Hi,
Thanks for your reply. I finally figured out that I mixed up the entropy of the environment with the entropy of the system, because my idea was that the total system, so environment + system, could be described by the microcanonical ensemble and I could use Boltzmann's formula, but then you will end up with something different:
The system including its environment can be described as a microcanonical ensemble. The number of possible configurations for this ensemble are ##\Omega_{total} = \sum_i w_i## where ##w_i## denotes the number of possible configurations given an ##\epsilon_i##.
We know $$w_i = \Omega (E\epsilon_i) \Omega (\epsilon_i)$$ (with ##\Omega (\epsilon_i) = 1##, ##\Omega (E\epsilon_i)## the number of microstates of the system when its energy equals ##E\epsilon_i## and ##\Omega (E)## the number of microstates of the environment, when it is not thermally connected to another system) and thus $$ \Omega_{total} = e^{S_{total}/k} = e^{S/k} e^{S_{env}/k} = e^{\beta (U  F)} \Omega_{env} = \sum_i \Omega (E  \epsilon_i) = \Omega (E) \sum_i e^{\beta \epsilon_i} = \Omega (E) e^{\beta F}$$
This simplifies to $$ \Omega_{env} = \Omega (E) e^{\beta U}$$
Do you know if this is correct, because I have never seen this result before. It does seem okay to me though.
