Statistical Mechanics in GR: Basics & Applications

[andresB]

Background: I'm just a guy who took some (very old fashioned) undergrad GR course some years ago. I'm only know about the basic stuff and nothing of the more advanced stuff.

Question: Is there an statistical mechanics for GR that resembles the one in classical mechanics?, I mean with Hamiltonians, phase space, and Liouville equation for probability density?

 

[bcrowell]

Some of the following may be useful:

http://physicsforums.com/showpost.php?p=3196058&postcount=6

http://math.ucr.edu/home/baez/entropy.html

http://physics.stackexchange.com/a/9309/4552

http://philsci-archive.pitt.edu/4744/1/gravent_archive.pdf

http://blogs.discovermagazine.com/c.../latest-declamations-about-the-arrow-of-time/

http://www.phy.olemiss.edu/~luca/Topics/grav/entropy.html

http://www.mth.uct.ac.za/%7Ehenk/ref_dir/chge.html

You're not going to get a Hamiltonian formulation, because the Hamiltonian formulation assigns a special role to time, whereas in relativity time is just another coordinate.

 

[andresB]

I think there is a fully covariant formulation for hamiltonian mechanics in special relativity where time is just another coordinate, so perhaps the problem in GR is in another place?

Anyways, thanks for the links, I Will read them, though I'm sad there is no Liouville equation in phase space for GR.

 

[bcrowell]

I think there is a fully covariant formulation for hamiltonian mechanics in special relativity where time is just another coordinate, so perhaps the problem in GR is in another place?

It may depend on exactly what you have in mind, but the Currie-Jordan-Sudarshan theorem is often interpreted as a no-go theorem for this.

 

[sweet springs]

There still is something unclear in statistical mechanics in SR, I have heard. For example transformation of temperature under Lorentz transformation is fully understood?

 

[andresB]

It may depend on exactly what you have in mind, but the Currie-Jordan-Sudarshan theorem is often interpreted as a no-go theorem for this.

Interesting, I will have to take a look at it. I was talking about the formulation in here

http://www.amazon.com/dp/0198766807/?tag=pfamazon01-20

 

[Demystifier]

One should distinguish equilibrium statistical mechanics from non-equilibrium statistical mechanics. The latter is much more complicated, so in the following I will restrict myself only to equilibrium statistical mechanics.

By definition, equilibrium is a state which does not change with time. Clearly, the notion of "not changing with time" cannot be invariant under general coordinate transformation, Lorentz transformations, or even non-relativistic Galilean transformations. Hence, in my opinion, it does not make much sense to look for a "covariant" formulation of equilibrium statistical physics. In equilibrium there is a preferred frame, the one in which the system is time-independent.

 

[Demystifier]

Question: Is there an statistical mechanics for GR that resembles the one in classical mechanics?, I mean with Hamiltonians, phase space, and Liouville equation for probability density?

If you don't insist that the formulation should be covariant (see my post above), then the answer is trivially - yes.

 

[Demystifier]

It may depend on exactly what you have in mind, but the Currie-Jordan-Sudarshan theorem is often interpreted as a no-go theorem for this.

I don't think that this theorem is relevant here. The theorem excludes relativistic-covariant theory of interacting point-like particles. However, if there are also fields, then a relativistic covariant interacting theory is possible. For example, classical electrodynamics is a relativistic-covariant theory of charged particles and EM fields. But you cannot describe interactions between charged particles in a covariant way without introducing EM fields.

 

[andresB]

If you don't insist that the formulation should be covariant (see my post above), then the answer is trivially - yes.

So, it exist but only works on a given reference system at a time?. It's something, do you have a link with more information?

EDIT: I see that you mentioned in your first post that you were talking about equilibrium statistical mechanics, I'm however more interested in a Liouville-like equation

 

[Demystifier]

So, it exist but only works on a given reference system at a time?. It's something, do you have a link with more information?

EDIT: I see that you mentioned in your first post that you were talking about equilibrium statistical mechanics, I'm however more interested in a Liouville-like equation

For equilibrium see the book
R.C. Tolman, Relativity, Thermodynamics and Cosmology (1934)
https://www.amazon.com/dp/0486653838/?tag=pfamazon01-20
The book is not only great, but also quite cheap.

I don't know much about the non-equilibrium case, but perhaps some of the following might be useful
http://pubman.mpdl.mpg.de/pubman/item/escidoc:153656:2/component/escidoc:1175603/Ehler_LNP28.pdf
http://relativity.livingreviews.org/Articles/lrr-2011-4/

 

[sweet springs]

In equilibrium there is a preferred frame, the one in which the system is time-independent.

In one frame let water, ice and vapor at triple point keep coexisting in a vessel. In other frame such coexistence are not observed?

 

[Demystifier]

In one frame let water, ice and vapor at triple point keep coexisting in a vessel. In other frame such coexistence are not observed?

Depends on what do you mean by "observed". In particular, you need to measure the temperature. But can you measure temperature in the ice if the thermometer moves with respect to ice? The problem, of course, is the fact that ice is solid, so thermometer cannot move through ice.

The situation is little less problematic with liquid and vapor phases, as thermometers can move through them. But the liquid phase will produce more friction, so the moving thermometer in the liquid will show larger temperature than that in vapor.

If you are a smart experimentalist, you will measure the temperature of radiation produced by those 3 phases. But then, philosophically, can you really say that the temperature of radiation is exactly the same thing as the temperature of its source? Moreover, what if the material does not radiate as a perfect black body? (In fact, no material does; there is always at least a small deviation, and for some materials, like those in LED bulbs, the deviation is large.) Furthermore, there are also some practical problems with this idea; if the thermometer moves very fast with constant velocity and the source is small, there will be not enough time to determine the temperature by a realistic thermometer.

 

[sweet springs]

Thank you. You teach me that gaseous, liquid and vapor phases of materials are not independent to frames. It is interesting if ice cube in my referegirator is vapor for an observer in another frame of reference.

 

[PeterDonis]

You teach me that gaseous, liquid and vapor phases of materials are not independent to frames.

That's not what he said. What he said was that the temperature of the different phases is frame-dependent--if you have solid, liquid, and vapor all in equilibrium at the same temperature in one frame, in a different frame you will have solid, liquid, and vapor at different temperatures. But what was solid, liquid, or vapor in one frame will still be solid, liquid, or vapor in the other frame; the phase doesn't change, the temperature does. (The pressure and density also change from one frame to another, so the physical laws that determine what phase the material is in are still valid.)

 

[sweet springs]

Thanks. Inhomogeneous temperature in the vessel according to occupation of ice, liquid water and vapor is quite interesting.
A SR simpler case temperature of river water puzzles me.