Until you do so, I don't think I can help.
Okay, well from the form of the chain rule you provided, it is pretty clear that it gives the correct answer as the solution does not compute anything more explicitly.

I have a couple of questions concerning notation however:
- so the chain rule is

Q1) Shouldn't one write rather than to distinguish that there is dependence on alone without ? It is clearer than that f takes the form of the chain rule you provided rather than the 'usual' one I initially went for?
Q2) concerning partial and total derivatives, I see you changed the to , to me f has both dependence on and , so I would have thought these need to be partials, however it would have made sense to me, since only depends on to change to ?
6. (Original post by xfootiecrazeesarax)
Okay, well from the form of the chain rule you provided,
It appears you're confused. I didn't give you a form of the chain rule, I showed you how to differentiate a particular expression involving a product between and a function f of .

Q1) Shouldn't one write rather than to distinguish that there is dependence on alone without ?
Again, f is just a function of . But the product obviously also has an explicit -dependence due to the first term () in the product.

Q2) concerning partial and total derivatives, I see you changed the to , to me f has both dependence on and
No, it doesn't. See above.

Edit: since you're obviously confused about all of this, I should perhaps add: I don't believe the first line I wrote has any ambiguities about what depends on what etc. However, that was in reference to a simple product, and as I said, what you have is a lot more complicated. The reason for the extra term is because you end up differentiating a product term involving , but because it's more complicated you'll need to be careful about what depends on what.

At postgraduate level I just expect to be giving hints and nudges, not needing to provide a step-by-step solution (which would be a lot of work).

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