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In the book of Intro. Statistical Mechanics by Huang, at page 156, while explaining macro canonical ensemble, the author states that

We are interested in the probability of occurrence of the microstate $\left\{p_{1}, q_{1}\right\}$ of subsystem 1 , regardless of the state of system 2 :

Probability that 1 is in $\mathrm{d} p_{1} \mathrm{~d} q_{1} \propto \mathrm{d} p_{1} \mathrm{~d} q_{1} \Gamma_{2}\left(E-E_{1}\right)$

where system 1 and 2 are the closed system and the heat reservoir, respectively. And $\Gamma_2 (E-E_1)=\Gamma_2(E_2)$ is the phase space volume of the reservoir having total internal energy $E_2$.

However, the above statement contains an implicity assumption that the probability of system one having a state of energy $E_1$ is proportional to phase space volume over the states with that energy. Meaning that the probability of a microstate being occupied by the system is proportional to macrostate of that microstate. Hence, the occurence of microstates aren't equal overall, but rather microstates have the same probability of occurence only within the set of microstates having the same macrostates. This is different than ergodic hypothesis, which assumes all microstates have the same probaility of occurence regarless of their macrostates.

Where does this assumption come from? Is there an explanation for why this must be the case?

Our
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  • This is called the ergodic hypothesis. – ACuriousMind Apr 04 '21 at 15:50
  • The links indicated as duplicate do not answer your question. However, you did get an answer to the same question a few days ago: https://physics.stackexchange.com/questions/625919/how-does-the-constancy-of-a-distribution-function-over-an-energy-surface-directl?rq=1 If something was not clear in the answer, you can ask for clarifications with comments or you may open a bounty. – GiorgioP-DoomsdayClockIsAt-90 Apr 04 '21 at 16:30
  • @GiorgioP that question asks about the contancy of the distribution function. – Our Apr 04 '21 at 17:03
  • @ACuriousMind see my edit. Tl;dr: The above assumption is different than ergodic hypothesis. – Our Apr 04 '21 at 17:08
  • In that case this is simply the definition of what it means for a system/macrostate to be in thermal equilibrium, see also https://physics.stackexchange.com/q/317563/50583 – ACuriousMind Apr 04 '21 at 17:13
  • @ACuriousMind that doesn't make more sense. The equilibrium is determined by the maximisation of entropy, but the entropy of a macrostate gets larger as the energy of the macrostate increases, so then why should the system settle down in a finite total energy when it can get more and more energy from the bath. – Our Apr 04 '21 at 17:53
  • @ACuriousMind see https://physics.stackexchange.com/questions/627098/why-does-a-canonical-ensemble-settle-to-a-finite-total-energy-in-equilibrium for a complete argument – Our Apr 04 '21 at 18:39

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