Talk:Continuity equation

Just to clarify the hidden note...

It was not intended to be hostile - I didn't say people can't change it... Never mind... Maschen (talk) 15:52, 2 September 2012 (UTC)

Extend scope?

I plan to make the following additions/changes:

• It says ${\mathbf {J}}=(J^{\alpha })=(c\rho ,{\mathbf {j}})$ can be any 4-current, not just for EM, so make the formulation using the 4-current: $\partial _{\alpha }J^{\alpha }=\sigma$, (same σ = generation rate per unit volume in the current article), a level-2 section in its own right.

This may be expanded to include differential forms later (i.e. mention the general 4-current in the language of diff forms, possibly also that Maxwell's equations automatically lead to ${\rm {d}}{\mathbf {J}}=0$ though not essential).

• Include a brief (level-2) section on general relativity and the stress-energy tensor $T^{\alpha \beta }$ since:

1. The 4-divergence of it is a continuity equation itself: ${\frac {\partial T^{\alpha \beta }}{\partial x^{\beta }}}=0$.
2. The equation is the divergence of a tensor field to yield a vector field (energy-momentum, hence energy-momentum is conserved), while the form in this article is the divergence of a vector field to obtain a scalar field (zero field for conservation obviously) - which breaks out of the dominating theme of this article.
3. It is a crucial constraint on the form of the Einstein field equations: $G^{\alpha \beta }=8\pi T^{\alpha \beta }$, this is easily sourced.

As we know this was mentioned above (in integral form).

• Add another (level-2) section which generalizes further the form of the RHS (i.e. what the divergence $\partial _{\alpha }J^{\alpha }=-\cdots$ actually can be for non-conserved currents) in the following way:

RHS = radial fields R (gravity/EM fields which act on the mass/charge elements forming the fluid)

+ surface effects Σ (shear stress, chemical/thermal conduction/diffusion through the surface of the region in consideration)

+ volume effects D (turbulence, viscosity, diffusive currents in the volume of the region)

+ generation rate G (production/removal of turbulent mass/energy, entropy, momentum)

where R, Σ, D, G can be tensors of any required order (including scalars), so in integral form the continuity equation would be:

$\int _{V}\left({\frac {\partial \rho }{\partial t}}+\nabla \cdot {\mathbf {j}}\right)dV=\int _{V}\left(R+D+G\right)+\int _{A}\Sigma dA$

as sourced from

J.R. Tyldesley, (1975). An introduction to Tensor Analysis: For Engineers and Applied Scientists. Longman. p. 35. ISBN 0-582-44355-5.{{cite book}}: CS1 maint: extra punctuation (link)

Here I used different letters from the book and article to prevent mutual conflicts, e.x. like changing S for vector area to A, use G for generation, Σ for surface effects etc. ... we may need to rename more letters later including f to j...).

• I know we have been through all this before... anyway...

1. blend all the specialized sections (EM, fluid mech, thermodynamics, QM) into one level-2 section. Right now, each section isn't much of a section with just a list of quantities and the equation all in the identical general form. IMO they can, and should, be summarized by that outcast table I added last year (cut and pasted from Laws of science to here), with
2. the paragraphs of text before the table, and
3. boxes of derivations after the table.

Forget this... not only does the attempt fail - some links would become broken (in particular to the QM section).

• Not include the convection–diffusion equation, Boltzmann transport equation, and Navier-Stokes equations, which are linked to in the lead.

Yes - I'm sorry being a nuisance to this article yet again, but in this way - it will have more scope. Favour or oppose? Maschen (talk) 00:07, 7 September 2012 (UTC)

It's easy to wrongly assume that ${\mathbf {J}}$ is a regular vector, not a 4-vector. Perhaps you should use a different font. For example, ${\mathfrak {J}}$ ,${\mathcal {J}}$ or $\mathbb {J}$ .--老陳 (talk) 02:35, 7 September 2012 (UTC)

Usually, bold letters are used for 4-vectors (here J = 4-current) while lower case bold for 3-vectors (here j = 3-current).

This is an easy but important issue to consider; to use \mathsf ${\mathsf {J}},{\boldsymbol {\mathsf {J}}}$, \mathcal ${\mathcal {J}},{\boldsymbol {\mathcal {J}}}$ etc. as you say for more general quantities (but preferably not \mathbbf $\mathbb {J} ,{\boldsymbol {\mathbb {J} }}$ which tends to be reserved for sets and definitely not \mathfrak ${\mathfrak {J}},{\boldsymbol {\mathfrak {J}}}$ !!... IMO looks very horrible and is difficult to reproduce by hand). My first choice would be \mathsf but you (and anyone) are welcome to disagree!

Thank you for raising the point. If there is favour to this change we can decide on all the new notation here along these lines before actually making the change. Maschen (talk) 08:30, 7 September 2012 (UTC)

\mathsf looks quite like \mathrm. It would be nice if we can use \mathcal, but the font can not handle lower case letters. By the way, where did you find the convention that bold letters are used for 4-vectors? I have not found such a convention used in many popular textbooks that I have read. For example, in Griffiths' "Introduction to Electrodynamics", the 4-vector potential is expressed as follows:

$A^{\mu }=(V/c,A_{x},A_{y},A_{z})$

--LaoChen (talk)12:56, 17 September 2012 (UTC)

Fonts: ..... \mathsf and \mathrm look completley different to me... caligraphic may be better to label objects like regions of space or flows or so on...

Letter-case convention: Ok the convention is not that common but at the top of my head e.x. J.R. Forshaw, A.G. Smith (2009). Dynamics and Relativity. Wiley. ISBN 978 0 470 01460 8. uses it, same with our article on four-vectors. As long as it's clear from the context it doesn't matter what letter-case is used.

You may have noticed the changes are essentially made now anyway, and the notation is self-consistent (or at least clear from context), with the exception of the non-conserved currents section above but that can wait till more refs are found for sufficient material. Maschen (talk) 15:13, 17 September 2012 (UTC)

I'm drafting things in a sandbox for now. It will not be added of course unless there is clear consensus for "yes" (or if no one opposes the changes within the next month or so). Some things may differ to the original plan (such as the table). Maschen (talk) 11:45, 7 September 2012 (UTC)

Conserved Charge

Concerning the last line, it is my understanding that to find the conserved charge from the conserved current the integrand should read ${\frac {J^{0}}{c}}$ and not $J^{\mu }$. As it is written now, the indices on the left and right side of the equation do not line up, which is nonsensical. For the life of me I cannot find a source for this, of course I do not own a dedicated QFT book. 174.78.149.150 (talk) 16:43, 29 March 2013 (UTC)

Yes - J⁰/c is the density. Typo well-spotted, it's fixed now. M∧Ŝc²ħεИ_τlk 03:59, 30 March 2013 (UTC)

confusion between current density of moving charges/mass/... and charge/mass/... density =

the current article brings the traditional confusion of the current density carrying "mobile" density (rho(r,t) with non null speed field v(r,t)) and the density that is the *sum* of both "static"(rho(r,t) with null speed field) and "mobile" density. In others words the current density j(r,t)= rho_mobile.v(r,t) =/= (rho_mobile + rho_static).v(r,t)= rho.v(r,t), where the last one is explicitly stated in different sections of this wikipage. can this be corrected/made explicit ? (remove/correct the definition of the current) See, for example, reference: Electromagnetic theory, 1941, J.A. Stratton, Mc Graw-Hills. wiki charge conservation seems OK — Preceding unsigned comment added by 81.245.114.72 (talk) 09:21, 19 April 2014 (UTC)

In the continuity equation for electric charge, v would be the average velocity of particles, weighted by the charge of each particle. In the continuity equation for mass, v would be the average velocity of particles, weighted by the mass of each particle.

Certainly there are situations with two subpopulations (e.g. conduction electrons vs immobile ion charges in a metal) with different average velocities, where it would be wise to treat each subpopulation separately instead of lumping them together into a single continuity equation. That is basically what you're suggesting by splitting the charge density into mobile vs immobile, each with its own velocity. I would not say that lumping them together is incorrect exactly. The lumped-together equation is true -- but not as useful as the two split-up equations. (The "v" in the lumped-together equation would be a weighted average of the mobile charge velocity and the immobile charge velocity (which is 0).)

Likewise, there are also situations with three subpopulations (e.g. conduction electrons vs holes vs immobile ions in a semiconductor), or four or five or more subpopulations (e.g. charge continuity in an electrolytic solution). In all these cases it is wise (but not strictly required) to write separate continuity equations instead of lumping them together.

If you're saying that the article does not make it clear that v is an average, then I agree! It's not mentioned at all in the article right now, but it's an important point. The multiple-subpopulation situation is one where it is especially important ... but even when there are not multiple subpopulations, there is still a distribution of different particle velocities, and v is still an average. :-D --Steve (talk) 03:09, 21 April 2014 (UTC)

[reply 01.05.2014] unfortunately, average speed does not resolve the highlighted problem. To make it more explicit than the mobile and immobile charges case (or mobile versus immobile matter, or ...), formally we have a scalar field (rho(r,t)) and a vector field (j(r,t)) and they are independant (in the general case). The continuity equation add a loose coupling between them. As soon as we implicitly say (for the general case) j(r,t)=rho.v, we just end with errors (see below,ohms law example). Taking formally an integral form of an average volume of the continuity equation will not change the point: we have a scalar field (rho(r,t)) and a vector field (j(r,t)).

To hilight my point:

first, we can have rho(r,t) = 0 and j(r,t)=/=0 and the continuity equation. This is incompatible with stating (in the "general case") j(r,t)=rho.v = 0.

Second, (our computers work, we have power): an electric conductor and the ohms law (finite conductivity): j=sigma.E. If one apply formally the continuity equation and maxwell equations and resolve it, one will get rho(r,t>>1) = 0, except at the boundary of the conductor, leading to an almost infinite resistance even for a copper conductor, a contradiction. On all my known good reference books, we can see how the author pay attention to the definition of the current (example reference: Electromagnetic theory, 1941, J.A. Stratton, Mc Graw-Hills) and avoid the trap (section 1.2, charge and current). — Preceding unsigned comment added by 81.247.97.77 (talk) 07:43, 1 May 2014 (UTC)

I was editing those sections recently. In the current version, I don't think the article implies that you can or should use the equation "j=rho.v" for electric current. It presents that equation essentially only in the context of fluid flow. The equation j=rho.v is not mentioned during the discussions of electric current--just j is used, not v. Do you agree that the article is OK now? --Steve (talk) 15:10, 1 May 2014 (UTC)

[reply 05.05.2014] unfortunatly no as long as there is j= rho v without the explicit restrictions statements. The main difference between elecotromangetism and fluid dynamics regarding the continuity equation is the positive only(or null) density (e.g. mass). Unfortunately, I have no references on my mind, but I assume (please note i have not checked this one so i can be wrong) if ones take a pipe where the moving fluid is with a partial phase transition (e.g. from a liquid to a non moving solid attached to some parts of the pipe), ones may end with the problem of j= rho_mobile.v(r,t) =/= (rho_mobile+rho_immobile).v(r,t).

If you really want to keep j=rho.v (in the section general equation),I will suggest to add a patch: to express the limitations (with the risk of creating another contradiction), something like: if there is a closed "sufficently regular" volume were the density has a non null speed for all t within a defined closed interval,then J= rho.v within this volume and time interval (i let you find a better wording / mathematical expression), however wihtout a proper reference this could be questionable. — Preceding unsigned comment added by 81.247.82.241 (talk) 19:47, 5 May 2014 (UTC)

It says "If there is a velocity field v which describes the relevant flow...". That is an "explicit restrictions statement", isn't it? Any example you come up with where j=rho v is not applicable is either a system where there is no velocity field, or else it's a system with a velocity field which does not describe the relevant flow.

If water freezes to the side of the pipe, then the velocity field is zero in the ice regions.

How about

"If there is a velocity field v which describes the relevant flow--in other words, if all of the quantity q at a point r is moving with velocity v(r)--then..."

maybe that helps? :-D --Steve (talk) 14:46, 6 May 2014 (UTC)

[reply 05.05.2014] ok for the comments on my example as I have to push too much in getting an analogy with the em (mixing the "mobile" matter with the non "mobile" matter and we will forget our main topic that is to avoid the reader to stick to the special case where j=rho.v. I really appreciate you effort to my comments so thanks for that. in conclusion, i will prefer to say, "if we have j = rho. v then ..." however looking at this wiki article it will change too many sections. so let's look more on this article.

from reading the continuity article, I understand the implicits (please correct me if you do not agree)

flow and density (of a quantity). -- it would be better enforce those labels along the whole article (or the ones you prefer like the current and the density,...) to ensure a better consistency (nice to have)

where the quanity is implicitly the physical observable

so rewording your "how about" i would try to propose the following based on the above labels

"If on a given volume, the density has a velocity field v which describes the flow then the flow j=rho.v --in other words, if all of the density around a point r and time t, rho(r,t) is moving with a velocity v(r,t)

however in that case you need a reference to proove it ... (it becomes a mathematical proof "if ...then")

reading again the article, i can see there is also the term "flux" used several time and that can contradicts (and it does not point to) the wiki page http://en.wikipedia.org/wiki/Flux. — Preceding unsigned comment added by 91.179.53.212 (talk) 16:03, 17 May 2014 (UTC)

Sorry, I don't really understand what you're saying...

What would "change too many sections"? Velocity is barely mentioned in the article right now. It's just mentioned in the fluid flow section (where it is clearly applicable), and in the section that says "if there is a velocity field...", where it defines exactly what that means.

I don't think the terminology is inconsistent or confusing. Can you clarify? "Flow" is a colloquial term that any English speaker will understand. "Flux" is a specific scientific term which is defined in the article. There is a whole section going through that definition of "flux" in great detail.

You are unhappy that there are no links to the Flux article. Actually, there are a couple of links. But there's no real point in linking. This article already has a thorough definition and explanation of flux, as thorough as the one in flux. I would say it's better. One reason it's better is because the flux article lists multiple definitions of flux, not only the definition that we use in the continuity equation, but also totally unrelated definitions. (Do you agree? You say that they "contradict", but I don't think they do. Why do you say that?) Therefore, a reader trying to understand the continuity equation is much better off learning about flux from the flux section of this article than from the flux article.

Are you complaining that the article uses several context-specific synonyms of "flux"? For example, "heat flow", "current density", etc. The reason for that is, that's how it is in the real world. When you're talking about electricity, everyone calls it "current"; when you're talking about heat, everyone calls it "flow" or "flux". We should use the terminology that everyone else uses, even if it's not as perfectly consistent as we might like. I think the article makes it quite clear that "current density" (for example) is a special case of the general concept of "flux". Do you agree? If you think something like that is not clear, please point it out, I can add more text.

When you say I "need a reference to prove it", it sounds like you are skeptical. Do you believe that the equation j = rho.v is never ever correct??? If not, what do you personally believe? If you just want a reference, you can open up any fluid dynamics book, and you will probably find an explanation of j=rho.v somewhere in the book. --Steve (talk) 20:41, 17 May 2014 (UTC)

Brownian motion and probability continuity

Suggest introducing Brownian motion as a first example of probability continuity, before bringing up quantum mechanics. A Brownian particle is closer to a classical one and its explanation would be more intuitive as to how mass flow becomes probabalized, and this example could then help clarify the QM example. --69.126.41.101 (talk) 15:48, 23 October 2014 (UTC)

I think that's a good idea. I rewrote it a bit, I hope you don't mind. :-D --Steve (talk) 02:05, 24 October 2014 (UTC)

Can we have a better sentence than the following?

<<<This statement does not immediately rule out the possibility that energy could disappear from a field in Canada while simultaneously appearing in a room in Indonesia.>>>

I am not saying that this is a bad sentence. It is perfectly okay when you are describing it to anyone. But I think it doesn't fit at the very beginning of an encyclopedic article. A good (and of course a simple) rephrasing would be preferable.

There are several other instances where it is advisable to revise the literature. Such as,

• nor can it "teleport" from one place to another
• the laws of physics in Brazil are the same as the laws of physics in Argentina. Ahmedafifkhan 07:07, 27 December 2017 (UTC)