Maximising entropy when energy is shared between systems

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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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. – Nanite Jan 29 '14 at 13:01
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 – el psy Congroo Jan 29 '14 at 13:21

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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HAHA thanks lol!!! – el psy Congroo Jan 29 '14 at 16:23