# Microstates and Macrostates in Canonical Ensemble

In the microcanonical ensemble for a system in equilibrium, the macrostate has a certain multiplicity, which represents the number of microstates that this system can be in.

For the canonical ensemble in Wikipedia the following is said:

In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature.The system can exchange energy with the heat bath, so that the states of the system will differ in total energy.

The way that I understand it is that in equilibrium, the macrostate has a certain multiplicity, which represents the number of microstates that this system can be in. BUT here the different microstates have different total energies. In other words these microstates of the canonical ensemble have their own microstates. An analogy would be that the system is in a statistical mixture of states, and these states are also mixtures of states. Is that the case? If not why is this part emphasized in Wikipedia:

...so that the states of the system will differ in total energy.

• Possible duplicate by OP: Microstates of the canonical ensemble Commented Dec 7, 2021 at 1:01
• Not really, since I am searching for a direct statistical difference and not a thermodynamic one, by including the Hamiltonian Commented Dec 7, 2021 at 1:04

The wikipedia emphasizes the difference, because in a microcanonical ensemble the total energy of the system is fixed. The "multiplicity" is the number of microstates with a given energy (or sometimes in a small window of energy). For canonical ensemble, it is still a probability distribution over microstates, but there is no restriction on the value of energy of the system, since the system can exchange energy with the bath. So one has to consider microstates with different energies.

An analogy would be that the system is in a statistical mixture of states, and these states are also mixtures of states. Is that the case?

In a sense, yes, but your language is imprecise. The energy of an isothermal system fluctuates as the system exchanges energy with the surroundings. At given energy $$E$$ the macroscopic state is represented by $$\Omega(E)$$ microstates. The canonical ensemble is the set of all these microstates, essentially a "statistical mixture" of microstates with different energies, in which the probability to observe a given microstate $$i$$ is $$p_i = e^{-\beta E_i}/Q$$, where $$Q$$ is the canonical partition function.

I would rephrase your comment to read (your version as written does not distinguish between macro- and micro-states):

The system is a statistical mixture of macrostates with different energies, each macrostate represented by a microcanonical ensemble of microstates

but personally I prefer to think of the canonical ensemble as a collection of microcanonical ensembles with energies $$E=0\cdots\infty$$, each weighted by a statistical weight $$e^{-\beta E}$$.

• Shouldn't the Boltzmann factor refer to the probability to observe a given macrostate? Commented Aug 7, 2022 at 17:05
• The Boltzmann factor refers to the probability of microstate. Macrostates don't have a probability, they are defined by their macroscopic variables, for example $(T,V,N)$. Given the microstate there is a large number of microstates, each characterized by their probability. Commented Aug 7, 2022 at 19:09