# Confused by the two uses of word "Macrostate"

After reading some question-answers about the term "macrostate" I am still confused. Macrostate of a thermodynamic system is generally defined as set of values of macroscopic state variables like P, V, T, N...corresponding to an equilibrium state. But sometimes Macrostate is referred to the different internal energy possible distributions between the particles of a system. For example:

N=3 particles; Internal energy=3 units

Macrostate A: 3 0 0 with 3 microstates: 3 0 0, 0 3 0, 0 0 3

Macrostate B: 2 1 0 with 6 microstates: 2 1 0, 2 0 1, etc.

Macrostate C: 1 1 1 With 1 microstate: 1 1 1

Total microstates: 10

Are these energy distributions properly named as macrostates? If yes, how must I name the state described by internal energy, volume and number of particles? also macrostate? Is it a word used to nominate two different concepts?

• The macrostate word can indeed be used in several different (but close and coinciding under certain conditions) senses. However, the second option you described is still somewhat strange. Where did you see it? Jan 2 at 4:05

Macrostates are summarized versions of the state of a physical system.

Imagine that you describe the reality of a system at some fundamental level by using some entities, and the properties of those entities. Examples of this are the description of a gas using particles as entities, and their positions and velocities as properties. Another example would be to use states as entities and their occupation status (number of particles in each state) as the property. At this fundamental level, the full description of the system is given by specifying the properties corresponding to all the fundamental entities.

At another level, we can provide a description of the same system using other properties. Those might be properties of other non-fundamental entities of the system (for example, describing the brain as made of neurons instead of u and d quarks and electrons). Also, they can be properties of the system as a whole (and not any sub-structure of it). These can be averages of the fundamental level quantities, or their sum (or any other mathematical combination, as far as I know). These usually constitute a much smaller set of properties that still provide a high predictive power, although not all of the possible choices of properties are equally useful.

The full list of properties at the first level of description could be called microstate, and the full list of properties at the second level could be called macrostate.

What I believe is the answer to your question is that the properties you might choose for the macroscopic description are kind of up to you. In one choice, you could choose the volume, internal energy and the number of particles as the properties that define a macrostate. Nevertheles, you could have chosen differently, and include a new property (maybe call it distribution type) which could take three possible values: $$A$$, $$B$$ or $$C$$. This would be totally valid. Specifying a macrostate in the first option would be less informative than in the second option, but since none of them are as informative as providing the full description of the microstate, they can be referred to as macrostates.

Besides all this, the only counting that would provide 10 as the number of associated microstates is the one in option one, where you fix only the total energy and total particles as $$(E,N)=(3,3)$$. In the case where you fix also the type of distribution (let's call it $$\mathcal T$$), a macrostate would be given by the three values $$(E,N,\mathcal T)$$ and, according to the information you give, there would be 3 possible macrostates: $$(3,3,A)$$, $$(3,3,B)$$ and $$(3,3,C)$$, with 3, 6 and 1 possible microstates each, so the question would not have one only macrostate (as seems to be proposed by the wording), but three different ones.

To conclude, I would say that even if the macrostates in your question seem to be labeled by $$(E,N,\mathcal T)$$, I believe they are just labeled by $$(E,N)$$, and naming the particle distributions as macrostates is erroneous given the solution of 10 microstates.