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Statistical mechanics is used to describe systems with large number of particles ~$10^{23}$.

The observable universe contains between $10^{22}$ to $10^{24}$ stars. Can we treat those many stars as a statistical mechanical system (for which one can define an entropy, temperature..etc)?

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Presumably related to the comments on physics.stackexchange.com/questions/32806/… . –  dmckee Jul 25 '12 at 19:12
I hope one will address the fact that $1/r$ in some sense is so different from $1/r^6$, I'm not capable of doing it the proper way. –  Yrogirg Jul 25 '12 at 19:31
Fourier transform and introduce a cutt-off. The problem is similar in plasma physics for charged particles. –  Shaktyai Jul 26 '12 at 13:42
@Shaktyai I meant that the system won't be extensive, unlike the gas and I guess the plasma. In plasma you have charges of both sign screening each other and you can do cut-off. If you have $N$ molecules and you add $N$ more nothing changes much, only doubles. But with gravity it's not just doubling. You can divide gas into two parts and they will interact only by their boundary, but gravitating gas would interact in bulk. I don't mean the statistical or hydrodynamic description is impossible (indeed I think it exists and used), I want to say it should differ in certain principal ways. –  Yrogirg Jul 26 '12 at 16:46
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up vote 8 down vote accepted

Yes. In general relativity and cosmology, the collection of galaxies is often even treated as an ideal fluid, in the thermodynamic sense, with temperature and pressure.

The standard textbook on general relativity, the book Gravitation by Misner, Thorne, and Wheeler, discusses this model in Section 27.2, though only on the thermodynamic level.

For the statistical mechanics of gravitation, see, e.g.,
W. Thirring, Systems with negative specific heat, Z. Physik 235 (1970), 339-352.
This is for inside a star, but a similar analysis works on the level of galaxies. See, e.g.,
Ahmad et al., Statistical Mechanics of the Cosmological Many-Body Problem, The Astrophysical Journal 571 (2002), 576-584.
A free online copy is at http://iopscience.iop.org/0004-637X/571/2/576

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I am perplexed on how you can define such macroscopic thermodynamics quantities for a "gas" of stars such as pressure and temperature. The stars are not flying around like the atoms of an ideal gas, or not contained in a box! Also I do not imagine how to define an equipartition theorem in this case for stars. Could you provide a reference please –  Revo Jul 26 '12 at 0:58
@Revo: acutally, it is galaxies, not stars that are moving randomly; I corrected the answer accordingly. See, e.g., the bibler on gravitation, Misner/Thorne/Wheeler, Section 27.2, for this approximation. –  Arnold Neumaier Jul 26 '12 at 12:50
On large time scale, even the solar system is chaotic, so yes stars can fly around like atoms in a gas. You just have to rescale the time –  Shaktyai Jul 26 '12 at 13:41
The equation of state is different that that for a simple gas model, but yes you can treat them statistically. The "box" is their mutual gravitational binding, and some can be ejected so it is the Grand Canonical Ensemble, but what is the problem? –  dmckee Jul 26 '12 at 14:34
I guess the problem is that: the collision operator diverges due to long range small angle numerous collisions. –  Shaktyai Jul 26 '12 at 16:17
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If you're prepared to regard the application of the virial theorem to large numbers of particles as statistical mechanics, then the answer to your question is definitely yes, because that's how Zwicky worked out that dark matter must be present in the Coma cluster of galaxies. See this article for details, or a quick Google on "Zwicky virial theorem" will find many such articles.

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