enter image description here

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*

  • $\begingroup$ 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

1 Answer 1


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.