I may well be all wrong about this but I am trying to understand from a microscopic point of view why Entropy is a concave function of internal energy. I found this in the following .pdf:
I started from this wikipedia article
and i understand why, if the particles composing the system have a limited number of available energy levels, then S(E) first increases and then decreases.
But saying that S(E) is concave should mean:
- when the temperature is T1, if i give a dE to the system its entropy increases of dS1
- when the tempereture is T2>T1, if I give the same dE to the system, its Entropy increases only of dS2 < dS1
I cannot see this with single particles.
If I have N particles in their lowest energy state there is only one microstate: all the particles are still.
If I give to this system the tiniest possible amount of energy, it will be taken by just one of the particle, so the possible microstates are N.
If I add another dE, the possible microstates should be N + N(N-1) = N^2 ... that is or one particle gets both dE or two different particles get it. Every time I add a dE I should increase the power of N.
Now, if the entropy is somehow proportional to the logarithm of the number of microstates, I should get S proportional to K ln(N^E), that is, something that is proportianl to E... taht is, no concavity
I am sure I am getting all this wrong... could you please help me understand this?