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If my closed system consists of 2 coins, I can define the microstates of my system to be HH, HT, TH, TT and the macostates to be HH, HT, TT. The entropy of the state HT would be $$S = k\ln\Omega= k\ln 2,$$ since the multiplicity (number of microstates) in the state HT is 2. The total entropy of the system of two coins is calculated from the total multiplicity, so $\Omega_{total}$=1+2+1 = 4, and $S_{total} = kln4$.

So what would happen if I defined my states differently? The closed system is still 2 coins, but I can define the microstates based on the position/momentum of all the particles inside each coin, or some other random way to define states for coins. The total multiplicity is different, so the entropy is different.

When people estimate the total entropy of the universe, what states are they using?

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  • $\begingroup$ Are you asking about the thermodynamic entropy or some other entropy (and if so, which?). The thermodynamic entropy of two coins is a function of the coin material and the temperature. Their macroscale positions are irrelevant. $\endgroup$ Commented Dec 1, 2022 at 4:35

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From my own point, entropy is well defined only for a given macrostate. However, the real meaning of the concept "macrostate" depends on what properties of the system you are interested in, which involves how to define "microstate".

In the 1st paragraph, you only care about which side each coin shows, the corresponding entropy would be a measure for the probability distribution of the microstates. In the 2nd paragraph, the definition of microstate through position and momentum would lead to an entropy concerning classical thermodynamic properties.

I would say the two kinds of entropy are essentially different notions.

p.s. the correct dimension of the 1st entropy is 1, no Boltzman constant needed because you don't care about the thermodynamic properties of the two coins.

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  • $\begingroup$ Your answer could be improved with additional supporting information. Please edit to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center. $\endgroup$
    – Community Bot
    Commented Dec 1, 2022 at 8:44
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Entropy counts microstates and a "microstate" is is an elementary description of the system at a level of details from which all other properties can be calculated. In the classical view, knowing the position and momentum of all particles is sufficient to calculate everything about the system. In this sense we are not free to choose what we call microstate.

In the example you bring, a material coin, you are considering two different notions both of which come under the same name of entropy. One is Shannon's entropy, which has to do with the probability that the coin will give Heads or Tails. The other is the physical entropy of the material of which the coin is made. When we talk of the entropy of the universe, it is the entropy of the material coin that matters. But it turns out that at the microscopic level systems act like coins in some sense, for example, a spin can go up or down. So there is deeper connection between the two entropies but they are not always the same thing.

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