# How does the number of microstates affect the entropy of an isolated system?

• Buzz Bloom
In summary, according to statistical mechanics, a system can have a finite but indefinite number of microstates, each with a specific energy. The number of microstates is proportional to the number of gas particles in the system, and the number of microstates is also proportional to the number of positions that the gas particles can occupy.
kimbyd said:

When one says comoving volume of radiation, is it the centroid of the volume that is comoving? I'm not confident I understand what that is meaning - can photons themselves be co-moving?

Grinkle said:
When one says comoving volume of radiation, is it the centroid of the volume that is comoving? I'm not confident I understand what that is meaning - can photons themselves be co-moving?
A comoving volume is a volume that increases in size along with the expansion. Think a hypothetical, finite-size cube whose sides scale along with the expansion factor.

In the case of the photon gas, the number of photons within a comoving volume is a constant. Photons exit the volume, but just as many photons enter it. The number density of photons decreases as the universe expands, and they also redshift as it expands.

Grinkle
Buzz Bloom said:
Hi @kimbyd:

I am confused by your comment since it restricts the issue to photons. The discussion I referenced was about a comoving system containing H2 gas in a state of thermodynamic equilibrium. I mentioned that photons were also included, since there was a uniform temperature, but I ignored calculating the entropy of the photons because the use of the
I mentioned photons because your argument assumes radiation domination. Thus photons make up nearly all of the entropy of the system.

I'm pretty sure that as long as clustering isn't happening, though, any non-relativistic gases will also expand in an adiabatic fashion. Without clustering, the full behavior of the system at any point in time is determined solely by the temperature and density of the gases. Because the temperature and density are purely functions of the expansion, the expansion is trivially a reversible process. Thus it must be adiabatic.

With clustering, things get messy and nobody knows how to calculate the entropy. So in the case of the above, my bet is that there genuinely is something wrong with your calculations.

Edit:
One possible resolution, without looking into it in detail, is that you have to include both the H2 gas and the photon gas in the calculation together, with changes in temperature of the H2 gas causing energy to be dumped into or extracted from the photon gas.

kimbyd said:
With clustering, things get messy and nobody knows how to calculate the entropy.

Is this because a clumped system is not at equilibrium, or for some other (or maybe additional) reasons?

kimbyd said:
In the case of the photon gas, the number of photons within a comoving volume is a constant.
Hi @kimbyd:

It is clear from the equation for N (the average number of photons) that the number does not change. It is also clear from the equation for S (entropy) that the entropy of the photons does not change as the hypothetical universe expands.

Am I correct that in a equilibrium mixture of H2 gas and photons, the entropy of the mixture is the sum of he entropy of the gas and the entropy of the photons? If that is correct, then the entropy of the mixture increases (using these photon gas formulas photons together with the Sackur-Tetrode equation for the gas) as the hypothetical universe expands, and decreases as it contracts as discussed in my post #33.

So it becomes necessary to have a specific reason why this does not violate the second law. I tend to like the reason based on your statement which I quoted about gravity at the end of post #33. What do think about this?

Regards,
Buzz

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