Talk:Boltzmann equation

Confusion

There is a confusion, very common, between Liouville and Boltzman. The Liouville Equation is a N-particle equation whereas the Boltzmann Equation is a 1-Particle Equation. The Boltzman Equation is an approximation. —Preceding unsigned comment added by Lionel sittler (talk • contribs) 13:28, 15 October 2007 (UTC) For a non-expert, the first sentence is confusing. How can there be a "statistical distribution of one particle"? Does this mean the distribution of positions that particle might occupy, the number of particles of a single type in a gas, or something else? — Preceding unsigned comment added by 96.241.52.108 (talk) 01:59, 29 May 2012 (UTC)

Another confusion in the introduction: the Boltzmann equation is not a linear equation at all! The collision term is quadratic in the particle distribution function, and in the case of plasma, the force is generated by the density itself... Dobmec (talk) 14:34, 5 January 2015 (UTC)

Technical

This page has no real description of the equation, lacks a good explanation of how it is used, and is otherwise in need of improvement. FrozenPurpleCube 16:19, 17 April 2007 (UTC)

There is no explanation of how the collision term is obtained as an integration of distributions over momentum space. Ntveem 20:03, 15 September 2007 (UTC)

Vlasov equation is different

Vlasov equation is completely different than Boltzman equation. The redirect should be detached. Temur (talk) 07:29, 14 January 2008 (UTC)

Molecular chaos and the collision term

Nowhere is it specified what the function g(p-p', q) is. Is it a Green's function? Could someone in the know correct this? 220.101.95.103 (talk) 06:54, 4 February 2010 (UTC)

Problems and rewrite

This article has numerous problems basically everywhere (and I haven’t the patience to list them all out in excruciating detail like I usually do). A rewrite is in order - I'll do it soon. F = q(E+v×B) ⇄ ∑_ic_i 10:30, 28 May 2012 (UTC)

"Too technical"...

If people find this article so technical, in spite of rewriting it, could they please point out which parts are "stumping"?

The lead says exactly what the equation is about, with no more than a description that the equation describes "transport" of physical properties (complicated to imagine?). The terminology and mathematics is inevitable. We can't just remove and dilute everything to a nursary rhyme.

Furthermore - if one understands "a decent proportion" of quantum mechanics (which isn't very easy), then why is this article any worse? F = q(E+v×B) ⇄ ∑_ic_i 09:14, 13 June 2012 (UTC)

Perhaps stating that I understand a "decent portion" of quantum mechanics was being a little cocky, but I quickly get lost in this article as there are many technical terms which are near impossible to understand without looking up what every one means. Perhaps if more internal links could be added it would better explain this article? I feel this article gets into the highly technical portions of the equation too quickly, perhaps a more gradual transition into the technical areas? Perhaps maybe some links to describe the mathematics on the subject, as I almost never can better understand the math, as it is difficult to Google search symbols. Perhaps I am simply not following enough existing links? I would really like to understand technical subjects, but this article is in my opinion fairly difficult to follow. Are there certain areas of physics which are prerequisites to be capable of learning this? (I edited the article without viewing its history, so I understand you may be bothered.) Googol30 (talk) 17:57, 13 June 2012 (UTC)

Hello - thank you for your response! =) I don't know statistical mechanics or this equation inside out yet either (stat. mechanics is basically the application of statistics to the random motion of a many-particle system), but I have some grip on the equation from the previous state of the article and a reference.

No need to feel guilty either, I'm not irritated at all. It's just that I can't think of anyway to reduce the terminology, and make a more gradual transition into the article, in fact I can't think of anything more to add to the article (apart from expanding other sections more, I'd do this myself but I don't understand enough yet). The heart of the equation is the probability density function, and the first section starts by its definition and then goes on to the equation.

All the same - I know that daunting feeling when reading an article on complex subjects and finding yourself having to click every link and flicking between articles just to get some vague idea from the intro, then quickly feeling no progress has been made...

It seems maybe the sentences can be filled out more with short phrases of explaining what the currently placed links mean and their relation in the article, also slotting more links in the process (especially to mathematical articles to explain the notation).

Thanks again for your response, deep interest, and good faith, much appreciated. =) F = q(E+v×B) ⇄ ∑_ic_i 21:43, 13 June 2012 (UTC)

For now the lead and 1st section has been rewritten. Hope this is better (but please don't say "yes" if it isn't...).

Given that my previous attempts were inconsistent - now I see what you meant... f is not simply the probability of N particles all "at" the same r and p, but importantly within the phase space volume element d³rd³p which is centred on r and p. You can just think of d³rd³p as a very tiny 6-dimensional "box" which is at the tip of the vectors, and the particles randomly have positions and momenta inside the tiny box, though unfortuatley 6d is not easy to visulize and nothing can be done about that...

I'll add images soon to try and illustrate the concepts, it takes me a long time to draw things digitally... F = q(E+v×B) ⇄ ∑_ic_i 23:08, 13 June 2012 (UTC)

Your recent work on the article seems to have made it easier to understand and is much appreciated, but my new concern is that the vocabulary may be watered down. What "grade level" is a subject like this usually taught and is this one of those topics that it is not possible to accurately describe the subject without oversimplifying the terminology? I appreciate your understanding, cooperation, hard work, and good faith so far and look forward to your response and the future of this article. Googol30 (talk) 20:42, 14 June 2012 (UTC)

I don't think the vocabulary is watered down or oversimplified... just padded. The technical terms are still there aren't they (now with at least some explaination)? The padding introduced (like "very small region" for differential volume element, linked to that article, then re-stated for what it is) was really the best I could do to frame the concepts in the mind of the public...

This material is somewhere between 2nd-4th year underdraguate level (geussing becuase my current understanding is only from reading outside of lectures, we havn't done this yet, and just finished 2nd year now, also course material varies from uni to uni)...

I think the guidelines are to "write one level down", so the re-writes were aimed around late college/1st year undergrad ("freshman" in USA).

Which specific phrases do you think are watered down, or is it throughout the first 1/2 of the article? F = q(E+v×B) ⇄ ∑_ic_i 21:01, 14 June 2012 (UTC)

I suppose I might have only said that it is "watered down" because I am beginning to understand it now, but looking through, nothing seems to have been changed too drastically, it does seem to have been somewhat simplified from its previous state. Perhaps I shall do my own research on the topic to see if I can find what prose it is usually accompanied with, and possibly find additional sources to get a better understanding of the topic. Googol30 (talk) 23:09, 14 June 2012 (UTC)

Fair eneogh. In maths and physics, it doesn't matter who anyone is. One always has to see things several times to gain even a partial understanding: first to see what the topic is vaguely, again to get some handle on the content, again to begin practicing to use to formalism, again and again and again to penentrate into further, deeper complexities and insights, tackling increasingly diffiult problems, and so on...

I'm very happy your'e beginning to understand this article - although by all means, do feel free highligh any problems. =) F = q(E+v×B) ⇄ ∑_ic_i 23:41, 14 June 2012 (UTC)

I removed the technical tag for now. F = q(E+v×B)⇄ ∑_ic_i 10:25, 28 June 2012 (UTC)

"Stronger form"

In what respect is the so-called "stronger form" stronger? Usually when this term is used, the weaker form and stronger forms are NOT equivalent, they will apply in different circumstances. But it seems to me that the "stronger form" is exactly equivalent to the weaker form, just written out in a more explicit way. Can someone please clarify? Thanks, --Steve (talk) 16:58, 7 December 2012 (UTC)

No clue. I changed to "General equation (principal form)" to "Principal statement" and "General equation (stronger form)" to "Final statement". Any better? Probably not, but not sure how else to re-title. No source as far as I can see even infers a "strong form". M∧Ŝc²ħεИ_τlk 14:36, 7 April 2013 (UTC)

Relative momentum problem

I'm fairly certain that the quantity "g" in the collision integral is always the relative velocity, and never the relative momentum, even for the case of a mixture. For one, the collision term does not have the proper units if "g" is a momentum. Secondly, in a two-body collision, the relevant quantities are center of mass momentum and relative velocity, not relative momentum. 70.124.92.52 (talk) 17:36, 25 June 2014 (UTC)

The Boltzmann equation is a nonlinear integro-differential equation,

The equation is a nonlinear integro-differential equation (it was erroneously called linear stochastic PDE, now it is corrected). See, for example the modern textbook "...solving Boltzmann's equation will be an arduous task since it is both nonlinear and has an integro-differential form..." (page 69 of Harris S. An introduction to the theory of the Boltzmann equation. Courier Corporation; 2004.)Agor153 (talk) 12:56, 16 April 2016 (UTC)

Solution of the Boltzmann equation

Someone has written "Until this breakthrough in analytical solution approaches, the various forms of the Boltzmann equation were solved using numerical methods". I have the feeling that the past tense alongside the word "until" imply that we have analytical solutions for the general Boltzmann equation on arbitrary initial distributions, so that numerics in this field are completely obsolete. I can't find a hint about this in the cited paper. I would suggest that someone gives a reference for at least one Boltzmann solver that actually calculates or at least uses the analytical solution, otherwise the first part of this sentence should be deleted and the tense should be adjusted. And sorry if my potential lack of knowledge about this language created a misunderstanding. 141.24.116.106 (talk) 16:38, 1 June 2016 (UTC)

The Title of the press release (reference 13) "Mathematicians Solve 140-Year-Old Boltzmann Equation" is not only misleading but plainly wrong. This paper https://arxiv.org/abs/1011.5441 does not "solve" the boltzmann equation, nor is it usable to construct solutions in "real world" problems. As Sbyrnes321 already correctly mentioned (back in 00:57, 31 July 2016), this is mainly an existence and uniqueness proof. Unfortunately even the title of the paper itself "Global Classical Solutions of the Boltzmann Equation without Angular Cut-off" can be misunderstood if the reader isn't working in the field. So I suggest to remove the citation & change the last sentence that implies that the analytical approaches could be used to do simulations (aka solve real world problems), because (to the best of my knowledge) this is currently not the case. Moreover the article history of this particular section makes me cringe, Line 220, 229 : https://en.wikipedia.org/w/index.php?title=Boltzmann_equation&type=revision&diff=819710956&oldid=732289428 .Sintun (talk) 23:15, 20 February 2018 (UTC)

I agree 100%. I edited according to your suggestion just now. --Steve (talk) 14:37, 21 February 2018 (UTC)

Mathematical equations do not parse

Can someone fix these equations? Beowulf (talk) 13:10, 4 April 2018 (UTC)

Are you sure? They all look fine on my computer, unless I'm missing something. Try reloading the page maybe? --Steve (talk) 16:14, 4 April 2018 (UTC)

Comparison of 'Principal statement' to 'Final statement'

The principal statement has the three terms (force, diff, and coll) and then says that 'Expressions for each term on the right side are provided below.' The final statement still has the collision term, but the force and diffusive terms are now replaced by three terms on the left of the final statement. Is there a way to connect this equation back to the principal statement, by stating which parts of the final statement refer to force term and which parts of the final statement refer to the diffusive term? Alamo NM (talk) 19:26, 10 September 2021 (UTC)

Missing i in integral equation

In section "General equation (for a mixture)" the collision integral should be written:

$\left({\frac {\partial f_{i}}{\partial t}}\right)_{\mathrm {coll} }=\sum _{j=1}^{n}\iint g_{ij}I_{ij}(g_{ij},\Omega )[f'_{i}f'_{j}-f_{i}f_{j}]\,d\Omega \,d^{3}\mathbf {p'_{i}} ,$

Concave Hull (talk) 23:15, 16 January 2022 (UTC)

"Force and Diffusion terms" Section

The portion reading

However, since collisions do occur, the particle density in the phase-space volume $d^{3}\mathbf {r} \,d^{3}\mathbf {p}$ changes, so $${\begin{aligned}dN_{\mathrm {coll} }&=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }\Delta t\,d^{3}\mathbf {r} \,d^{3}\mathbf {p} \\[5pt]&=f\left(\mathbf {r} +{\frac {\mathbf {p} }{m}}\Delta t,\mathbf {p} +\mathbf {F} \Delta t,t+\Delta t\right)d^{3}\mathbf {r} \,d^{3}\mathbf {p} -f(\mathbf {r} ,\mathbf {p} ,t)\,d^{3}\mathbf {r} \,d^{3}\mathbf {p} \\[5pt]&=\Delta f\,d^{3}\mathbf {r} \,d^{3}\mathbf {p} \end{aligned}}$$

Is essentially a non-sequitor, or at the very least it is "begging the question." If we are already going to assume

$\Delta f=f\left(\mathbf {r} +{\frac {\mathbf {p} }{m}}\Delta t,\mathbf {p} +\mathbf {F} \Delta t,t+\Delta t\right)-f(\mathbf {r} ,\mathbf {p} ,t)$

Then why not just start with ${\frac {df}{dt}}=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }$?

At present, this "derivation" is very confusing. The section is called "force and diffsion terms" and it is never even said which terms those are. 72.33.0.26 (talk) 19:02, 24 September 2024 (UTC)