Title: Entropy of a system: microstates and macrostates
Tags: homework-and-exercises;thermodynamics
Question: Currently we're deriving the Gibbs expression for the entropy during an introductory thermal physics course. We're dealing with a system with $N$ different equally likely microstates that are "partially" distinguishable. Indistinguishable microstates are in the same macrostate (whatever that means).
Now, apparently, both the entropy of the micro- as well as the macrostates contribute to the total entropy of the system,
\begin{equation}
S=S_\mathrm{micro}+S_\mathrm{macro}.
\end{equation}
I can squeeze out the right answer all right, but I don't understand what I'm doing here.
So could anyone elaborate how I am supposed to interpret this? I always thought of entropy as a way of expressing in how many different states a system can arrange itself. I understand how microstates can contribute to this, but I don't understand how to rationalize that it must equal a certain theoretical expression.
My exercise then proceeds to mention that I should take the statistical average of the entropy over each macrostate $i$, $\langle S_i\rangle$ to get the contribution of all microstates to the total entropy. What? This really seems out of the blue. I'd very much like to gain an understanding of this.
And how should I differentiate between the contribution of the macrostates? I'm starting to believe I don't understand the difference anymore...
Comment: I think you are not clear on what a macrostate is.  A macrostate is a set (ensemble) of microstates, all of which are constrained to have the same value of some macroscopic property:  volume, temperature ...  In @Steeven excellent answer, the macroscopic property is: the total number of heads is $N$.  The only time you consider "the contribution of the macrostates" is in deciding which one has the highest entropy, not in the calculation of entropy itself.  In a particular ensemble, all states are in the same macrostate.

Comment: I see.  I missed that on my first reading.  I'm not sure from the context of your problem, but it might be this:  If, for example, your system is immersed in a bath of constant temperature,  then the energy of the system is not exactly known.  There is a distribution of energies among all of the macrostates.  To find the entropy one may calculate the entropy for each macrostate of energy $E_i$, and then average over all $i$.  In this case, the average would be weighted by the Boltzmann factor.

Comment: I don't understand that expression either.  It may be correct in some context.  I don't suppose there's anything else in the wording of the problem that might help?

Comment: I'm stumped.  If someone else here can't figure it out, you may have to ask your instructor.

Comment: @garyp How do I fit the statistical average of the entropy over each macrostate into his explanation?

Comment: @garyp "There is a distribution of energies among all the macrostates." - The system is *in* a certain macrostate. Do you mean here that the system can have a certain amount of energy depending on the macrostate?

Comment: @garyp Rereading what you said: "You don't consider the contribution of macrostates" in the calculation of entropy itself". My exercise gave that expression of $S_\mathrm{total}=S_\mathrm{micro}+S_\mathrm{macro}$ and I have to admit, most of my confusion stems from this expression. I did not come up with this relation myself.

Comment: @garyp "The total entropy is a sum of the contribution of the macrostates and the microstates of the system. $S_\mathrm{total} = S_\mathrm{macro} + S_\mathrm{micro}$, The entropy of macrostates $S_\mathrm{macro}$ is a measurable quantity because the macrostates are distinguishable. The entropies $S_\mathrm{total}$ and $S_\mathrm{micro}$ cannot easily be measured (we have no direct
access to the microstates), but we can find theoretical expressions in terms of
the number of microstates."

Comment: and from this I have to find what expression to use for the $\langle S_i\rangle$ ?! And the total entropy of the system (which apparently is $S_\mathrm{total}/k=\ln N$).


Here is another answer: Flip four coins. Coins have heads H and tails T. What are the possible results?
1: HHHH
2: HHHT, HHTH, HTHH, THHH
3: HHTT, HTHT, THHT, HTTH, THTH, TTHH
4: HTTT, THTT, TTHT, TTTH
5: TTTT
There are five different outcomes. That is, five different macro-states.
Macro-state 3 happens for most different micro-states. It is the most probable outcome.
If the micro-states were not all equally probable as here, then their probabilities have to be included when finding the most probable macro-state. It would then not be a simple count as here but a weighed count.
The macro-state with the the highest probability is said to have the highest entropy and that is no. 3. In that case we know the least about how the micro-state outcome looks. There we have more chaos than in the other cases.
Comment for this answer: @Steeven The coin flip example is very illustrative. Could we think of the microstates as the outcumes and macrostates as the events?
Comment for this answer: I am not sure about the question of entropy of macro-states. Macro-states are defined as you want to define it. In my example Macro-states are the number of head and tails. Another situation could be all cases of less than half heads. That would be one macro-state while the other would be half or more heads. Macro-states are just a way to group the micro-states. And usually in physics Macro-states are defined as the actual property we can measure or see (temperature e.g.)
Comment for this answer: @Vass I don't understand your question, sorry. What exactly do you wish to achieve?
Comment for this answer: @Vass Are you asking about the symbols used? I am not familiar with neither the $\omega$ nor the $\Omega$ here. The indexing of microstates into macrostates does not follow any particular general notation, as far as I am aware.
Comment for this answer: @AntoniosSarikas I follow you on those word choices. Your point being that an event occours regardless of the order of the parts of the outcome. You might have to fully define "outcome" and "event" in any case.
Comment for this answer: @Steeven, is there a convention to formally define the set for the `macrostate` property the `microstates` follow?
Comment for this answer: @Steeven, from your example *1:* corresponds to a macrostate of which *HHHH* is a member, and *2:* is another macrostate of which microstates  *HHHT, HHTH, HTHH, THHH* are members. These macrostates *1:* and *2:* are each an $\omega$, correct? But if we want to describe a set of these macrostates for which the space of microstates map to, do we say there is an index for them like $\omega_m$ or $\Omega$? How do we define $HHHH$ being part of a specific $\omega_m$ or $\Omega$? Or maybe $\mathbf{\Omega}$ is used for the multiple macrostates?
Comment for this answer: I'm still confused about two things. It is given that all probabilities are equally probable (in my case as well). However, for calculation of the entropy of the microstates I suddenly *do* have to use probabilities. Big contradiction there.
Comment for this answer: And I'm still not sure how I distinguish the entropy contribution of macrostates versus microstates. What's the fundamental difference?

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