Where is the critical moment where the microcanonical ensemble enters the justification for the equilibium state?

Question

As explained in many books, for the microscopic justification of the second law of thermodynamics (lets formulate it as the total entropy takes maximum among all possible exchanges of two systems), you don't have to enter the realm of the canonical ensamble.

Where is the microcanonical phase space density $$\varrho=const. \ \ \ \ \text{if}\ \ \ \ E<H<E+\Delta,$$ used in the computation of the phase space which comes from the composition of two other systems?

Let $E=E_1+E_2$. For fixed energies $E_1$ and $E_2$, the new volume is given by $\Gamma(E)=\Gamma(E_1)\Gamma(E_2)$ and at the intermediate point where one considers all the possible energy exchanges, one writes $\Gamma(E)=\sum_{\epsilon}\Gamma(E_1+\epsilon)\Gamma(E_2-\epsilon)$.

I don't see how the construction of the composed phase space is computationally influenced by $\varrho$, and it also seems to me that this composed space would be computable without stating $\varrho$ explicitly. It's a volume after all, it shold just be the product in any case.

Furthermore, is $\varrho$ involved in the derivation that the maximum among the possible composed phase volumes is very sharp? (Is there a general derivation?)

Answer 1

The "microcanonical ensemble" is just saying each of the $\Gamma(E)$ states is equally likely. When you multiply the volumes at energy $\pm\epsilon$ and add over $\epsilon$, you are using the microcanonical ensemble assumption that all the states are equally likely, to get that the probability is the volume.

As for the sharpness, this is from the thermodynamic observation that one of the systems 1 or 2 is very big, so that it has a value of $\partial_U S $, which tells you how the volume changes with energy. This volume changes in a way proportional to the macroscopic size, since the S is extensive, and there are Avogadro's number of particles.