Talk:Liouville's theorem (Hamiltonian)

message: You're staring right at the equation! What do you mean state the theorem? Phys 19:52, 5 Sep 2003 (UTC)
replies: text: |dp^dq^...|
in usual notations. Then H means Hμ really, and the 'base case' H = 1 of the theorem is saying that the background Liouville measure μ is invariant under the flow. Well, I'm not saying this very well, but I think that's the setting.
Charles Matthews 16:47, 13 Sep 2004 (UTC)
thread_title:  State the theorem

message: Could someone please expand this article to help explain the theorem more fully. What is it used for? How is it used? What year did was it brought to light? How did it change or effect physics? Why is it significant? What are the practical ueses? Kingturtle 02:49, 4 Jun 2004 (UTC)
thread_title:  help 

message: I think it would be nice if we could have a version of the proof which does not rely on Darboux coordinates, which only happens to be local in some symplectic manifolds. Phys 23:16, 11 Nov 2004 (UTC)
thread_title:  version w/o Darboux coords 

message: Can we have a discussion on your recent mods to this article? - whilst I'm sure many are useful, some of them seem to put the cart before the horse (if I were a 3rd year physicist I don't think I'd get past your early description in the phase-space section); and in one or two places seem to me to muddle things (eg in _my_ world the convective derivative is NOT a definition it's a deduction ; and again your statement on Hamiltonian and non-Hamiltonian systems is rather unclear). 
Phase space (section 2) is in any case discussed elsewhere. Avogadro's number is an irrelevance - stat mechanics applies to much larger and smaller system's than the 1 gram-mol scale! (is this perhaps a typo?)
The equation (section 3)  you call Liouville's appears to me be 'just' a conservation of D equation found by integrating system-point flows over a control volume (ie the RHS is div (D 'v') where 'v' is a p,q 'velocity') - it's NOT the equation (presently your second) which physicists and theoretical chemists look to to eg justify the weighting on the Boltzmann distro, - this has some real-world dynamics in it. This first equation is indeed used later on as the (nearly self-evident :-)) starting point for the 'fluids' proof of LT (this is not by any means original with me - I can point you to at least 2 similar proofs on the web, from Harvard and Oxford TP departments IIRC).
(In passing - Is there a subtle distinction between LE (which already has an article) and LT which we should strive to maintain?)
Can we avoid the use of the term symplectic system until later - it may well be idea du jour to mathematicians and advanced physicists - but Wiki still lacks an adequate succinct expostulation accessible to a  graduand-physicist (of my generation.) At present all symplectic space refs spin off into ideas like 1-forms, tangent bundles etc., which may be simple but are not presently explained in terms accessible to those with only a standard 'maths-methods' background.
The dynamics of non-Hamiltonian systems and non-conservation of heat are of course important,  but most physicists sidestep them, being interested in micro-systems, or treating the system as embedded in a larger system obeying conservation.
It may be that a dual approach combining the 'physics' and symplectic systems ways is doomed not to work, like
the tensors fiasco, but we can at least try.
Can we talk? Your place or mine, or in the appropriate talk page? Linuxlad 15:14, 13 Feb 2005 (UTC)
thread_title: Open note to Denevans

message: Having struggled through several hundred pages of Penrose (ref. 1) I find that the 'informal proof' (taught IIRC on Natural Sciences Tripos) is essentially the same as the symplectic geometry statement given in 20.4 ('Hamiltonian dynamics as symplectic geometry') of that book . Since one of the main problems for eg physicists is in reading across terms like '2-form', I have added a paragraph to 'guide' those who follow. Please feel free to amend or correct, but remember the intended audience for this section is from the physical sciences.
Linuxlad 09:23, 8 Mar 2005 (UTC)
thread_title: Informal Proof & Symplectic Geometry equivalent

message: I tried to revise this completely. There was a lot that was redundant on this page, and it wasn't ever clear where the main statement of the theorem was, or what it had to do with physics, so I tried to clean it up a little, and removed the excessive subsectioning. I don't think it is perfect, but I hope you'll agree its an improvement. –Joke137 22:47, 4 Jun 2005 (UTC)
thread_title:  revision 

message: IS this a MATHEMATICAL theorem? I think not. It's a theorem about the way the real world behaves. It may follow trivially from the property of symplectic spaces (as I'm always told), but you have to show  a 1:1 correspondence between such spaces and real phase space coordinates - or do mathematicians assert that Hamilton's relations, or equivalently, Newton's laws, are a mathematical inevitability. If so, I would welcome a demonstration.:-) Bob aka Linuxlad 08:45, 7 November 2005 (UTC)
replies: text: Yes, it's a mathematical theorem. The article needs modifications. The theorem state that the natural measure on the phase space(the Lebesuge measure, locally) is invariant under the Hamiltonian flow. More general, it gives nec and suff conditions when a smooth measure on a mannifold is invariant under a flow governed by a system of differential equations. Mct mht 04:57, 4 April 2006 (UTC)

text: Sorry, beg to differ - them's there are fightin' words. To physicists, its a "real world theorem". To mathematicians, its an important theorem in the general study of differential equations in the large. Its not far away from Frobenius theorem, which connects up several huge areas of mathematics. Your POV is showing: you think the physics explanation is clear, presumably because you have a physics education. Your "simple" explanation would leave the math undergrad scratching their head going "what the heck is this S**T?" Its a different language. In the end, you must know both languages.linas 05:52, 13 June 2006 (UTC)
thread_title: category

message: from the article:
which was Gibbs's name for the theorem
err...what was? Deepak 00:38, 31 May 2006 (UTC)
thread_title: clarification

message: We should note somewhere (not least in the keywords!) that Joseph's name and the eponymous theorem are extensively spelled/misspelled as Louiville in the literature.  This includes not only the scientific literature but even an item in the existing Wikipedia entry!  (try searching the page).  I assume that this is because the latter spelling is more natural (indeed I only assume that the former is actually correct!)
:I think that would belong in the article on the man himself. Deepak 16:08, 31 May 2006 (UTC)
thread_title:  Liouville - Louiville 

message: Curiously, the article now points out that the tuplet (\rho, \rho\dot{q}^i,\rho\dot{p}_i) is a conserved current; this is a corollary of the proof. But this implies that Noether's theorem applies. What, exactly, in this context, is the set of symmetries implied by Noether's theorem?  This article should make this connection explicit; Noether's theorem is a major landmark in physics, so this is just begging the question. linas 05:37, 13 June 2006 (UTC)
replies: text: Never mind. linas 14:17, 13 June 2006 (UTC)
replies: text: Noether's Theorem states that symmetry implies conservation.  In the article it is said "Equivalently, the existence of a conserved current implies, via Noether's theorem, the existence of a symmetry. The symmetry is invariance under time translations, and the generator (or Noether charge) of the symmetry is the Hamiltonian."  This needs to be changed, conserved quantities do not, at least by Noether's Theorem, imply symmetry.  I'm guessing this should read something like "The symmetry of invariance under time translations implies, via Noether's Theorem, that current is conserved."  However, since I'm not sure this is true, I didn't want to change this myself. 208.124.63.250 (talk) 14:03, 7 April 2008 (UTC)
replies: text: The statement that "the conservation of rho comes from symmetry in time" confuses me, because time symmetry leads to conservation of the Hamiltonian, which seems completely different than rho.Chris2crawford (talk) 02:34, 21 August 2020 (UTC)

text: Also I don't see how "as well as conservation of the Hamiltonian along the flow have been used."  AFAIK all you need is Hamilton's equations and equality of mixed partials.Chris2crawford (talk) 02:41, 21 August 2020 (UTC)
replies: text: Hmm. Unstated assumptions are that (1) the number of p,q variables is constant in time (this assumption is violated in chemical reactions) and that (2) the total energy aka Hamiltonian is constant in time. The conventional, standard derivation of Hamilton's equations is to write an expression for the energy, and then to minimize the energy over all possible trajectories, which then gives Hamilton's equations as providing that minimum. This is, for example, how one finds geodesics in Riemannian manifolds, in general: geodesics have the lowest energy. Similar ideas for symplectic manifolds. For this reason, Hamiltons eqns are not considered to be "fundamental", but rather follow from the variational minimization of energy. Not sure how to shoe-horn this additional information into this article. I'll look. Hmmm. 67.198.37.16 (talk) 22:33, 19 May 2024 (UTC)
thread_title:  Noether's theorem 

message: The Liouville equation as stated is only true when \frac{\partial H}{\partial t} = 0.--Eh9 (talk) 11:28, 26 March 2008 (UTC)
thread_title:  Missing premise 

message: Under "remarks" it reads "The Liouville equation is valid for both equilibrium and nonequilibrium systems."
Can anyone make this more clear? Because I have textbooks here, which say that the equation only holds for 
equilibrium systems..  —Preceding unsigned comment added by 130.60.5.218 (talk) 09:17, 19 October 2009 (UTC)
thread_title:  non-equilibrium system 

message: The questionable reference is http://www.lecture-notes.co.uk/susskind/classical-mechanics/lecture-7/liouvilles-theorem/
I hate to remove it because it is well-written and because it takes a fresh approach to showing the incompressible nature of flow in Hamiltonian phase space.  But one sentence bothers me:
Notice that, in many dimensions, we don't require that for each i,
\frac{\partial v_{qi}}{\partial q_i}+\frac{\partial v_{pi}}{\partial p_i}=0
meaning that there isn't conservation of points in a particular region, but we can say that if the velocity increases in one direction, then it must decrease in another direction by the same factor.
Am I missing something here?-- guyvan52 (talk) 14:48, 8 January 2014 (UTC)
thread_title:  Questionable reference 

message: I am not so sure Gibbs was the first. Two points of evidence:
* Gibbs says in 1884 in the abstract of a paper titled "On the Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics.", the first time he used a primitive form of LT: "The object of the paper is to establish this proposition (which is not claimed as new, but which has hardly received the recognition which it deserves) and to show its applications to astronomy and thermodynamics."
* Gibbs in his 1902 book, third chapter, cites the entire chapter (which relies on LT) as a follow up to a paper of Boltzmann from 1871. Cercignani's Ludwig Boltzmann: The Man Who Trusted Atoms interprets this as meaning that Boltzmann was already using LT in 1871. Can't confirm this one unfortunately, I can't read German.
Probably it's true that Gibbs was the first to appreciate the full LT application to thermodynamics, and very likely we can say Gibbs was the first to formulate it in the modern way (in 1902). I've updated the language in the text a bit to reflect this. Probably would be good to have one more citation (from a modern stat mech textbook) on the origin of LT.
Nanite (talk) 21:55, 17 February 2014 (UTC)
replies: text: Yes, this article sorely needs a history section. 67.198.37.16 (talk) 00:09, 20 May 2024 (UTC)
thread_title:  who did it first 

message: What is given in the article are only applications of the theorem, but not the theorem itself. The original Liouville's theorem states that the phase space volume remains invariant upon canonical transformations of the generalised coordinates and momenta of a given mechanical system. In its proof a couple of properties of the Jacobian are used. In formulas:
: \int d \Gamma = \text{const}
where 
: d \Gamma = dq_1 \cdots dq_s dp_1 \cdots dp_s
is the volume element of the phase space. This general form can be used in many more fields that just statistical physics, where it has been used by Gibbs. Nothing of this is in the article. The true (general) Liouville theorem and its proof are given in vol. 1 Classical Mechanics of the Landau & Lifshitz series. In the original Liouville paper from 1838, the proof uses the same idea but realizes it in a more complicated and roundabout way (the Jacobian was not yet invented then). Lantonov (talk) 12:18, 27 October 2020 (UTC)
thread_title:  Liouville's theorem 

message: The `Damped Harmonic Oscillator' section makes the claim that for Liouville's theorem to hold energy must be conserved, but this is at best misleading. The only true requirement is that the phase space evolves following Hamilton's equations (which the damped harmonic oscillator doesn't), but if we have a time-dependent hamiltonian Liouville's theorem will still work. --J B 14:56, 13 December 2022 (UTC)
thread_title:  Energy conservation requirements 

message: This article is missing a discussion & treatment of the general statistical/ergodic setting. All we've got is just one sentence, at the end:
:The Liouville equation is integral to the proof of the fluctuation theorem from which the second law of thermodynamics can be derived.  It is also the key component of the derivation of Green–Kubo relations for linear transport coefficients such as shear viscosity, thermal conductivity or electrical conductivity.
The primary issue here is that these topics all require discussions of internal (microscopic) degrees of freedom, whereas the theorem as stated applies only to conservative systems, where friction is absent. This is a huge gap in the current article. 67.198.37.16 (talk) 00:02, 20 May 2024 (UTC)
thread_title:  Missing treatment of statistical systems