# What is the difference between a macrostate and multiplicity?

The entropy of a system of an ideal gas depends on the external parameters $$U, V, N$$.

I always thought entropy is defined by a certain macrostate, which is a set of given external conditions like fixed $$U,V,N$$. This is one defined macrostate. The macrostate is the framework for a certain amount of accessible microstates.

But now I read that there is not only a single macrostate of a system, but that there can be various macrostates. The multiplicity of a macrostate is the probability of a certain macrostate to occur. So if there are various macrostates that can occur (regardless of how unlikely) what does that mean for our external parameters? How can there be different macrostates with different $$U, V, N$$?

Or is it just misleading that the multiplicity of indistinguishable particles is also called a macrostate, although there is only one "real" macrostate (defined by the external parameters)?

"But now i read that there is not only a single macrostate of a system, but that there can be various macrostates."

This is not in conflict with your previous understanding that a fixed U,V,N defines a macrostate. I think you just misinterpreted the statement. If a particular set of values for U,V,N denotes a particular macrostate, that means that if you change any of the values, you have another, different macrostate. The above statement just says that there are different macrostates (that U,V,N can take different values; i.e. the system can be found in different macrostates).

"The multipicity of a macrostate is the probability of a certain macrostate to occur."

No reason to assume that all macrostates will occur with equal probability. A macrostate is an abstraction that ignores the particular microscopic configuration of particles (microstate), meaning that there can be a number of microstates that all manifest as the same macrostate. That's where the multiplicity comes from.

In other words, it's not talking about a multiplicity of macrostates that are somehow equivalent. It's a multiplicity of microstates that give rise to the same macrostate. Some macrostates can be "manifested" by many different microstates, while others by only a small number of them - which then ties into the probability (greater if there are more ways for the same (macro-)thing to occur).

"So if there are various macrostates that can occur (regardless of how unlikely) what does that mean for our external parameters?"

It means that the values of the parameters have a probability distribution.

So if there are various macrostates that can occur (regardless of how unlikely) what does that mean for our external parameters? How can there be different macrostates with different U, V, N?

It means that the quantities which define the macrostate of the system (in this case, $$U,V,$$ and $$N$$) are not being held fixed.

• If your system is in thermal contact with a heat bath, then $$U$$ is not fixed because heat can flow between the system and its environment.
• If the system can expand or contract, then $$V$$ can change.
• If the system is in contact with a particle reservoir (e.g. a semi-permeable membrane), then $$N$$ is not fixed because particles can flow in and out.

Consider a system with fixed $$U,V,$$ and $$N$$. The values of $$(U,V,N)$$ define a macrostate of the system. The number of microstates which correspond to this macrostate, also called the multiplicity, is $$\Omega(U,V,N)$$; since all microstates are equally likely, the probability that the system is in any particular microstate is simply $$\frac{1}{\Omega(U,V,N)}$$. The collection of all such microstates and the associated uniform probability distribution is called the microcanonical ensemble.

Now consider a system with fixed $$V$$ and $$N$$ but which is in contact with a heat bath at temperature $$T$$. The system can exchange energy with the heat bath, which means that its total energy is not fixed; in principle, it could have any value of $$U$$. The probability that a system is in a particular microstate now depends on the total energy $$U$$ of that microstate.

Explicitly, the probability that the system occupies a microstate with energy $$U$$ is given by $$P(\mu) = e^{-U/kT}/Z(T)$$, where $$Z(T) = \sum_{\text{Microstates}}e^{-U_i/kT}$$ is called the partition function of the system. This set of possible microstates (along with this probability distribution) is called the canonical ensemble.

• ah okay, so it is only useful to talk about the multiplicity of macrostates if U, V, N are not fixed right? If all of those three are fixed there is only one macrostate. Thanks for the clarification. It makes sense to get multiple macrostates in the canonical and grandcanonical ensemble, but not in the microcanonical! Oct 4, 2020 at 12:37

Multiplicity tells how many microstates have a macrostate. E.g. how many possible multiparticle configurations have a glass of 293K water on surface pressure. Entropy is the logarithm of multiplicity. (multiplied by k)