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Can't understand one step in derivation (partition function)

  • Thread starter tomyuey938
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  • #1
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This is from self-study coursework rather than homework. I hope it's ok in this forum.

I'm following a statistical mechanics lecture on youtube, and the professor is deriving the average energy as a function of the partition function. He goes:

-1/Z dZ(beta)/d beta = -dlnZ(beta) / d beta

where Z(beta) is the partition function = e^{-(beta) E}.

I spent an hour trying to figure out why the LHS equals the RHS (from both directions, including using differentiation by substitution) but I just can't figure out how he does it.

Perhaps this is a really simple question (and if so, I apologies for my stupidity) but can anyone tell me how to go between the two steps?

Thank you in advance!
 

Answers and Replies

  • #2
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As best as I can figure out (don't know anything about statistical mechanics) this is just a simple derivative. If the LHS reads -1/Z(beta) dZ(beta)/d beta (which I imagine it should) then this is just an application of the chain rule.
To make it easier to think about consider LHS and RHS as positive instead of negative (which we can do by multiplying by -1) then look at the RHS. The derivative of ln(Z(beta)) is found by the Calc 1 rule "derivative of the inside times the derivative of the outside" where the inside is Z(beta) and the outside is ln(Z(beta))
thus you have 1/Z(beta) (the direct derivative of ln(Z(beta))) times dZ(beta)/dbeta (the derivative of the argument of ln)
 
  • #3
14
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dot.hack, thank you so much, I've got it now. I made such a trivial mistake. I really appreciate your help!
 

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