Talk:Liouville's theorem (Hamiltonian)

784:

theory like pretty much everything else, but in the right setup - one that historically first found use in physics, but gets its definitive form in mathematics. As it is understood and set up now, this is a mathematical theorem, which if stated correctly has to define its terms as mathematical structures which were first `invented' for physical purposes, but now have a precise definition. One day maybe they'll be seen as non-physical, and some deeper insight may override the whole of Hamiltonian mechanics in a fundamental way as far as this universe is concerned (not in the foreseeable future) - but this will remain mathematically sound. I agree the article does not capture the precise mathematics involved very well - there are more physicists than pure mathematicians out there, and this is far more important to the former.

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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.

173:, and that means it's not the theorem itself. Secondly, nothing is actually stated! You've left everything undefined, and no assumptions are given. If that's your idea of stating a theorem, you'll get straight Fs in any theoretical math course. The equation is a differential equation in which the notation { ρ, H } is not given any definition. "How probability distributions evolve under time evolutions"? What does that mean? One may assign to each instant 74: 53: 563:

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).

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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.

758:, 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. 1682:'s Mathematical Methods of Classical Mechanics (in the 2nd edition, it's in section 16). It does not use differential forms but only basic calculus. The version with the Hamiltonian flow and the canonical symplectic form comes later in that book. The language of this proof can be extended to provide a disproof of the challenged assertion. 1495: 633:

particular the Jacobian of the time evolution, in "other statements." It seems no easier to formulate precisely than the usual way, in terms of Liouville's equation. In particular, I don't think that dq_i always compensates for dp_i: it could be that dp_j compensates, or even dq_j... (but see below - linuxlad)

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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 &

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Ta - (Lets continue the discussion of the double pendulum on your user page first). On the main issues, as I've written on your userpage, I think the statement on dpi v dqi is true to first order, for small times, for the faces of a suitably aligned small hypercube in phase space. Which is all that's

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Hamiltonian flow preserves the symplectic leaves of the canonical coordinates. It's total phase volume that's preserved, not phase volume projected down to individual leaves. The condition stated is sufficient, but it's certainly not necessary. It might even be worthwhile to state a counterexample.

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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

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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

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Yes, something odd here about the formulation, anyway. A smooth measure on a manifold has a different variance property to a function, no? So, is the point that in the Hamiltonian setting the background top-level differential form (wedge all the dp and dq) is invariant under the flow, meaning that we

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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

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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

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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

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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

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You can say that a single article got it wrong, but you can't say an entire field "got the terminology wrong" as there's no reason to prefer the terminology of one field over that of another. I can assure you that I've read dozens of articles in which this terminology is used. I would argue however,

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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

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I just now removed the troublesome sentence. I don't know why y'all didn't remove this 16 years ago. Yes, I suppose if you are able to find suitably diagonalized conjugate coordinates, this statement would be true (it is generally true in symplectic geometry, if I'm not mistaken. but ...). But that

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I have removed a couple of typos - in the equation of continuity, (which says nothing about convective derivatives being zero), and the role of N. Added a brief reference to the 'fluids analogy' (which is essentially what your proof is) because it's easier not to make mistakes (for me anyway) that

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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

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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

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I added a proof as a link to Wikiversity a while back. Looking at I realized I made two mistakes, both due to my inexperience with such issues: First I made it a Permalink, which would have served to suppress edits to that link. Second, I should created a Wikiversity page devoted exclusively to

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Well, I tried to not introduce any unnecessary formalism, but of course a better writer could still simplify it. Liouville's theorem is a mathematical statement, so it is hard to imagine stating it in a non-mathematical way. I think that the statement (1) you make is the statement about volume, in

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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

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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

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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

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I agree, if possible, a clean proof ought to be added. In general I'm not such a big fan of tedious proofs on wikipedia (especially not if they involve many tedious steps or sneaky approximations), however short and elegant ones can be illuminating more than words. Exactly as you say, I think it

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joke137 - I'm sorry I don't agree with your last few points - the phase space is system points of potentially quite complex entities - any thing you can write a Hamiltonian for. (Eg a million flying bedsteads tied together with string) Mixing of variables is by the by - the cancellation clearly

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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 Knowledge entry! (try searching the page). I

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There is a branch of mathematical called 'symplectic geometry', within which manifolds in the main with such defining equations are fundamental are the study (sort of) - purely mathematical. The theorem is understood in terms of this, and has a mathematical proof, that can be traced back to set

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Later:- No, the effects must cancel in (p,q) pairs (because of the Hamilton relations - see my original para, which you've cut) - isn't this what you'ld expect? (the 'system' could comprise two pendulums at opposite ends of the universe, so p1,q1 and p2,q2 have no coupling) It's essentially the

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From a pedagogical point of view I think that there should a clear and early statement that the theorem is 0) that the phase space density around a state point (ie convective derivative) is constant, and that this follows from 1) if you follow a set of points, the stretch in dp is offset by the

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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

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Sorry, beg to differ - it is a theorem about a property of the real world, like Newtonian mechanics, and as such needs to be understood by physicists, physical chemists and astronomers. The clearest explanations are found on these sites. Knowledge needs to keep the need for a physically-based

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I don't know exactly what you mean by "system points." The theorem is about a phase space distribution function, which could either, physically, represent one particle (with some uncertainty about its position in phase space) or several, non-interacting particles, as in the case of the Vlasov

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divergence statement (where the cancellation is also in p,q pairs) in a Lagrangian frame (as I think You're saying above), I think. The 'proof' is standard - I was given it in 3rd year physics many years ago, (and found it helpful and memorable) - and it's also on the web. :-)

193:... and is ρ supposed to be (1) a probability density function, or (2) a cumulative probability distribution function, or (3) a probability measure, or (4) something else? That's one thing you'd obviously need to state here in order to have a statement of the theorem. 500:

I think in essence V doesn't change because the shrinkage of V in the q coordinate direction is exactly offset by an equivalent increase in the conjugate p direction, and this RELIES on Hamilton's relationships between the p's and q's and their time derivatives.

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The double-linked pendulum provides a counterexample. In general, there are always periods of counter-phase amplitude oscillations in each pendulum as seen against the other. This corresponds to an quasi-oscillating transfer of phase volume between the

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Incidentally I think you need care when using for single particles (as when you introduce Mr Vlasov) - since the theorem is about 'system points', and classically there could be any number of those, provided you choose an appropriate relative density.

2403:, I guess. However, both of these are rather abstract concepts; I don't know if there is a simpler way to put it. I also don't know if there are examples that are not both measure-preserving and also symplectic. Perhaps there would be examples from 224:

I think the point here is that they want to talk about 'smooth measures' on the phase space, whether or not those are probability measures. 'Smooth measure' is a measure absolutely continuous with respect to the background measure μ, which is like

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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?

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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.

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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.

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Thank you. I completely agree with your objections, and I have very similar statements in my stat. mech notes. Why I didn't get things right the first time, I can't tell you—but hey, the article is much better now, so it's all okay in the

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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..

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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.

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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.

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The information paradox article points here to find out why information is never destroyed. But I cannot remotely figure it out. This idea about information seems so important it would be nice to find a detailed explanation. Manuel

459:"Distribution function" also seems to be the preference among my quantum and stat. mech. professors, but I'd have to dig out my lecture notes to be sure. What can I say? Physics people are strange. (But they lead charmed lives.) 768:

Returning to this after a long break - sorry, the theorem only holds in worlds where Hamilton's relations between the ps and qs hold, or equivalently where Newton's laws are true - its key insight is thus a physical one. Bob aka

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Classically, the use of the theorem is to justify some version of the ergodic hypothesis ie in a time-stationary system the probability of a system being in a small phase space volume is proportional to the phase space volume. .

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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

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contraction in dq and/or 2) viewed as a fluid in phase space, the flow has zero divergence. The basic statement of the theorem and the key reasons should be accessible to a competent final-year physics or chemistry 'major'

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I changed the lede to say the above. I punted on the chemistry angle, though. A single sentence at the end of the article mentions this in passing. It needs expansion. In the meanwhile, I think the current lede will do.

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products (no summation convention) stay constant as the volume flows through phase space. Go back to say March 2005, or look on the web - not anything clever, but something for us simple physicists, or chemists. Bob aka

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that what currently (before my edit) appears in the article is incorrect: What I call the quantum Liouville equation is certainly not THE master equation. Any linear 1st order differential equation is A master equation.

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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?)

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The fact that proofs are found in physics textbooks is not helpful for those without convenient access to those books. For better of for worse, Knowledge is the prime source of information for lots of

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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.

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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.

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in many ways, and there is no differential equation that this assignment must satisfy merely as a result of this thing's being an assignment of probability distributions to instants in time.

1902:, is something I do not recall. It might be true, but a proof would clarify the question without ambiguity; it would be nice to see what goes wrong if the partial derivative does not vanish. 655:

That is only true for infinitesimal times. Of course in your example of seperated pendulums, it holds for infinite times, but for a general system there is mixing between all the variables. –

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One commenter felt confused by the statement that the theorem is true for non-equilibrium cases. I believe that the statement in Knowledge is true, but must confess that I could not define

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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).

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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:

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of the Hamiltonian system that factorize it appropriately, a la symplectic geometry. At any rate, it is not "straightforward", and not something that is "simply found with a web-search."

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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).

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invented differential forms, and, as I understand it, he did so for Einstein to do GR with (although I don't believe he ever switched), and, again as I recall, this was in the 1920's.

1490:{\displaystyle {\tfrac {\partial ^{2}H}{\partial q_{i}\partial q_{j}}}={\tfrac {\partial ^{2}H}{\partial q_{i}\partial p_{j}}}={\tfrac {\partial ^{2}H}{\partial p_{i}\partial p_{j}}}=0} 1226: 1180: 1900: 494:

The local density in phase space D=N/V. You need to argue that N is constant within the marked volume, and this I think is what your argument on non-crossing of trajectories does.

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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.

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to represent the motion between collisions of the drifting electrons. In reality, these velocities are distributed around a mean value according to a distribution function

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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.

1949: 1521: 1125: 1098: 300:"Distribution function" seems to be the standard usage in the literature my job entails reading, even though the functions under consideration certainly aren't the 2428:). The original intent was to also exclude chemical reactions, which would change the number of particles, and/or transfer energy to internal degrees of freedom ( 2529:

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

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Footnote - we've been here before, in both my physicists' 'baby notation' and Penrose's 2-forms - see March 2005 versions and p484 in the Penrose tome. Bob aka

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in its original form is from 1882, and recognized as a 2n-form shortly after those objects were invented. It's not like this stuff is particularly recent.

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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.

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The previous two posts have me convinced that we need the actual proof. The article does outline the proof but I think we need a wee more. Here's why:

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assert that Hamilton's relations, or equivalently, Newton's laws, are a mathematical inevitability. If so, I would welcome a demonstration.:-) Bob aka

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Yes, thanks for this. The total derivative popped up in the equation of continuity – I must have accidentially copied it from the Liouville equaton. –

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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

130: 2420:(red link), which in simple terms says that Liouville's thm applies only after you've waited long enough for all friction effects to wander away ( 361:

Vol. 7, No. 15, 1997) derives the Boltzmann transport equation from Liouville's theorem, as given in this article. In the introduction, they say,

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This article is missing a discussion & treatment of the general statistical/ergodic setting. All we've got is just one sentence, at the end:

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This is one case where I'm inclined not to be tolerant of terminology differences between fields, and say simply that they've got it wrong.

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preserved in leaves. If not, however, the mixed partial derivatives mean that cross-leaf dependencies appear in Hamilton's equations.--

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In addition, the relationship is between generalised coordinates and their conjugate momenta, so the use of q rather than x is usual.

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Nolte has a good historical paper on this. It is probably Jacobi who should be credited with recognizing LT in a mechanical sense.

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interprets this as meaning that Boltzmann was already using LT in 1871. Can't confirm this one unfortunately, I can't read German.

566:(In passing - Is there a subtle distinction between LE (which already has an article) and LT which we should strive to maintain?) 621:

My first impression is that it's mathematically top-heavy for my taste (Though I agree that it was in urgent need of a cleanup).

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without giving that word a great deal of thought. A simple algebraic proof would allow me to stop wondering what a word means.

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assume that this is because the latter spelling is more natural (indeed I only assume that the former is actually correct!)

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which has hardly received the recognition which it deserves) and to show its applications to astronomy and thermodynamics."

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way. And also on the motion of of a cloud of points, because I think it gives a feel for what's going on in phase space.

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be simple but are not presently explained in terms accessible to those with only a standard 'maths-methods' background.

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Certainly there are many dynamical systems that do not preserve a measure that is positive on all non-empty open sets.

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which represents the isotropic probability density of finding the electron in the three-dimensional velocity interval

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It would be a very good idea if the article specified what type of dynamical system these equations apply to.

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equation. This can then be generalized to the Boltzmann equation, in which a collision operator is added. –

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Lets take this off-line rather than slog it out here - my email is fizziks@n-cantrell.demon.co.uk Bob

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The relationship holds because Hamilton's relations apply to pi, qi pairs, and so this ensures the

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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statement occurs before any discussion of being able to diagonalize, or more generally to find

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You also need to show that V doesn't change as you follow the test particle (total dV/dt =0).

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_Your_ last step (using the Hamilton's relations) _is_ showing that the divergence is zero.

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http://www.lecture-notes.co.uk/susskind/classical-mechanics/lecture-7/liouvilles-theorem/

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Instead, it just plunges into describing the equations relevant to Liouville's theorem.

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Mathematically, if all the mixed partials of the Hamiltonian are zero, that is, if

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Restored this - my fault it went before. The page needs to be brought in line with

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really being claimed. The longer term gyrations and twists are by the by. Bob aka

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A four-dimensional phase space is minimal. You can have Hamiltonians both where

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It is straightforward to show that as the cloud stretches in one coordinate –

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Sorry I don't agree that determinism alone is sufficient to give the theorem!

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You're staring right at the equation! What do you mean state the theorem?

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have a time-dependent hamiltonian Liouville's theorem will still work. --

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The principal approximation of Sect. 2.2 was to take a single velocity

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The distribution function is assumed to satisfy the Liouville equation

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Firstly, the equation I am staring at is something you have called a

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The second comment, that a condition of the theorem to be valid is,

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can be derived. It is also the key component of the derivation of

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Notice that, in many dimensions, we don't require that for each i,

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Can we talk? Your place or mine, or in the appropriate talk page?

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citation (from a modern stat mech textbook) on the origin of LT.

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and the bracket on the right-hand side denotes a Poisson bracket.

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This is tantamount to saying that the diffeomorphism induced by

2416:(aka "incompressible dynamical system") which then invokes the 2337:{\displaystyle d\Gamma =dq_{1}\cdots dq_{s}dp_{1}\cdots dp_{s}} 929:{\displaystyle (\rho ,\rho {\dot {q}}^{i},\rho {\dot {p}}_{i})} 2116:

I am not so sure Gibbs was the first. Two points of evidence:

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I think that would belong in the article on the man himself.

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under a flow governed by a system of differential equations.

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couple of properties of the Jacobian are used. In formulas:

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To all who have corrected my mistakes, both major and minor:

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Nolte, David D. (2010). "The tangled tale of phase space".

1228:. Anybody want to illustrate the Hamiltonian functions? -- 940:; this is a corollary of the proof. But this implies that 2399:, for starters. But it would have to be one defined on a 1945:

that link. I am now attempting to rectify both errors.

825:'Conservation of density in phase (space)', as it says. 702:

arises from the relation between pairs of ps & qs.

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The Liouville equation is integral to the proof of the

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Curiously, the article now points out that the tuplet

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and distribution function ρ, the theorem states that

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I will now insert this new link into the article.--

1762:The Liouville equation as stated is only true when 1736:Also, p.s. the two-form you are looking for is the 1678:There's a good treatment of Liouville's theorem in 1599:{\displaystyle H=\sum _{k=0}^{n}H_{k}(q_{k},p_{k})} 596:

Informal Proof & Symplectic Geometry equivalent

2336: 2246: 2196:Yes, this article sorely needs a history section. 2083: 1894: 1798:{\displaystyle {\frac {\partial H}{\partial t}}=0} 1797: 1598: 1515: 1489: 1294: 1265: 1220: 1174: 1119: 1092: 928: 427: 304:distributions. For example, Blum and Rolandi's 284:"? If so, it should say that, because the term 357:An interesting little paper by Blatt and Opie ( 640:of the flow vanishes. (see below - linuxlad) – 8: 2247:{\displaystyle \int d\Gamma ={\text{const}}} 2126:Ludwig Boltzmann: The Man Who Trusted Atoms 1621:There's already a Knowledge article on the 2567:C-Class physics articles of Mid-importance 1069:I believe the following claim to be false: 47: 2328: 2312: 2299: 2283: 2265: 2239: 2225: 2066: 2048: 2038: 2026: 2008: 1998: 1996: 1875: 1867: 1769: 1767: 1587: 1574: 1561: 1551: 1540: 1528: 1502: 1471: 1458: 1440: 1432: 1419: 1406: 1388: 1380: 1367: 1354: 1336: 1328: 1326: 1286: 1277: 1257: 1248: 1206: 1193: 1187: 1160: 1147: 1141: 1111: 1105: 1084: 1078: 917: 906: 905: 892: 881: 880: 865: 416: 377: 375: 2487:Missing treatment of statistical systems 1221:{\displaystyle q_{0},q_{1}\rightarrow 0} 1175:{\displaystyle q_{0},p_{0}\rightarrow 0} 743:understtanding in mind, please. Bob aka 292:, which is of course a different thing. 177:in time a probability distribution, say 2155: 1909:It is a beautiful proof, as I recall.-- 1895:{\displaystyle \partial H/\partial t=0} 813:which was Gibbs's name for the theorem 614:hope you'll agree its an improvement. – 508:I shall edit accordingly if you agree. 49: 19: 1926:would clear up the range of validity. 1100:say – it shrinks in the corresponding 306:Particle Detection with Drift Chambers 7: 2382:2601:200:C000:1A0:494:DE11:8C09:9B16 95:This article is within the scope of 2397:measure-preserving dynamical system 1065:Probable false claim requires proof 280:Does "distribution function" mean " 38:It is of interest to the following 2270: 2233: 2059: 2041: 2019: 2001: 1880: 1869: 1780: 1772: 1464: 1451: 1437: 1412: 1399: 1385: 1360: 1347: 1333: 1279: 1250: 388: 380: 14: 569:Can we avoid the use of the term 486: 2469:Energy conservation requirements 2371:Liouville's theorem applies to. 290:cumulative distribution function 82: 72: 51: 20: 2562:Mid-importance physics articles 2097:Am I missing something here?-- 135:This article has been rated as 2390:16:24, 30 September 2021 (UTC) 1979:The questionable reference is 1685:Penrose simply used 2-forms. 1593: 1567: 1212: 1166: 923: 867: 422: 403: 1: 2482:14:56, 13 December 2022 (UTC) 2418:ergodic decomposition theorem 2369:what kind of dynamical system 2143:21:55, 17 February 2014 (UTC) 1967:02:06, 18 February 2014 (UTC) 1936:21:59, 17 February 2014 (UTC) 1919:15:52, 22 December 2013 (UTC) 1606:. In this case phase volume 779:20:50, 28 November 2007 (UTC) 270:In a system with Hamiltonian 115:Knowledge:WikiProject Physics 109:and see a list of open tasks. 2502:second law of thermodynamics 2367:The article never describes 2358:12:18, 27 October 2020 (UTC) 1841:09:17, 19 October 2009 (UTC) 1295:{\displaystyle \Delta q_{i}} 1266:{\displaystyle \Delta p_{i}} 726:08:45, 7 November 2005 (UTC) 282:probability density function 118:Template:WikiProject Physics 2107:14:48, 8 January 2014 (UTC) 1026:02:41, 21 August 2020 (UTC) 1002:02:34, 21 August 2020 (UTC) 487:Anville's change 29/10/2004 2583: 1815:11:28, 26 March 2008 (UTC) 1713:15:52, 27 March 2008 (UTC) 1673:13:58, 26 March 2008 (UTC) 1656:08:47, 24 March 2008 (UTC) 1635:02:51, 24 March 2008 (UTC) 1313:20:17, 21 March 2008 (UTC) 1238:19:12, 21 March 2008 (UTC) 538:version w/o Darboux coords 482:12:51, 17 April 2007 (UTC) 141:project's importance scale 980:14:03, 7 April 2008 (UTC) 961:14:17, 13 June 2006 (UTC) 950:05:37, 13 June 2006 (UTC) 800:16:08, 18 July 2010 (UTC) 763:05:52, 13 June 2006 (UTC) 748:10:51, 4 April 2006 (UTC) 737:04:57, 4 April 2006 (UTC) 589:15:14, 13 Feb 2005 (UTC) 546:23:16, 11 Nov 2004 (UTC) 533:14:48, 27 Nov 2004 (UTC). 514:17:20, 29 Oct 2004 (UTC) 473:00:32, 19 Sep 2004 (UTC) 296:00:42, 12 Sep 2004 (UTC) 240:16:47, 13 Sep 2004 (UTC) 220:20:09, 19 Sep 2003 (UTC) 207:08:52, 10 Sep 2003 (UTC) 201:can blur the difference? 134: 67: 46: 2557:C-Class physics articles 2543:00:02, 20 May 2024 (UTC) 2461:00:05, 20 May 2024 (UTC) 2442:22:47, 19 May 2024 (UTC) 2430:grand canonical ensemble 2206:00:09, 20 May 2024 (UTC) 1847:Add more complete proof? 1750:23:48, 19 May 2024 (UTC) 1053:22:33, 19 May 2024 (UTC) 849:16:08, 31 May 2006 (UTC) 830:08:27, 31 May 2006 (UTC) 821:00:38, 31 May 2006 (UTC) 697:03:30, 6 Jun 2005 (UTC) 690:14:18, 5 Jun 2005 (UTC) 680:15:48, 9 Jun 2005 (UTC) 672:08:01, 6 Jun 2005 (UTC) 659:15:48, 9 Jun 2005 (UTC) 652:08:01, 6 Jun 2005 (UTC) 644:03:30, 6 Jun 2005 (UTC) 629:09:40, 5 Jun 2005 (UTC) 618:22:47, 4 Jun 2005 (UTC) 605:09:23, 8 Mar 2005 (UTC) 465:15:43, 13 Sep 2004 (UTC) 261:"distribution function"? 253:02:49, 4 Jun 2004 (UTC) 214:, amongst other things. 197:23:26, 5 Sep 2003 (UTC) 190:23:23, 5 Sep 2003 (UTC) 166:19:52, 5 Sep 2003 (UTC) 2522:electrical conductivity 2405:sub-Riemannian geometry 2211: 1516:{\displaystyle i\neq j} 854: 288:is often taken to mean 2510:transport coefficients 2412:The lede does link to 2338: 2248: 2095: 2085: 1975:Questionable reference 1896: 1820:non-equilibrium system 1799: 1600: 1556: 1517: 1491: 1296: 1267: 1222: 1176: 1129: 1121: 1094: 930: 429: 28:This article is rated 2339: 2249: 2086: 1987: 1897: 1800: 1601: 1536: 1518: 1492: 1297: 1268: 1223: 1177: 1122: 1120:{\displaystyle q_{i}} 1095: 1093:{\displaystyle p_{i}} 1071: 931: 835:Liouville - Louiville 550:Open note to Denevans 430: 286:distribution function 2531:conservative systems 2518:thermal conductivity 2506:Green–Kubo relations 2264: 2224: 1995: 1866: 1766: 1527: 1501: 1325: 1276: 1247: 1186: 1140: 1104: 1077: 864: 374: 2498:fluctuation theorem 2414:conservative system 2401:symplectic manifold 2212:Liouville's theorem 1952:to the sister wiki. 212:symplectic topology 98:WikiProject Physics 2422:dissipative system 2334: 2244: 2081: 1892: 1795: 1596: 1513: 1487: 1479: 1427: 1375: 1292: 1263: 1218: 1172: 1117: 1090: 926: 810:from the article: 425: 265:The article says: 34:content assessment 2407:but I don't know. 2395:Good question. A 2242: 2177:10.1063/1.3397041 2073: 2033: 1831:comment added by 1787: 1715: 1699:comment added by 1691:Darboux's theorem 1478: 1426: 1374: 942:Noether's theorem 938:conserved current 914: 889: 855:Noether's theorem 790:comment added by 756:Frobenius theorem 571:symplectic system 395: 158:State the theorem 155: 154: 151: 150: 147: 146: 2574: 2343: 2341: 2340: 2335: 2333: 2332: 2317: 2316: 2304: 2303: 2288: 2287: 2253: 2251: 2250: 2245: 2243: 2240: 2188: 2187: 2160: 2112:who did it first 2090: 2088: 2087: 2082: 2074: 2072: 2071: 2070: 2057: 2056: 2055: 2039: 2034: 2032: 2031: 2030: 2017: 2016: 2015: 1999: 1901: 1899: 1898: 1893: 1879: 1843: 1804: 1802: 1801: 1796: 1788: 1786: 1778: 1770: 1694: 1605: 1603: 1602: 1597: 1592: 1591: 1579: 1578: 1566: 1565: 1555: 1550: 1522: 1520: 1519: 1514: 1496: 1494: 1493: 1488: 1480: 1477: 1476: 1475: 1463: 1462: 1449: 1445: 1444: 1434: 1428: 1425: 1424: 1423: 1411: 1410: 1397: 1393: 1392: 1382: 1376: 1373: 1372: 1371: 1359: 1358: 1345: 1341: 1340: 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