As stars contract, why does total entropy rise?

[Ken G]

TL;DR Summary

In the PF FAQ by @bcrowell, the remarkable point is made that the early universe had to be in a low entropy state or structure could not form spontaneously, but how does entropy rise when structure forms?

The FAQ by @bcrowell cites an explanation by physics netizen John Baez as to how entropy rises when a star loses heat and contracts. However, the linked explanation falls short of describing the key role that gravity must be playing. The FAQ by @bcrowell discusses why a low-entropy state of universal gravity must be playing a key role, since the matter in the universe is nearly thermalized (so is by itself in a state of high entropy). But the cited accounting of entropy rise in a contracting star by Baez includes gravity only as a source of energy, and talks about the entropy associated with the lost heat to the environment, but gives no direct entropic significance to the gravity itself.

Something must be missing here, because it is well known (via the Jeans instability) that isothermal gas can be made to spontaneously contract if the mass is high enough. In that situation, there is no entropy increase associated with heat transport from the ball of gas to its (isothermal) surroundings, yet there is the same entropy drop seen in the gas itself that Baez described (in essence, the loss in volume accessible to the particles reduces their entropy more than the increase in velocity space that their higher energy gives them access to). So the only thing left to account for the spontaneous entropy rise that must appear during an isothermal Jeans instability has to have something directly to do with entropy associated with gravity. What is this missing element, that surely must also be present in structure formation in the early universe?

 

[Demystifier]

For a conceptual discussion of entropy in gravitating systems see e.g. https://arxiv.org/abs/0907.0659.

 

[Ken G]

Believe it or not I have read that, but I don't think it really addresses the issue. Like Baez, Wallace seems to be saying that if you track the radiation lost by the contracting star (or the kinetic energy given to the outer regions of a globular cluster as its inner core of stars contracts), then you can see that the total entropy rises. But neither Baez nor Wallace actually show this is true. Indeed, I believe I offered a counterexample already: the Jeans instability. Normally that is viewed as a purely isothermal example of gravitational collapse, so one imagines that there must be an entropy rise there-- without any heat transport from hot to cold, or kinetic energy from interior to exterior. So although Wallace criticizes Penrose for claiming that one needs some explicit dependence of entropy on gravity itself, it seems to me this has to be the case. Wallace correctly points out that the entropy generated by a black hole is both spectacularly large, and quite irrelevant for the contraction of suns and solar systems, but what is not clear is if gravity that is not associated with black holes carries an analogous contribution to the entropy that is much smaller but still quite important.

 

[Ken G]

I think I see it. When gravity releases energy via contraction, it will be passed to the gas in the form of heat, and that generates entropy immediately. This is not the same as the transport of the radiation, or indeed anything that has to do with heat loss from the system, it is inherent to the contraction and would be there even if there is no heat transport (though adiabatic contraction rarely occurs, since the total entropy usually drops). Nor does it require radiation, it's just the heat that the gas acquires when it contracts. So in the isothermal situation, the gas must pass this heat to its surroundings, but that all happens at the same T so there is not any additional entropic considerations in the transport-- but there is still the original entropic effect of the initial gravitational energy release. In short, when a gas at temperature T releases gravitational energy dU by contraction, it experiences an increase in entropy of dU/T, in addition to whatever other entropy changes occur due to dV and dT (the latter of which can be found from the entropy of an ideal gas). This could be regarded as the entropy from the gravity itself, since it does not require transport through a T gradient.

This must relate to the interesting point Wallace makes that the virial theorem, which is normally thought of as a force-balance constraint, can also be thought of as a maximum of the entropy. So as net heat is lost from a system, it must contract to find a new entropy maximum. In the Jeans instability, the heat loss occurs due to the contraction, rather than the other way around. This would seem to give a fundamentally different role for gravity in the entropy calculation.

 

[Ken G]

On another point, for those who don't want to read the Wallace article but are interesting in gravitational involvement in entropy, a key conclusion Wallace reaches is that the entropy generated by nuclear fusion in the Sun is crucial to why "second law processes" (the key processes on Earth that exhibit entropy increase, i.e., the things that make stuff happen) do occur here. This is a fully justified view, made clear from the fact that the entropic maximum for the conditions in the core of our Sun is that it should be essentially pure helium, and it is headed there, but will take a long time to get there.

However, this same argument also supports the very viewpoint he disputes, which is Penrose's claim that gravitational involvement in entropy is what was originally needed to get any of that to happen, hence is the crucial agent for the whole business. We can say that a uniform plasma is also not a state of maximum entropy in the presence of gravity, just like 25% helium is not the state of maximum entropy in the solar core. And so even though the entropy increase from fusion is central to how life has evolved on Earth, the entropy increase associated with gravitational energy release (the U/T I mentioned above) was the original "prime mover" that allowed the Sun to contract enough to initiate fusion in the first place. Wallace is aware of this, but doesn't seem to see how it undercuts his claim that the ability of gravity to generate entropy was not the essential (or how I might put it, original) cause of why we are all here.

So I conclude that Wallace makes some very interesting points that are well worth consideration, but he ultimately does not justify his attack on the idea that without gravity, the universe would be a place where "second law processes" are very rare indeed-- which is the crux of Penrose's argument. Thus I would recommend merging the points made by both Penrose and Wallace, rather than placing them in opposition.