# Statistical Mechanics

1. Feb 26, 2012

### hansbahia

$\Omega$^(0)(E) = $\Omega$(E)$\Omega$(E^(0) - E),

a) Write this equation in terms of entropy

b)Taylor series expand this resulting equation to 2nd order in the individual energies.Use the fact that the subsystems are in equilibrium with a total xed energy to simplify the resulting expression.

c) How can you show the resulting expression for problem b that the variance of (E) is proportional to N, the number of particles in the system.

The way I approached this question was...

a) I know that Helmholtz free energy is A = U - TS, where U= (E), the total energy and S is the entropy, than A = E - TS, therefore S=-kTlnZ, however how can I apply when E^(0) is the total energy of the two subsystems. In other words how can I mathematically solve this problem?

b) I assumed subsystem A has some mean energy E about which there are fluctuations. Therefore since the macroscopic state of maximal entropy for the system is the one where all micro-states are equally likely to occur, with probability 1 / $\Omega$(E), during the system's fluctuations, so S is equal kln ($\Omega$(E)). Am I right? how can I compute this?

c) No clue. So far I could establish that the fluctuations
from equilibrium, of the number of accessible states scales could be 1/$\sqrt{}N$.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution