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I'm not entirely sure about the difference between a macrostate and an ensemble, though I think they are different. To me, it seems correct that both terms can be used to refer to a collection of microstates that share the same set of macroscopic properties (like temperature, volume, etc.), which makes me wonder how these two terms can be different.

For example, suppose we have a gas confined to a container with a fixed volume of $V = 1\,\text{L}$, a temperature of $T = 300$ K, and $N = 1000$ particles. This would correspond to a specific macrostate of the gas. However, this is also an example of an $NVT$ ensemble, isn't it? And if we increase the temperature to 400 K, we would get a different $NVT$ ensemble and also a different macrostate. It looks like these two terms can be used interchangeably. Are they actually the same?

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  • $\begingroup$ en.wikipedia.org/wiki/Ensemble_(mathematical_physics) , There can be an ensemble of macrostates, it is a more general term , imi, $\endgroup$
    – anna v
    Commented Mar 13, 2023 at 4:44
  • $\begingroup$ Thank you for your reply. Could you possibly indicate the relevant paragraphs in Wikipedia about how there can be an ensemble of macrostates? From Wikipedia I'm not sure how to reach this conclusion. Or could you possibly provide an example of a macrostate in the scenario I mentioned in the question, i.e. a macrostate in the NVT ensemble where (V=1, N=1000, and T=300)? $\endgroup$
    – Jack
    Commented Mar 13, 2023 at 4:58
  • $\begingroup$ I think if you read the article you will see that the term "ensemble" is used in many ways in physics, that have nothing to do with macrostates of thermodynamics. It is a more general term. $\endgroup$
    – anna v
    Commented Mar 13, 2023 at 5:21

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In my preferred terminology, a statistical ensemble consists of the set of all possible microstates of the system and an associated probability distribution over that set. A macrostate is a collection of microstates which is defined by specific values of macroscopic variables (energy, temperature, etc) or, in a looser sense, by some kind of macroscopic condition that individual microstates may or may not satisfy.

For example, suppose we have a gas confined to a container with a fixed volume of $V = 1\,\text{L}$, a temperature of $T = 300$ K, and $N = 1000$ particles. This would correspond to a specific macrostate of the gas. However, this is also an example of an $NVT$ ensemble, isn't it?

The $NVT$ (or canonical) ensemble assigns to each microstate $x$ the probability $$\rho_{NVT}(x) = e^{-E[x]/kT}/ Z_{NVT} \qquad Z_{NVT} = \sum_x e^{-E[x]/kT}$$

where $E[x]$ is the total energy of the microstate $x$ and $Z_{NVT}$ is the canonical partition function. It is convenient to imagine the particles' position and momentum variables to be discretized for ease of calculation, but one can work with continuous variables as well - it simply introduces technical complications without altering the spirit of the process.

We might ask for the probability that the system has some particular value (or range, in the continuous case) for the total energy, pressure, or chemical potential - all of which generally differ from microstate to microstate. We could ask for the probability that the particles in the gas all occupy the upper half of their container. These conditions would all correspond to different macrostates in the same ensemble, which could be assigned probabilities via the probability distribution given above.

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  • The macrostate is the set of macroscopic variables that fix the macroscopic state of the system, for example, $(E,V,N)$.

  • The ensemble is the set of possible microstates accessible to the macroscopic state. The microcanonical ensemble, for example, is the set of microstates with fixed $E$, $V$ and $N$.

If we view the macrostate as a container of microstates, i.e., a bucket with "size" $(E,V,N)$, then the ensemble is the set of microstates it contains. Since the macrostate fixes the corresponding ensemble we may view the two entities, "macrostate" and "ensemble", as pointing to the same "thing". However, we should distinguish them semenatically because the former is a state and the latter is a set.

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