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So I'm currently studying statistical mechanics from different textbooks, but my professor suggested Kerson-Huang for a general derivation of entropy in microcanonical ensembles. In chapter 6.2 is introduced an ensemble occupying a phase space volume $\Omega(E)$ corresponding to an energy $E$, which is later partitioned in two microcanonical ensembles denoted by $N_{1}$ and $N_{2}$ numbers of particles and energies $E_{1}$ and $E_{2}$, with $E=E_{1}+E_{2}$ fixed, while the phase space volumes occupied by each of them are $\Omega_{1}(E_{1})$ and $\Omega_{2}(E_{2})$. In order to find the phase space volume of the ensamble including both of them the book performs the following calculation:

$ \Omega(E) =\sum_{i=1}^{\ E/{\Delta}} \Omega_{1}(E_{i})\Omega_{1}(E-E_{i})\label{eq1}$

$\Delta$ being the arbitrary discrete unit of energy established. My question is, shouldn't $\Omega(E)$, which I think represents the volume in the phase space occupied by the ensamble associated to both of the sub-systems, be equal to $\Omega_{1}(E_{1})\Omega_{1}(E-E_{1})$ alone? Summing over all possible values of $E_{1}$ should mean summing over all possible partitions of the initial energy $E$, but in that way the obtained volume would be way bigger than the actual volume of the composite ensemble.

I'm afraid I'm missing some kind of meaning of $\Omega(E)$ and, if I am, I would like to understand why is the presented equation correct. I apologize in advance for any grammar mistakes since English isn't my first language.

Edit

I found some hits of solutions in some questions posted by other users at NVE Ensemble and Probability and in the answer to the second question at Entropy definition, additivity, laws in different ensembles but I still can't quite grasp the difference between the omega term used in Kerson Huang and $\Omega_{1}(E_{1})\Omega_{1}(E-E_{1})$.

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  • $\begingroup$ If the subsystems do not exchange energy (are separated by an adiathermal wall), then simple product gives correctly the phase space volume of the combined system. If the subsystems exchange energy, then it does not, because $E_1$ is just the expected average of energy in the first subsystem, which is supposed to fluctuate in time due to exchange. Thus energy of the first subsystem can be $E_1 + \delta E_1$ and in that case energy of the second subsystem has to be $E_2 = E - \delta E_1$. This imbalance in energy implies different phase volumes, and in principle has to be taken into account. $\endgroup$
    – Ján Lalinský
    Commented Apr 11 at 22:47
  • $\begingroup$ Thank you so much for your answer, however I think I'm missing how energy exchange between the two systems implies the necessity to sum over every energy partition. Could you give me a hint on how is that so? $\endgroup$
    – Matteo Grandinetti
    Commented Jun 4 at 14:32
  • $\begingroup$ Well if the two systems exchange energy, then various values of $E_1$ for energy in system 1 are possible, and different values of $E_1$ imply different sets of microstates. Sum of numbers of these microstates implied by all possible values of $E_1$ looks like a natural way to take into account all compatible microstates. $\endgroup$
    – Ján Lalinský
    Commented Jun 4 at 14:59
  • $\begingroup$ Thanks you so much, it Is much more clear now! $\endgroup$
    – Matteo Grandinetti
    Commented Jun 5 at 15:11

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