I am currently taking a Physics course Statistical Physics, and this pop up regularly:
this is from Fundamental of statistical and thermal physics
My question is: I understand Z is the partition theorem equal to the summation from zero to infinity of exponential to the - beta times the energy. How is the derivative related to the summation? why can I go from the left side of the equation meaning the side where both summations are present to the right side where theres a partial derivative of z with respect to beta.
what is the math process to go from summations to derivatives?
In physics, $Z$ is the partition function given by $$Z = \sum_{i}e^{-\beta E_i}$$ where i = 1,2,3.. $\infty$ is the index of the microstates of the system.
Now if you differentiate w.r.t. $\beta$, you get $$\frac{\delta Z}{\delta \beta} = -E_i Z$$ This is true because the derivative of the sum is equal to sum of derivatives. And the derivative of each term in the summation is equal to $E_i e^{-\beta E_i}$, for a given microstate $i$.
$$ E_i = -\frac{1}{Z}\frac{\delta Z}{\delta \beta}$$
$\frac{1}{Z}$ can also be written as $\frac{\delta}{\delta \beta} ln Z$. Hence the relation. Hope this helps.