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Phase space theory suggests that the largest course-graining region, $p$, in a phase space, $P$, is the point in the phase space with the highest entropy. As such, it is in thermal equilibrium with the course-graining regions surrounding it.

Now, when an intensity vs frequency graph is obtained for the CMB (Cosmic Microwave Background), a curve almost identical to Planck's curve representing black-body radiation is seen. The temperature of the CMB is observed to be almost uniform across the universe, and the curve of black-body radiation represents a point of thermal equilibrium for a system. Since the CMB has the same curve as black-body radiation, this would suggest that at the birth of the cosmic microwave background, the universe was in a state of thermal equilibrium.

Referring back to the phase space spoken of in the first paragraph, since the universe was in a state of thermal equilibrium, it was at a point of highest entropy.

However, surely due to the Second Law of Thermodynamics, the entropy of the universe would need to decrease - so it wouldn't be possible for the entropy of the universe to be in such a high state so soon after its birth - so it should surely begin in a low state of entropy.

So what's the deal here? Did the universe begin in a really low state of entropy, or a really high one? Are all my explanations here logical or quite inaccurate?

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    $\begingroup$ why do you think the universe was in equilibrium state any time after the big bang? $\endgroup$ – Aksakal Mar 27 '14 at 0:32
  • $\begingroup$ Isn't the question of large vs small entropy in the early universe an open question? $\endgroup$ – Kyle Kanos Mar 27 '14 at 0:43
  • $\begingroup$ In a gravitational system (such as the universe) there is not state of thermal equilibrium, because gravitational systems have negative specific heat (so things don't evolve towards uniform temperature which corresponds to an unlikely state). Once the universe has cooled enough for gravity to play a role, it can explore many more likely states than uniform density, via gravitational collapse; entropy, which counts these states hence increases. $\endgroup$ – chris Apr 21 '14 at 16:03
  • $\begingroup$ More on low entropy at Big Bang. $\endgroup$ – Qmechanic Jul 26 '16 at 20:30
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You've overlooked gravitational entropy.

The entropy of a black hole horizon is given by:

$$ S = \frac{kA}{4 \ell_p^2} $$

where $A$ is the area of the horizon, $k$ is Boltzmann's constant and $\ell_p$ is Planck's constant. This entropy is absolutely huge, so if you take a uniformly distributed gas in thermal equilibrium and concentrate it into a black hole you vastly increase the entropy.

So you are quite correct that ignoring gravitational entropy the entropy of matter/energy was initially at a maximum, but include gravitational entropy and you add lots of scope for the entropy to increase.

There is much debate about this area, and in particular it's one of Roger Penrose's favourite topics. Googling for penrose universe entropy or something like that will find lots of relevant articles such as this one.

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