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I was wondering about the hypothetical - and apparently improbable - heat death of the Universe when I stumbled upon this seeming contradiction. A certain volume of space with a uniform distribution of particles has maximum entropy. However, the action of gravity would condense these particles, decreasing the entropy of the system, which would violate the second law of thermodynamics.

My question is simply: what am I missing here? What is the solution to this contradiction?

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    $\begingroup$ You aren't missing anything. The universe is simply very far from equilibrium, at this point. Locally entropy can decrease, and does. Hawing said that, when matter gets compressed by gravity, a lot of radiation escapes, which you have to add to your entropy calculation. $\endgroup$
    – CuriousOne
    Aug 26, 2014 at 1:13
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    $\begingroup$ See en.wikipedia.org/wiki/Black_hole_thermodynamics $\endgroup$
    – user4552
    Aug 26, 2014 at 1:40
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    $\begingroup$ possible duplicate of How does the evolution of a solar system not break the second law of thermodynamics? $\endgroup$
    – John Rennie
    Aug 26, 2014 at 5:16
  • $\begingroup$ @JohnRennie that question doesn't really seem to be a duplicate of this one. Perhaps you mean an answer to that question is similar to the answer to this question? Or perhaps planets coalescing is analogous to a big crunch? $\endgroup$
    – Brandon Enright
    Aug 26, 2014 at 5:56
  • $\begingroup$ @BrandonEnright: both deal with the question of why entropy increases in a gravitational collapse. In both cases it's because you need to include the entropy of the gravitational field. $\endgroup$
    – John Rennie
    Aug 26, 2014 at 6:09

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A certain volume of space with a uniform distribution of particles has maximum entropy.

That is correct for non-interacting particles, but wrong for particles with the gravitational interaction. When gravity condenses these particles, it increases the entropy of the system, not decreases it, at least when the Jeans instability condition is satisfied.

To calculate entropy properly, you should consider the phase space volume, and the phase space is built with taking into account all interactions in the system.

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  • $\begingroup$ But as Hawking radiation and black hole evaporation prove, gravity wells are, by far, not the highest entropic states, still. $\endgroup$
    – CuriousOne
    Aug 26, 2014 at 1:22
  • $\begingroup$ @CuriousOne This is a speculation by Penrose, not a rigorous proof. You can take my answer as restricted to classical mechanics and Newtonian gravitation. $\endgroup$
    – firtree
    Aug 26, 2014 at 1:23
  • $\begingroup$ I must have missed the latest science, where it was proven experimentally, that the universe cares about classical mechanics. $\endgroup$
    – CuriousOne
    Aug 26, 2014 at 1:23
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    $\begingroup$ @CuriousOne It was proven at least in 17-19th centuries, not the very latest science. $\endgroup$
    – firtree
    Aug 26, 2014 at 1:25
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    $\begingroup$ Firtree is correct that gravitational collapse of massive particles to a black hole leads to an increase in entropy. That doesn't contradict the statement that Hawking evaporation into (mostly) zero-mass particles also leads to an increase in entropy. Here is a paper on the latter issue: arxiv.org/abs/gr-qc/0609022 $\endgroup$
    – user4552
    Aug 26, 2014 at 1:39

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