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Let's say that the universe in time zero consists just of a cloud of matter. Now as the time progresses, and the matter interacts gravitationally, it will gradually collapse into a sphere. Will the entropy of the universe really increase?

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Let's say that the universe in time zero consists just of a cloud of matter. Now as the time progresses, and the matter interacts gravitationally, it will gradually collapse into a sphere. Will the entropy of the universe really increase?

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Mapes

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The total entropy in the universe tends to increase for every spontaneous process. http://math.ucr.edu/home/baez/entropy.html" [Broken] might interest you.

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atyy

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I always thought site that was rather diabolical!http://math.ucr.edu/home/baez/entropy.html" [Broken] might interest you.

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Mapes

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Also see atyy's references in https://www.physicsforums.com/showthread.php?t=224800". I haven't looked at them in detail--my research lies far away from the topic--but I'm glad to know they're there for reference.

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That's not exactly correct; the change in entropy is positive for all irreversible processes. The change in free energy is negative for spontaneous processes. Also, it is not clear how to define an isolated system in the OP- is the entire universe an isolated system? What are the boundary conditions? Has any work been performed during the process of collapse?The total entropy in the universe tends to increase for every spontaneous process. http://math.ucr.edu/home/baez/entropy.html" [Broken] might interest you.

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The system is thus kept at some constant energy. Then the total entropy (defined in a suitable way, like minus the sum over p_i Log(P_i) ), can only increase.

Free energy does not apply here, because the system is not kept at constant temperature. Even if the system is approximately at consant temperature, one has to be careful with applying the usual results of thermodynamics, because of the hidden assumptions like there being only short range interactions. That's clearly not the case if gravity is relevant.

So, due to the long range gravitational interaction, the entropy and internal energy won't be the usual extensive functions and thus the standard results like E = T S - P V + mu N are not going to be valid anymore.

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But this is exactly what Baez calculates in his article - and he shows that even if we take into acount the entropy of the velocity distribution, it will still give us smaller entropy than at the beginning. So where is the remaining entropy?

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Read the whole article. Distinguish between the initial phase of collapse (that occurs even if the system is isolated) and its continuation (in which the total energy of the system is allowed to decrease). Note Baez is focusing on the latter (by beginning with the virial theorem).But [Baez] shows that even if we take into acount the entropy of the velocity distribution, it will still give us smaller entropy than at the beginning. So where is the remaining entropy?

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Mapes

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The total entropy in the universe tends to increase for every spontaneous process.

Minimization of free energy for isothermal systems is equivalent to maximization of entropy for constant-energy systems. (See Callen'sThat's not exactly correct; the change in entropy is positive for all irreversible processes. The change in free energy is negative for spontaneous processes.

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If we consider the system to be a closed system then the boundary conditions are adiabatic boundary counditions and no work is performed/extracted.

The system is thus kept at some constant energy. Then the total entropy (defined in a suitable way, like minus the sum over p_i Log(P_i) ), can only increase.

<snip>

I agree that for a static universe consisting only of a uniform subvolume of dust the results are reasonably well-understood. But reconciling thermodynamics with GR is not as straightforward. Assigning a temperature can be ambiguous, for example: Unruh radiation (http://en.wikipedia.org/wiki/Unruh_radiation). And I don't think we can simply demand that the universe has an adiabatic boundary, see for example black hole thermodynamics (http://en.wikipedia.org/wiki/Black_hole_thermodynamics).Minimization of free energy for isothermal systems is equivalent to maximization of entropy for constant-energy systems. (See Callen'sThermodynamicson the extremum principle, for example.) As Count Iblis pointed out, surely it makes more sense to model the universe to have constant energy as opposed to being isothermal.

One possible reason for the difficult nature of this problem is that the initial conditions are (perhaps) ill-posed- we did not completely specify the initial state. How can a closed universe filled uniformly with dust come to be? What is the initial total energy, momentum, angular momentum? Can they be unambiguously assigned in a way consistent with GR?

I don't know the answers to these questions- it's way outside my expertise. Tolman's book "Relativity, Thermodynamics and Cosmology" has a chapter on this subject- this could be a good excuse for me to read it.

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Mapes

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