# Gravitational Contraction and Entropy

## Main Question or Discussion Point

"Does the spontaneous contraction of a cloud of gas due to gravity violate the second law of thermodynamics? In other words, does entropy decrease as a result of the gravitational contraction?"

This question is posed in interesting article here:
http://www.mathpages.com/home/kmath573/kmath573.htm

If it was answered, it went right by me. In any case, it seems a conundrum.
Thanks

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Sick question!

I would be inclined to say that the entropy of the gas decreases, and that's ok, a system's entropy can decrease, just as long as the big picture's entropy increases. So I would think about if the entropy of the universe increased even though part of it decreased... that might be a good place to start...

I would be inclined to say no. Take the case of a piston acting on a completely insulated gas cylinder. You can compress the gas, which is an adiabatic process (entropy = const.). I would argue that gravity is a similar mechanism, so entropy would be constant. Another way of looking at it, even though the volume is decreasing, the total kinetic energy is increasing (gravitational potential -> kinetic energy), so the number of accessible states does not actually decrease. If you include the case of the gas cloud radiating thermal energy, then entropy would not be const, just as an non-insulated piston would.

"Does the spontaneous contraction of a cloud of gas due to gravity violate the second law of thermodynamics? In other words, does entropy decrease as a result of the gravitational contraction?"

...

If it was answered, it went right by me. In any case, it seems a conundrum.
Thanks
It's only a conundrum because you are assuming that: entropy increase = spread out.
This is only the case when there is no force involved. With a force like gravity, the state where particles are spread out is actually a low-entropy state; the more they are concentrated together the higher the entropy becomes. Under the influence of gravity, there are many more microstates when the system has collapsed.
A black hole, for instance, being created out of a large amount of matter condensed into a very small area, has a huge entropy. There was a paper a while back where some guys claimed to have calculated the approximate entropy of the universe and the number turned out to be almost entirely due to black holes.

I'd say there are many instances where the second law is useless. If you look at the derivation of entropy, you'll find many assumptions that are only satisfied for kind of homogeneous systems which are completely randomized. Maybe an expert on ergodic theory can phrase this more mathematically correct.

The second law is no more than saying for flipping a coin the ratio between heads and tails tends to 1. But this doesn't have to be true if the coin is unfair or if the coin properties change with time. This second law is really only such a trivial probabilistic statement.

PS: Actually in this particular case you can probably regain the second law, but you have to redefine some sort of dynamic phase space. Therefore just being close together does not correspond to a low entropy. But maybe being in a special place or having a special velocity means low entropy.

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You have to be very careful about your intuition in this case. The heat capacity of a gravitationally bound system is negative. The first consequence is that every derived equation that assumes it's positive is wrong and you need to go back to first principles. The second consequence is that it cannot collapse unless it can lose energy to an external system. Therefore, it is not a closed system, full stop. The second law does not apply.

"It's only a conundrum because you are assuming that: entropy increase = spread out."

Yeah, it is dispersed in micro-states but not necessarily spread out in space as with background radiation.
I am distressed that such a fundamental concept is so difficult for me to get a firm handle on.

Thanks all.

Here's an interesting article on just this subject: http://math.ucr.edu/home/baez/entropy.html" [Broken]

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You have to be very careful about your intuition in this case. The heat capacity of a gravitationally bound system is negative. The first consequence is that every derived equation that assumes it's positive is wrong and you need to go back to first principles. The second consequence is that it cannot collapse unless it can lose energy to an external system. Therefore, it is not a closed system, full stop. The second law does not apply.
Dear me, I am conumdrumed again. If the system is the universe as a whole (that is, no external sink), is it not true that there is a general trend toward increasing entropy?

Here's an interesting article on just this subject: http://math.ucr.edu/home/baez/entropy.html" [Broken]
Thanks, that is a great article.

He states that:

" * The energy of a gravitationally bound cloud of gas decreases as the cloud shrinks.
* The entropy of a gravitionally bound cloud of gas decreases as the cloud shrinks.
* A gravitationally bound cloud of gas has a negative specific heat. "

And playfully ends with:

"Well, I said I'd tell you the answer to this puzzle, but by now I've had a change of heart. I had so much fun figuring out the answer myself that I'd hate to deprive *you* of this pleasure. So go ahead, figure it out. But if you give up, click here for a hint. "