Is the Hubble Constant Linked to the Second Law of Thermodynamics?

[martinbn]

Depends on the measure you choose for your phase space. In classical mechanics (and general relativity is an example of classical mechanics) it is canonical to take a measure which conserves phase-space volume by the Liouville theorem
http://en.wikipedia.org/wiki/Liouville's_theorem_(Hamiltonian) .
With such a measure, the volume of the phase space does not depend on the spacetime metric, so the expansion of the universe does not expand the phase space.

How does this work in field theories, where you have infinitely many degrees of freedom!

 

[Jimster41]

Depends on the measure you choose for your phase space. In classical mechanics (and general relativity is an example of classical mechanics) it is canonical to take a measure which conserves phase-space volume by the Liouville theorem
http://en.wikipedia.org/wiki/Liouville's_theorem_(Hamiltonian) .
With such a measure, the volume of the phase space does not depend on the spacetime metric, so the expansion of the universe does not expand the phase space.

Thanks Demystifyer. That is helpful. I have seen the Hamiltonian. I had tagged it with "surfaces of conservation". I had not seen the Louiville's Theorem in Hamiltonian Mechanics. This seems Really relevant. I can get that to solve I'll posed problems like what's the shape of the universe, we have only a limited set of tools. One of which is setting initial conditions and constraints of assumption.

my question is still in my mind, in spirit, but it now has, important red x's and a lot more hair I need to pull. I'd like understand how field theories have infinite degrees of freedom, to Martinbn's question, but I don't so I'd be interested to know more what he means.

Friedman seems descriptive, but for me at least doesn't yet seem like an explanation of how things work. And I keep wondering what happens if you start without matter and pressure as postulate, start without Hamiltonian assumption for Louiville, or even start with an explicit growth or expansion requirement for phase space, and picture s.t. geometry like a spin foamy gas. This is a naive question of course. I can imagine it's annoying. Sorry about that. I appreciate the patience. Luckily I have Susskind and Penrose and that dude Unger-Smolin, and a long boring plane ride ahead of me today. What could be better.

 

[Demystifier]

How does this work in field theories, where you have infinitely many degrees of freedom!

That's a good question. Classical statistical mechanics (including the definition of entropy and the second law) is based on the measure in phase space which satisfies the Liouville's theorem. A naive generalization to a classical field theory such as electromagnetism leads to the celebrated UV catastrophe, which is related to the infinite number of UV degrees of freedom. However, the problem resolves when one replaces classical electrodynamics with quantum electrodynamics, because the latter involves quantization of energy with given frequency. But to avoid the UV catastrophe, it is not necessary to apply the whole quantum-field-theory machinery. One can use a semi-classical approach first proposed by Planck, just by assuming that energy of given frequency $\omega$ must be an integer times $\hbar\omega$. With such a modified classical theory, one can make sensitive physical predictions despite the infinite number of degrees of freedom.

Alternatively, a more intuitive approach is to work with field theory on a space lattice. The number of lattice points is finite and fixed, which corresponds to a classical system with a finite and fixed number of degrees of freedom. In particular, one can study a discrete version of classical general relativity on such a lattice. With such an approach, the expansion of the Universe does not increase the number of points on the space lattice. What increases is the metric (tensor), not the number of points. That's how expansion of the Universe is compatible with the constant volume of the phase space.

 

[Demystifier]

I had not seen the Louiville's Theorem in Hamiltonian Mechanics. This seems Really relevant.

Let me just add that the Louiville's theorem is a part of the Susskind's "theoretical minimum":
https://www.amazon.com/dp/0465075681/?tag=pfamazon01-20

 

[wabbit]

What increases is the metric (tensor), not the number of points. That's how expansion of the Universe is compatible with the constant volume of the phase space.

This is a striking formulation.

Is the following version valid ?
- Change units of distance to comoving distance, keeping units of time unchanged. The phase space is now unchanged by expansion.

 

[Demystifier]

Another related point:
It is shown in the MTW book [Misner, Thorne, Wheeler: Gravitation], Sec. 22.6, that the Liouville's theorem is valid for classical particles in an arbitrary curved spacetime.

 

[Demystifier]

This is a striking formulation.

Is the following version valid ?
- Change units of distance to comoving distance, keeping units of time unchanged. The phase space is now unchanged by expansion.

That should be correct.

 

[wabbit]

That should be correct.

Many thanks. So there simply is no "phase space paradox", just an artefact/illusion in one approach. This doesn't exhaust the topic of entropy in the presence of gravity of course, but at least this settles (I think) one aspect of our discussion so far.

 

[Jimster41]

Let me just add that the Louiville's theorem is a part of the Susskind's "theoretical minimum":
https://www.amazon.com/dp/0465075681/?tag=pfamazon01-20

Starting ch 5 on Energy now. Spent the first leg doing a pain soak in "Loop Quantum Gravity" by Gambini and Pullin. This is a really great resource.

 

[wabbit]

I wonder if the "Shape dynamics" approach of Koslowki & al. that you mentioned elsewhere might provide a basis for an interesting alternative way of looking at this ?
http://arxiv.org/abs/1302.6264
http://arxiv.org/abs/1501.03007

Sorry, bumping my own post here, but doesn't this dimensionless approach yield an easier description of phase space in GR, and possibly helps in thermodynamics ? I must admit these papers are a little difficult to parse for me.

Otherwise, Rovelli's approach and his description of equilibrium and thermal time sound very interesting ; these seem to lead to a formulation of thermodynamics that is better suited to gravity than the usual one (e.g, temperature is not uniform in equilibrium in GR, so dS=dQ/T for instance isn't that useful, but more general formulas may replace the usual one). I need to re-read his papers more carefully...

 

[marcus]

Depends on the measure you choose for your phase space. In classical mechanics (and general relativity is an example of classical mechanics) it is canonical to take a measure which conserves phase-space volume by the Liouville theorem
http://en.wikipedia.org/wiki/Liouville's_theorem_(Hamiltonian) .
With such a measure, the volume of the phase space does not depend on the spacetime metric, so the expansion of the universe does not expand the phase space.

Thanks for weighing in on Liouville measure! Sorely needed. Hoping you would.
An interesting aspect of the "compact phase space from positive curvature constant" paper:
http://arxiv.org/pdf/1502.00278v2.pdf
==quote page 2 section III "Compact Phase Spaces==
A compact phase space is the classical limit of a quantum system with a finite dimensional Hilbert space. This can be seen in many ways; the simplest is to notice that a compact phase space has a finite (Liouville) volume, and therefore can accommodate a finite number of Planck size cells, and therefore a finite number of orthogonal quantum states.
==endquote==
Positive Lambda let's them change over from a phase space su(2) x SU(2) built with a flat infinite copy of the Lie algebra su(2)---from that over to a phase space SU(2) x SU(2) where both components are curved and compact. So finite Liouville volume.

This is the phase space of the geometry itself not the particles swimming in it. It seems to offer the hope that particle field theory built on such a geometry could itself be more manageable, more finite. It is a preliminary result in 2+1 dimensions so it remains to be seen if it can be extended.

Wabbit, it's good to keep shape dynamics in sight too! I'm glad to see you brought your post about it forward to keep it current. Too much going on to deal with every interesting issue. Demystifier may want to take that topic up---his choice. I'm mostly in wait-and-see mode---this is a nest of inter-related fertile topics

 

[wabbit]

Let me just add that the Louiville's theorem is a part of the Susskind's "theoretical minimum":
https://www.amazon.com/dp/0465075681/?tag=pfamazon01-20

Thanks for the reference, reading list increased, disregarding threat of gravitational collapse.

 

[Jimster41]

Such a day reading I can hardly see.

I am interested to see what the shape Dynamics Thing is. Once I rest my eyes.

Kind of hoping to loop back to a confusion I have... How likely is it for a system in a constant volume phase space to have increasing entropy. Maybe this is just the antimony of cosmogenesis I'm bouncing off of, but how is it we happen to be on a journey from one apparently staggeringly low entropy (improbable) location toward the inevitable high entropy floor... If we've been in the same phase space the whole time. Why isn't it just as natural, if not more, when trying to explain a system with such a de facto arrow of time, to imagine an interaction of phase spaces as the (first) cause of system dynamics instead of a system just doing something dynamic spontaneously in its same old phase space. If the phase space is fixed (closed) it seems like the most likely thing for it to be doing spontaneously (according to the second law) it sitting dead in the highest entropy corner.

 

[wabbit]

How likely is it for a system in a constant volume phase space to have increasing entropy.

Not related to gravity or cosmology in particular, but my understanding is you just "spread out" in that phase space. Many (most ?) examples of increasing entropy do not involve any change of phase space, rather how a process starting in a small corner of phase space will likely end up visiting a large part of it.

In a gravitational collapse, part of the story as described by Baez is that everything ends up zipping around at high speeds, like a hot gas.

As to why our universe started out in a low entropy state, this is often described as an unsolved mystery, though analysis of the bounce process in LQC at least appears to suggest that the quantum transition through the bounce might somewhat smooth out everything (sorry, don't have the reference at hand but I think that's from one of the recent papers highlighted by marcus).

Edit: this might be Wilson-Ewing and not LQC but effective QG cosmology nonetheless. Quite a few other papers also discuss cosmological bounce perturbation spectra, e.g. Biswas, Mayes, Lattyak: Perturbations in Bouncing and Cyclic Models, a General Study

 

[wabbit]

Just saw this: Marozzi, Uzan, Umeh, Clarkson: Cosmological evolution of the gravitational entropy of the large-scale structure which looks very relevant for this thread:

This article derives the entropy associated with the large-scale structure of the Universe in the linear regime, where the Universe can be described by a perturbed Friedmann-Lemaître spacetime. In particular, it compares two different definitions proposed in the literature for the entropy using a spatial averaging prescription. For one definition, the entropy of the large-scale structure and for a given comoving volume always grows with time, both for a CDM and a ΛCDM model. In particular, while it diverges for a CDM model, it saturates to a constant value in the presence of a cosmological constant. The use of a light-cone averaging prescription in the context of the evaluation of the entropy is also discussed.

 

[Jimster41]

Not related to gravity or cosmology in particular, but my understanding is you just "spread out" in that phase space. Many (most ?) examples of increasing entropy do not involve any change of phase space, rather how a process starting in a small corner of phase space will likely end up visiting a large part of it.

In a gravitational collapse, part of the story as described by Baez is that everything ends up zipping around at high speeds, like a hot gas.

As to why our universe started out in a low entropy state, this is often described as an unsolved mystery, though analysis of the bounce process in LQC at least appears to suggest that the quantum transition through the bounce might somewhat smooth out everything (sorry, don't have the reference at hand but I think that's from one of the recent papers highlighted by marcus).

Edit: this might be Wilson-Ewing and not LQC but effective QG cosmology nonetheless. Quite a few other papers also discuss cosmological bounce perturbation spectra, e.g. Biswas, Mayes, Lattyak: Perturbations in Bouncing and Cyclic Models, a General Study

Just saw this: Marozzi, Uzan, Umeh, Clarkson: Cosmological evolution of the gravitational entropy of the large-scale structure which looks very relevant for this thread:

Yes, yes, I realized this morning that the corner was exactly the wrong metaphor.
I'm hoping to check out these reading and others you have posted.

I do feel confident I 'm going to be able to get a better picture. I know that this question has been asked, and there are illuminating answers, or at least a well defined overlook on the majesterial void. Just a frustratingly slow climb.

 

[Jimster41]

That paper, (Wabbit) and the WIki drill into CDM may be the answer I was looking for, or had forgotten. If you include CDM in the phase space of the universe (along with the Entanglement stuff, black holes and other stuff so extreme as to seem orthogonal) - it seems equivalent to a closed phase space, with major problems, that just seemed so self imposed, and maybe is missing an opportunity.

I think this is related to a past train of thought and sources where I got the compelling cartoon of colliding sheets or Branes - and the idea that and proposing the Union of colliding sets as observable directly (in the aforementioned closed phase space, from our perspective) was to rush back toward the cosmogenesis problem. Whereas, imagining that we can only observe the intersection, or the intersection plus one remainder, but not the other remainder, was a step-wise (if desperate) solution to that humiliating problem (cosmogenesis). It leaves the container of cosmogenesis one step removed, as it must be, but at least two new features of it are at least potentially still accessible to inquiry through indirect inference and deduction from the intersection we inhabit and observe.

It seemed at the time a bit (understandably) anthropomorphic to propose when we see that 80% of the "mass" or gravitational structure is not explained by our definitions of matter, that it's still more matter and not to include that it might be a projection of something we are at least partly a step genuinely removed from (truly not material by the standard), but that isn't therefore the garden of Eden, and is rather a similar but different intersecting set (for there must be two sets for an intersection).

I am content to believe that the formalism captures all these semantics quite nicely. Sure wish I could freaking follow it.

 

[Jimster41]

Sorry to bump my own thread, but I just wanted to close it productively.

I realize now that I had read the chapters on the very subjects of Louivilles Theorem and Entropy in Penrose' "Cycles of Time" with the notion of universal expansion (conformal scaling as I know understand it) already tangled up with those. Going back and re-reading I realize I completely missed the sentence where he states that Louville's Theorem says that the size of phase space doesn't change. I didn't know it said that, read right past it, and latched exclusively onto the explication he does so well (with pictures) of how a system configuration trajectory must go from smaller areas of coarse-graining to larger. This is "equivalent times of sojourn" I think, which I thought was Louivilles theorem. I thought he was referring only to subregions of the phase space of the universe and the idea was soon to be revealed that this local behavior, things moving from smaller to larger coarse gaining regions of phase space was due to the fact the universe was expanding in a way that made each new phase space set, somehow more coarse grained than old phase space. Or something like that... I wasn't clear. But I had my mind made up.

So now I'm back to square one (maybe two), just trying to follow him and Susskind.

Also the phrase I got from the Smolin Unger Book, maybe totally familiar to you all, is "the antinomy of cosmogenesis", not Antimony, the periodic element