Liouville Theorem and the Expanding Universe

[Irishdoug]

I've a general question. I'm self-studying classical mechanics using various means one of which is Leonard Susskind's Theoretical Minimum lecture series.

I'm on Lecture 7 and we are doing Liouville's Theorem. My understanding of it so far is that in phase space as something expands in, say, the x-direction, it must contract by an equal amount in the y-direction. This is no different in 3-D.

A question was asked in the lecture if this theory holds for the expanding universe. The answer was given as yes. This would thus mean that the universe is expanding and contracting at the same time, which obviously does not make sense. I presume I am just viewing this all wrong so what is the "answer" so to speak? How does the theorem still hold?

Thankyou

 

[fresh_42]

I've a general question. I'm self-studying classical mechanics using various means one of which is Leonard Susskind's Theoretical Minimum lecture series.

I'm on Lecture 7 and we are doing Liouville's Theorem. My understanding of it so far is that in phase space as something expands in, say, the x-direction, it must contract by an equal amount in the y-direction. This is no different in 3-D.

A question was asked in the lecture if this theory holds for the expanding universe. The answer was given as yes. This would thus mean that the universe is expanding and contracting at the same time, which obviously does not make sense. I presume I am just viewing this all wrong so what is the "answer" so to speak? How does the theorem still hold?

Thankyou

Liouville says that if a certain nice (entire, i.e. holomorphic on the complex plane) function is bounded, then it is already constant.

So all I can see is, that from non constant we can conclude unboundedness. Can you elaborate what this extraction - contraction thingy is?

 

[Irishdoug]

So I'm doing very basic classical mechanics, just so you are aware!

Liouville theory states that in phase space volume is conserved, provided the system can be described by using the Hamiltonion.

Liouville says that if a certain nice (entire, i.e. holomorphic on the complex plane) function is bounded, then it is already constant.

So all I can see is, that from non constant we can conclude unboundedness. Can you elaborate what this extraction - contraction thingy is?

So I'm doing basic Classical mechanics just do you are aware :)

Liouville Theorem states that in phase space volume is conserved, provided the system can be described using the Hamilton ion formulation. So volume is conserved however the shape of the system can change ie if the system was a fluid, the fluids shape can change but not its volume.
I've uploaded pictures so you can see the basic maths.

In the lecture I gathered Leonard Susskind said that Liouville's Theorem hols for the expansion of the universe i.e it's volume is constant. I just don't understand how this could be?

If you skip to 53:27 roughly is when the question is asked and answered.

I vaguely understand your response, however could you try reword it in more simpler language if possible?

 

[fresh_42]

Sorry, for confusing you. I thought of another theorem by Liouville. I will have to consider the mechanical context first. Maybe someone else steps in.

 

[anorlunda]

So volume is conserved however the shape of the system can change ie if the system was a fluid, the fluids shape can change but not its volume.
I've uploaded pictures so you can see the basic maths.

That's oversimplifying. Phase space has 3 position and 3 velocity degrees of freedom for each particle in a multi particle system. It is the volume of that multidimensional space that is conserved.

Susskind said that he was attempting to explain using a 2D whiteboard. By analogy only, the 2D area on the whiteboard is conserved as time evolves.

By the way, Susskind and others, also point out the relationship between Liouville's theorem and conservation of information.

A proof of Liouville's theorem uses the n-dimensional divergence theorem. This proof is based on the fact that the evolution of $\rho$ obeys an n-dimensional version of the continuity equation:

IMO, application of this theorem to an expanding universe, would be cosmology.

 

[Irishdoug]

Sorry, for confusing you. I thought of another theorem by Liouville. I will have to consider the mechanical context first. Maybe someone else steps in.

No it's fine. Thanks for the taking the time to reply.

 

[Irishdoug]

That's oversimplifying. Phase space has 3 position and 3 velocity degrees of freedom for each particle in a multi particle system. It is the volume of that multidimensional space that is conserved.

Susskind said that he was attempting to explain using a 2D whiteboard. By analogy only, the 2D area on the whiteboard is conserved as time evolves.

By the way, Susskind and others, also point out the relationship between Liouville's theorem and conservation of information.
IMO, application of this theorem to an expanding universe, would be cosmology.

Ok but in his explanation he said that as the q-space increases, the p-space (momentum) decreses. And he says that this holds true for the expanding Universe. Thus the overall momentum of the Universe must be decreasing. I'm guessing that the Theorem is maybe too simplistic to be applied to the expansion of the Universe? Or is it that it can only be applied to certain particle systems as oppsoed to Universe as a whole?

Thankyou for the reply.

 

[anorlunda]

I think the answer lies in the difference between motion through space and the expansion of space. That is how distant galaxies can recede from us faster than c without any FTL motion.

 

[Irishdoug]

I think the answer lies in the difference between motion through space and the expansion of space. That is how distant galaxies can recede from us faster than c without any FTL motion.

Potentially. I think this blog post about a particular scientific paper may explain what I am asking. I just have to try and figure out exactly what it is saying! http://www.preposterousuniverse.com/blog/2014/09/10/cosmological-attractors/

 

[PeterDonis]

in phase space as something expands in, say, the x-direction, it must contract by an equal amount in the y-direction

In phase space, yes. But in phase space, in "1D", if the "x" direction is position, the "y" direction is momentum. (In "3D", phase space is 6-dimensional, not 3-dimensional.)

Also, "something expands" is vague. What the theorem actually says is that, if you pick a set of points in phase space at a given time, the set of points that those points evolve into over time keeps the same volume. But the set of points does not have to remain convex or even continuous.

Finally, the formulation of Liouville's Theorem you're describing is for classical, pre-relativity mechanics. I'm not aware of a formulation that works for general relativity. So I don't think you can apply it to our expanding universe, since that requires GR to describe.

 

[Irishdoug]

In phase space, yes. But in phase space, in "1D", if the "x" direction is position, the "y" direction is momentum. (In "3D", phase space is 6-dimensional, not 3-dimensional.)

Also, "something expands" is vague. What the theorem actually says is that, if you pick a set of points in phase space at a given time, the set of points that those points evolve into over time keeps the same volume. But the set of points does not have to remain convex or even continuous.

Can I ask what you mean by convex in this context? Although that helps, as I thought the set of points did have to remain continuous, and could not "break away" from each other as such.

Finally, the formulation of Liouville's Theorem you're describing is for classical, pre-relativity mechanics. I'm not aware of a formulation that works for general relativity. So I don't think you can apply it to our expanding universe, since that requires GR to describe.

Maybe I've found my future PHD topic haha!

 

[PeterDonis]

in his explanation he said that as the q-space increases, the p-space (momentum) decreses. And he says that this holds true for the expanding Universe. Thus the overall momentum of the Universe must be decreasing.

This is one of those times when I really wish Susskind would be more precise. You can apply the theorem locally to a portion of the universe. But I'm not aware of any formulation that works for the universe as a whole.

 

[PeterDonis]

Can I ask what you mean by convex in this context?

Just what your intuition would think: that the region in phase space occupied by the points is convex. Heuristically, a straight line connecting any two points should not go outside the region if the region is convex. But the region in phase space occupied by the points does not have to have this property.

I thought the set of points did have to remain continuous

The whole set of points in phase space occupied by the system at all times, not just one instant of time, probably does meet the technical definition of "continuous". But AFAIK there is no requirement that the set of points in phase space occupied by the system at one instant of time must be continuous for all instants.

 

[Irishdoug]

Just what your intuition would think: that the region in phase space occupied by the points is convex. Heuristically, a straight line connecting any two points should not go outside the region if the region is convex. But the region in phase space occupied by the points does not have to have this property.

Oh ok.

This is one of those times when I really wish Susskind would be more precise. You can apply the theorem locally to a portion of the universe. But I'm not aware of any formulation that works for the universe as a whole.

You not a fan? Ok I think that satisfies my yearning. Thanks.

If anyone else would like to add something please do though.

 

[kimbyd]

I've a general question. I'm self-studying classical mechanics using various means one of which is Leonard Susskind's Theoretical Minimum lecture series.

I'm on Lecture 7 and we are doing Liouville's Theorem. My understanding of it so far is that in phase space as something expands in, say, the x-direction, it must contract by an equal amount in the y-direction. This is no different in 3-D.

A question was asked in the lecture if this theory holds for the expanding universe. The answer was given as yes. This would thus mean that the universe is expanding and contracting at the same time, which obviously does not make sense. I presume I am just viewing this all wrong so what is the "answer" so to speak? How does the theorem still hold?

Thankyou

As I understand it, Liouville's Theorem in this context relates to systems described by a Hamiltonian. For an expanding universe, to my knowledge it is only possible to define a Hamiltonian if that universe is closed. In that case, the theorem certainly applies, and is represented by the fact that the Hamlitonian formulation of General Relativity applied to a closed universe results in conserved energy. You end up with a term which can be represented as gravitational potential energy, such that as the energy in the matter/radiation/dark energy of the universe changes, this gravitational potential energy changes to compensate.

However, there is no way to represent this conservation of energy in open or flat universes, and I don't think the theorem applies in those cases.

Edit:
I'm pretty sure that PeterDonis is correct: you can apply the theorem to a local portion of the universe. Just not the whole thing.

 

[Irishdoug]

Brilliant, cheers for that.

 

[TEFLing]

Ok but in his explanation he said that as the q-space increases, the p-space (momentum) decreses. And he says that this holds true for the expanding Universe. Thus the overall momentum of the Universe must be decreasing. I'm guessing that the Theorem is maybe too simplistic to be applied to the expansion of the Universe? Or is it that it can only be applied to certain particle systems as oppsoed to Universe as a whole?

Thankyou for the reply.

Doesn't that apply to the particles of plasma permeating space?

It sounds like it applies to matter within the fabric of space time...

Not to space time itself

As space like slices of space time progressively enlarge, photons in space time redshift, losing momentum and energy...

And apparently similar for massive particles as well

Think you're talking about the abstract PHASE space occupied by matter and energy quanta residing in space time ("the actors on the stage")...

Not about the physical space time of the universe itself ("the stage itself")

 

[PeterDonis]

As space like slices of space time progressively enlarge, photons in space time redshift, losing momentum and energy...

With appropriate definitions of "redshift", "momentum", and "energy", yes.

And apparently similar for massive particles as well

No. Massive particles do not "redshift" like photons do as the universe expands.

 

[kimbyd]

No. Massive particles do not "redshift" like photons do as the universe expands.

That's not strictly true. You can represent the redshift as a loss of momentum relative to the background expansion. Massive particles do experience just such a loss in momentum relative to the background (because as they move they catch up to parts of the universe that are moving away from them).

 

[PeterDonis]

That's not strictly true. You can represent the redshift as a loss of momentum relative to the background expansion.

Yes, that's true in general. In our actual model of our universe, all of the matter (i.e., stuff with nonzero rest mass) is assumed to be cold, which means its momentum relative to the background expansion is already negligible, so it does not redshift.

If we wanted to make a more general definition of the difference between "massive" and "massless" particles in an expanding universe, we could say that massless particles can redshift forever, while massive particles can't; the latter will eventually become cold and their momentum will become negligible (which of course has already long since happened to the massive particles in our universe).

 

[kimbyd]

Yes, that's true in general. In our actual model of our universe, all of the matter (i.e., stuff with nonzero rest mass) is assumed to be cold, which means its momentum relative to the background expansion is already negligible, so it does not redshift.

If we wanted to make a more general definition of the difference between "massive" and "massless" particles in an expanding universe, we could say that massless particles can redshift forever, while massive particles can't; the latter will eventually become cold and their momentum will become negligible (which of course has already long since happened to the massive particles in our universe).

As a fraction of energy density, yes, this is definitely the case: nearly all energy in matter is now in mass-energy, not in kinetic energy.

However, this can still be a significant effect on the motions of galaxies and the growth of large scale structure.