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This is a problem to do with statistical physics, and the exchange of energy when we have two microcanonical ensemble.

I don't understand why there should be a minus sign in the middle, if Energy* is the energy of system 1 , Energy total - E* is the energy of another system, Two system were put together and interact, I was told this equation can maximise entropy, but I can't really see why?

I thought S = (S1(E*))+S(E total - E*) is the total entropy. therefore it should be a positive sign when you differentiate..

Just to clarify, S(E*) and S(E total - E*) , they are just labels, not function of E* and E total - E*

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  • $\begingroup$ Your equations don't quite make sense, you are differentiating a function of E* with respect to E. How does E* depend on E? Also, how do you define entropy of microcanonical ensemble? There are at least two ways. $\endgroup$
    – Nanite
    Jan 29, 2014 at 13:01
  • $\begingroup$ Quoting from damtp.cam.ac.uk/user/tong/statphys/sp.pdf I think (E) has nothing to do with functions. It means the entropy of system with Energy 1 , but not function of E1 $\endgroup$ Jan 29, 2014 at 13:21

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A quantity is maximized or minimized when its derivative is zero ...

You want to maximize $S_1(E) + S_2(E_{total} - E)$ as $E$ varies, and you are defining $E_*$ to be the location of that maximum.

So, you can take the derivative with respect to $E$, and set it to zero at $E_*$.

$$ 0 = \frac{d}{dE} (S_1(E) + S_2(E_{total} - E))|_{E = E_*}$$ $$ = (\frac{dS_1}{dE}|_{E = E_*}) - (\frac{dS_2}{dE}|_{E = E_{total} - E_*})$$

The negative sign comes from the chain rule. :-D

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