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I am reading Landau and Lifshitz and I am confused about two steps in the Fluctuation theory chapter. They occur just before Eqn. 3 in "Fluctuations of the fundamental thermodynamic quantities". Here they are:

First we expand $\Delta E$ in a power series, I'm good with this: $$\Delta E -T\Delta S+ P \Delta V = \frac{1}{2}\left(\frac{\partial^2 E}{\partial S^2}\Delta S^2 + 2\frac{\partial^2 E}{\partial S \partial V}\Delta S\Delta V + \frac{\partial^2 E}{\partial V^2}\Delta V^2\right)$$

but then they rewrite this as

$$\frac{1}{2}\left(\Delta S\Delta\left(\frac{\partial E}{\partial S}\right)_V+\Delta V\Delta\left(\frac{\partial E}{\partial V}\right)_S\right)$$

This part doesn't make sense to me. And lastly they rewrite this as

$$\frac{1}{2}(\Delta S \Delta T - \Delta P\Delta V)$$

This seems plausible given the previous equation, I just don't know why the deltas are there. In other words, why isn't it

$$\frac{1}{2}(T \Delta S - P\Delta V)$$

Any help would be greatly appreciated.

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    $\begingroup$ Hint: $\Delta x\partial_x y\approx \Delta y$ $\endgroup$ – user12029 Apr 6 '18 at 23:14
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For constant volume, $1/T=(\partial S/\partial U)_V$ (from $dS = dQ/T)

For constant entropy, we know $dS = dQ/T = (dU+PdV)/T$ , therefore, $(dU-PdV)/T=0$ or $P=-(\partial U/\partial V)$

So $$\frac{1}{2}\left(\Delta S\Delta\left(\frac{\partial E}{\partial S}\right)_V+\Delta V\Delta\left(\frac{\partial E}{\partial V}\right)_S\right)=\frac{1}{2}(\Delta S \Delta T - \Delta P\Delta V)$$

For your first question,

$$\frac{\partial^2 E}{\partial S^2}\Delta S^2 + \frac{\partial^2 E}{\partial S \partial V}\Delta S\Delta V= \Delta S\frac{\partial}{\partial S}(\frac{\partial E}{\partial S}\Delta S+\frac{\partial E}{\partial V}\Delta V) =\Delta S\frac{\partial}{\partial S}(\Delta E) =\Delta S \Delta(\frac{\partial E}{\partial S})$$

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