Entropy: Gravity

Gravity presents some new wrinkles for entropy. Even the classical version has a surprise. John Baez has a very nice discussion of the fundamental details.

If you weren't careful, you might think gravity could violate the 2nd law of thermodynamics. Start with a bunch of gas in outer space. Suppose it's homogeneously distributed. If it's big enough, it will start clumping up thanks to its gravitational self-attraction. So starting from complete disorder, it looks like we're getting some order! Doesn't this mean that the entropy of the gas is dropping?

Well, it's a bit trickier than you might think. First of all, you have to remember that a gas cloud heats up as it collapses gravitationally! The clumping means you know more and more about the positions of the atoms in the cloud. But the heating up means you know less and less about their velocities. So there are two competing effects. It's not obvious which one wins!

Baez proceeds to do the calculations for an only slightly idealized case. I recommend these. They are a nice exercise. Mostly algebra with a bit of mechanics and statistical mechanics, but nothing fancy.

The bottom line is this: as the cloud shrinks under gravitational contraction, the cloud heats up - the molecules move faster in their tighter orbits. John's math shows that the net effect is a decrease in entropy. Meanwhile, kinetic energy has increased, but potential energy has decreased even more. Consequently the cloud has heated up even though it has lost energy. This is opposite to the behavior of familiar thermal systems!

In other words, the specific heat of the cloud is negative! Adding heat cools the cloud and subtracting heat makes it hotter! This fact turns out to be central to star formation as well as to understanding gravity and entropy. In order for a cloud of gas to contract, it needs to be able to give off heat. Sun sized stars typically form in dust clouds, with the dust grains serving as minature radiators to cool the collapsing cloud.

Prof Baez leaves us with the question he started with: if the entropy decreased, what does that say about the second law?

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