Given the mean energy of a system in a heat bath is(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\bar{E} = - \frac{\partial ln(Z)}{\partial \beta}

[/tex]

Where Z is the partition function and [tex]\beta = k_BT[/tex]

Why is the standard deviation of E defined by:

[tex]

(\Delta E)^2 = \frac{\partial^2 ln(Z)}{\partial \beta ^2}

[/tex]

I can't seem to find any proof of how the second derivative is related to the standard deviation.

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# Standard deviation of the energy of a system in a heat bath

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