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In the micro canonical ensemble the microstates of a system in an arbitrary macrostate, are also eigenstates of the Hamiltonian. Does the same apply to the microstates of the canonical ensemble? Are they eigenstates of the the Hamiltonian? I would expect them not to be, since here the energy is not constant. But I am not sure

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When talking about the canonical ensemble, one has to distinguish

  • the Hamiltonian of the system of interest, $H_S$
  • the Hamiltonian of the system of interest + bath/thermostat/reservoir, $H_{tot} = H_S + H_B + V_{SB}$

$H_{tot}$ is treated in a microcanonical ensemble framework, and hence we are discussing its eigenstates. Generally it will not commute with $H_S$, since there is some interaction between the system and the bath. The derivation of the canonical ensemble is however based on solid reasoning that the interaction energy is smaller than the energy of the system, and can be neglected in tehrmodynamic limit (roughly speaking, the energy of the system is proportional to its volume, whereas the interaction energy is proportional to its surface, i.e., scales as volume to power $2/3$.)

Hence, once this logic is accepted and we talk about microcanonical ensemble, the microstates of the system are its eigenstates.

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  • $\begingroup$ And the microstates of the canonical ensemble, are the same as the ones in the MCE? $\endgroup$
    – imbAF
    Commented Dec 7, 2021 at 10:46
  • $\begingroup$ In MCE you consider microstates for all possible particle numbers $N$, whereas in CE the particle number is fixed. But if you consider the same bath in both cases, assuming only no particle exchange between the system and the bath for CE, and allowing particle exchange for MCE, then the microstates of the total Hamiltonian are the same (system+bath has a fixed particle number). $\endgroup$
    – Roger V.
    Commented Dec 7, 2021 at 10:54
  • $\begingroup$ One more question, In wikipedia it says: " The system can exchange energy with the heat bath, so that the states of the system will differ in total energy." And then here en.wikipedia.org/wiki/… it says : "A canonical ensemble does not evolve over time, despite the fact that the underlying system is in constant motion. This is because the ensemble is only a function of a conserved quantity of the system (energy).[" Aren't these two sentences contradicting each other? $\endgroup$
    – imbAF
    Commented Dec 7, 2021 at 12:10
  • $\begingroup$ It is the probabilities that do not evolve in time, but not the system itself. E.g., if we toss a coin, the probability of it landing heads/tails is $1/2$ - as we keep tossing, it evolves landing sometimes this way, sometimes the other, but the probability remains unchanged. $\endgroup$
    – Roger V.
    Commented Dec 7, 2021 at 12:12
  • $\begingroup$ I understand that but I am asking about the part it mentions energy. In the first sentence we say that "The system can exchange energy " and in the 2nd sentence :" This is because the ensemble is only a function of a conserved quantity of the system (energy)". Isn't this contradictory? Initially we say that the system changes energy and later on that the ensemble is a fucntion of the conserved quantity of the system, which is energy, and we just said that this changes for the system $\endgroup$
    – imbAF
    Commented Dec 7, 2021 at 12:26
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At the core of the statistical mechanics using ensembles, there is the possibility of assigning a probability to the set of all possible mechanical states of the system (microstates).

Therefore, the starting point is the identification of such microstates. In principle, any complete set of commuting observables could be used. However, for equilibrium macrostates, one knows (von Neumann's equation) that it is possible and convenient to use eigenstates of the energy for all possible energies of the mechanical system.

Such a statement does not depend on which energy eigenstates contribute to a macrostate. Therefore, they are used to label the microstates, in the case of a microcanonical ensemble, where only one particular value of the energy is picked up, but also for canonical and garn-canonical ensembles, where some finite probability is assigned to the microstates of any possible value of the energy.

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