Does Gravity Affect Entropy Levels in Self-Gravitating Systems?

[yuiop]

Hi,

John Baez has written an interesting article on a subject I was looking for the answer to ( http://math.ucr.edu/home/baez/entropy.html ), but frustratingly he does not give the final answer!

I have tried to figure it out and sadly failed. I even wrote to Baez but got no reply

Does anyone here know how to work out the answer?

Thanks

 

[yuiop]

Sorry to bump this, but I would really love to know the answer :)

 

[Mapes]

I believe Baez is hinting that some material is unavoidably flung off, carrying energy and entropy. As a result,

$$\frac{\partial E}{\partial V}\propto\frac{N^2}{V^{4/3}}\neq 0$$

for the bounded collection of gas.

 

[cesiumfrog]

I believe Baez is hinting that some material is unavoidably flung off

By "flung off", surely you mean "radiated out" (so as not to contradict the initial construction)?

Baez is pointing at the virial theorem, which says that on average 2.KE + PE = 0 (not PE + KE = const). In other words, gravitational collapse can only occur if the system somehow loses energy to its environment.

It is this heating of the surrounding space (in addition to the raising internal temperature) that is hinted to increase universal entropy enough to justify the ordered gravitational clumping of matter.

To me it seems peculiar that so little fuss is made of this conclusion; many authors do write of gravity as negative entropy. (I also found non-obvious the equation of KE and temperature, lacking consideration of ordered average motion.)

 

[Mapes]

Good points. I see now that my earlier answer violates an initial assumption.

Now I'm curious: A photon can leave the bounded area, but one of the original particles can't. Would it violate the assumptions of the virial theorem if a neutron flew out (generated in some atom-atom collision) as long as the number of original gas atoms were unchanged?

 

[Deadshade]

No it cannot - at least not globally .
It however does so LOCALLY in a collapsing gaz cloud which is , of course , not a closed system and therefore it is not farther bothering that its entropy decreases .
As the temperature increases , the contracting cloud radiates more and more (that is btw where the "missing" energy goes) .
And as radiation has the highest symmetry it has a high entropy .
The decrease of entropy of the contracting gas cloud is compensated by the huge increase of entropy that the radiation carries away and the second law is safe .

So actually it is the second law that is responsible for the fact that gas clouds have to light up and become stars .

 

[yuiop]

There appears to be a correspondance between thermal entropy and gravitational entropy with one converting into the other interchangeably, just like kinetic energy can be converted into potential energy and and then back again, indefinitely in an ideal oscillating system. Is it possible that there is an interplay between gravitational entropy and thermal entropy (and the various other forms of entropy such as entropy of mixing and information entropy etc) that conserves total entropy? How well do we understand all the various forms of entropy and how they interelate?

Take a cloud of gas molecules in open space. They expand and we say that is example of increasing entropy because the molecules are dispersing. Then take a cloud of gas molecules on a larger scale and they gravitationally collapse and we also say that is example of increasing entropy. So whether molecules disperse or clump together we say it is increasing entropy. Of coure the difference is due to scale and gravity but that is the important point. Entropy is not just about thermal entropy. Now let's take the extreme example of a large cloud of gas molecules clumping to form a black hole. Hawking provides us with an equation for the entropy of a black hole and shows it has greater entropy than the dispersed cloud of molecules. Fine, but hang around a while and the black hole evaporates to become a ...wait for it... a dispersed cloud of molecules. Now that seems to contradict my previous statement. Whether a cloud of molecules clumps to form a black hole or black hole evaporates to become a cloud of molecules, we say both are examples of increasing entropy. Of course, black holes only evaporate when the universe has expanded and cooled suffiently for the the black hole to lose more due to radiation than it gains from absorbing the background CMB. The important thing to notice is the interplay between the size of the universe, gravity and entropy. Imagine a very late stage universe where all the black holes have evaporated to form a very dispersed cloud of radiation and molecules and then starts collapsing. (If I recall correctly the gravitation of a radiation dominated universe is greater than that of a matter dominated universe). So if the universe is one dispersed cloud of atoms and radiation and it collapses to form one big black hole then the collapse of the universe is an example of increasing entropy because that is what Hawking's equation says.

 

[atyy]

Reconciliation of Statistical Mechanics and Astro-Physical Statistics. The errors of conventional canonical thermostatistics
D.H.E.Gross
http://arxiv.org/abs/astro-ph/0511716

Microcanonical mean-field thermodynamics of self-gravitating and rotating systems
E.V. Votyakov, H.I. Hidmi, A. De Martino, D.H.E. Gross
http://arxiv.org/abs/cond-mat/0202140

Thermodynamics of rotating self-gravitating systems
E.V. Votyakov, A. De Martino, D.H.E. Gross
http://arxiv.org/abs/cond-mat/0207153

Thermodynamics of self-gravitating systems
P.H. Chavanis, C. Rosier, C. Sire
http://arxiv.org/abs/cond-mat/0107345

 

[atyy]

Thermodynamics of self-gravitating systems
P.H. Chavanis, C. Rosier, C. Sire
http://arxiv.org/abs/cond-mat/0107345

Chavanis et al: a maximum entropy state does not always exist and the system may undergo a "gravothermal catastrophe": it can achieve ever increasing values of entropy by developing a dense and hot "core" surrounded by a low density "halo".

So maybe your question should have been "Can gravity not increase entropy?"?