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Consider a system taking on N states with energies $\epsilon \sim \mathcal{N}(\mu,\sigma^2)$. Are such systems well-studied in any context? I ask because I'd like to be able to take certain expectations related to the partition function. For example, one can show that:

$<Z> = Ne^{\mu + \frac{\sigma^2}{2}}$

without too much trouble (in units where $\beta = 1$). What I'd really like to be able to find, though, is $<\log Z>$. To start, one could try Taylor-expanding $\log Z$ about $<Z>$, obtaining:

$<\log Z> \approx \log <Z> - \frac{1}{<Z>^2}Var(Z)$,

where:

$Var(Z) = Ne^{2\mu + 2\sigma^2} + N(N-1)e^{2\mu + \sigma^2} - <Z>^2.$

But my expression for $Var(Z)$ doesn't seem to agree very well with numerical simulations, perhaps because $Z$ itself is so right-skewed. So, in sum, have such systems been studied before? Thanks in advance-- statistical mechanics isn't quite my field.

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It seems that there is, at least, a notation problem. $Z$ is a partition function, not a random variable. So, notations $<Z>$ or $<Log Z>$ or $Var(Z)$, make no sense. –  Trimok Nov 1 '13 at 9:56
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This looks pretty much like the random energy model. See, for example, Chapter 9 in Anton Bovier's book Statistical Mechanics of Disordered Systems: A Mathematical Perspective. –  Yvan Velenik Nov 2 '13 at 10:38
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@Trimok: This makes sense for a disordered model, such as the Edwards-Anderson model, the Sherrington-Kirkpatrick model, or the random energy (REM) model. In that case, the partition function is itself a random variable (depending on the realization of the disorder). What he asks for seems actually similar to the REM. –  Yvan Velenik Nov 2 '13 at 10:41
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@wvoq: another place you can consult is Mezard+Montanari's book *Information, Physics and Computation", Chapter 5. –  Yvan Velenik Nov 2 '13 at 10:44
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@YvanVelenik : Good point, so wiki Spin glass seems to be a good entry point for these models. –  Trimok Nov 2 '13 at 10:55

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