Liouville, the second law and chaos

[haushofer]

Dear all,

I've never really understood how exactly Liouville's theorem about time-conservation of phase space volume can be reconciled with the second law of thermodynamics. Recently I came across this popular article,

http://www.necsi.edu/projects/baranger/cce.pdf

"Chaos, Complexity, and Entropy" by Michel Baranger. In this paper, from page 13 on, he explains this 'paradox' with chaos: slightly different initial conditions in phase space 'fractalize' the volume, which remains constant due to Liouville. But by coarse graining we approach this fractalized volume by a solid volume ("bagging the mess"), and it is this coarse-grained volume which is bigger than the fractalized volume. Hence the second law of thermodynamics.

Is anyone familiar with this view? At the end Baranger comments on the subjectivity of this coarse-grained volume and relates it to the laws of big numbers.

Any comments are appreciated :)

(edit: no idea why the text color has become blue, but never mind)

 

[JorisL]

I know the first 2 authors of this article H-theorems from macroscopic autonomous equations ( open access http://fys.kuleuven.be/itf/staff/christ/files/pdf/pub/2006/hthmsfinal.pdf ) have surely considered this question. Especially C. Maes has a knack for this type of questions.

At first glance they seem to treat the question in the article they use their usual high level mathematics. (they are mathemagical physicists after all.

 

[Demystifier]

Yes, coarse-graining is the standard explanation of the second law.
See e.g. Fig. 3.1 in
http://www-physics.ucsd.edu/students/courses/spring2010/physics210a/LECTURES/210_COURSE.pdf

 

[haushofer]

Yes, coarse-graining is the standard explanation of the second law.
See e.g. Fig. 3.1 in
http://www-physics.ucsd.edu/students/courses/spring2010/physics210a/LECTURES/210_COURSE.pdf

Great notes. But somehow I don't yet grasp the idea of the entropy of the second law being a subjective notion. Is this comparable with renormalization, where physical parameters depend on the energy scale one is looking at?and is Invoking chaos necessary in this coarse-graning picture of Liouville's theorem vs the 2nd law?

 

[Demystifier]

But somehow I don't yet grasp the idea of the entropy of the second law being a subjective notion.

This is explained even in popular science books, such as
https://www.amazon.com/dp/9812832254/?tag=pfamazon01-20

In short, entropy counts the number of microscopic realizations of a given macroscopic state. But the notion of the "macroscopic" state is very subjective.

Is this comparable with renormalization, where physical parameters depend on the energy scale one is looking at?

Yes, that's a good analogy.

and is chaos necessary in this coarse-graning picture of Liouville's theorem vs the 2nd law?

It's usually associated with chaos, but not always. In principle, it is possible to have a second law due to coarse graining without chaos.

 

[haushofer]

Thanks. Your book recommendation seems curious. I've "The physics of information" by Bais and Farmer in my closet and will read up with this coarse graining.

 

[Demystifier]

"The physics of information" by Bais and Farmer

My google search does no find such a book. Are you sure about the title and authors?

 

[haushofer]

https://arxiv.org/abs/0708.2837

 

[Demystifier]

https://arxiv.org/abs/0708.2837

Ah, it's a paper.
You said that it is in your closet, so I assumed that it is a book. Whenever possible, I like to keep papers as pdf's, not as papers in a literal sense.

Anyway, the paper seems great!

EDIT: Now I have realized that the paper is published as a chapter in the book "Philosophy of Information", which is a book that I already have.