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We know that $$ \langle ( E_j - \langle E_j \rangle )^2 \rangle = - \frac{\partial}{\partial \beta} \langle E_j \rangle $$ And in the thermodynamic limit $ \langle E_j \rangle \longrightarrow U $

In the book Introduction to Statistical Physics of Silvio R. A. Salinas, there's the following text:

Making the identification between the expected value of energy and the internal thermal dynamic energy, we have $$ \langle ( E_j - \langle E_j \rangle )^2 \rangle = -\frac{\partial}{\partial \beta} U = k_B T^2 \frac{\partial U}{\partial T} = Nk_B T^2 c_V \geq 0 $$

Where did that $ k_B T^2 \frac{\partial U}{\partial T} $ come from?

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$$ T= \frac{1}{\beta} \implies \frac {\partial T}{\partial \beta} = - \frac {1}{\beta^2} = - T^2$$

hence, using the chain rule:

$$ - \frac {\partial U}{\partial \beta} = - \frac {\partial U}{\partial T} \frac {\partial T}{\partial \beta} = T^2 \frac {\partial U}{\partial T} $$

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