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I'm looking at a specific derivation on wikipedia relevant to statistical mechanics and I don't understand a step.

$ Z = \sum_s{e^{-\beta E_s}} $

$Z$ (the partition function) encodes information about a physical system. $E_s$ is the energy of a particular system state. $Z$ is found by summing over all possible system states.

The expected value of $E$ is found to be:

$ \langle E \rangle = -\frac{\partial \ln Z}{\partial \beta} $

Why is the variance of $E$ simply defined as:

$ \langle(E - \langle E\rangle)^2\rangle = \frac{\partial^2 \ln Z}{\partial \beta^2} $

just a partial derivative of the mean.

What about this problem links the variance and mean in this way?

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    @Akhil : First of all I didn't intend to be hostile and I know it wouldn't be appropriate to get into any arguments on this. I felt that the OP didn't put effort into formulating the question in a readable form for a general audience. I think it would've been great if it was that way.2011-03-29

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The answer is valid for the partition sum $Z$ (which is closely related to the moment generating function). The reason is the special structure of the partition sum $Z = \sum_s e^{-\beta E_s}.$ The system is characterized with probability $P_s=\frac{e^{-\beta E_s}}{Z}$ that a state $s$ with energy $E_s$ is attained.

Given this definition it is easy to see that $-\partial_\beta \ln Z = -\frac{\partial_\beta Z}{Z} = \sum_s E_s \frac{e^{-\beta E_s}}{Z}= \sum_s P_s E_s =\langle E \rangle .$

Similarly, one can easily convince oneself that \begin{align*} \partial_\beta^2 \ln Z &= -\partial_\beta \left[ \sum_s E_s \frac{e^{-\beta E_s}}{Z} \right] =\sum_s E_s^2 \frac{e^{-\beta E_s}}{Z} - \left[ \sum_s E_s \frac{e^{-\beta E_s}}{Z}\right] \left[\sum_{s'} E_{s'} \frac{e^{-\beta E_{s'}}}{Z}\right]\\ &= \langle E^2\rangle -\langle E\rangle^2 = \langle (E- \langle E\rangle)^2\rangle, \end{align*} i.e., the variance is given by the second derivative of $\ln Z$.

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Have you seen the link of the definition of variance or expected value in wiki, here?

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The variance of a random variable $X$ is always defined as $<(X - )^2>$; this is the expected square of the difference between the expected and actual values.