Talk:Diffusion equation

Diffusion in inhomogeneous environment correct?

The article Fick's law and Fokker–Planck equation in inhomogeneous environments examines the validity of different versions of the diffusion equation. In particular, it looks at the Fick's law variant (as presented in the current version on wikipedia), a Fokker-Planck equation approach, and a Master equation. The article shows that the Fick's law is the least accurate description for inhomogeneous diffusion constants. Therefore, the current version of this article should be changed.

I think the bigger picture can be made much more consistent if the diffusion equation is derived via Ito calculus. From the Ito SDE $dX=\sigma (x,t)dW$ immediately follows the Fokker-Planck equation ${\frac {\partial }{\partial t}}\phi (x,t)={\frac {\partial ^{2}}{\partial x^{2}}}\left(D(x,t)\phi (x,t)\right)$, where $D(x,t)=\sigma (x,t)^{2}/2$. — Preceding unsigned comment added by 141.84.42.172 (talk) 17:24, 27 November 2015 (UTC)

Another form

Feynman gives the diffusion equation (Volume II 3-4) as

${\frac {\partial \phi }{\partial t}}=k\nabla ^{2}\phi$

Is this equivalent? Or should it be added? It is more understandable to me at a high school level.

I think we should do this too, it's much more recognizable. Isn't this:

${\frac {\partial \phi }{\partial t}}=\nabla \cdot D(\phi )\nabla \phi ({\vec {r}},t)$

just the same as this:

${\frac {\partial \phi }{\partial t}}=D(\phi )\nabla ^{2}\phi ({\vec {r}},t)$

In that case, the latter form is much preferred. For example, this is how Diffusion equation at scienceworld.wolfram.com puts it.

— Sverdrup 23:50, 18 November 2005 (UTC)

They are not equivalent, since

$\nabla \cdot D(\phi )\nabla \phi ({\vec {r}},t)=(\nabla D(\phi ))\cdot (\nabla \phi ({\vec {r}},t))+D(\phi )\nabla ^{2}\phi ({\vec {r}},t).$

However, if the diffusion coefficient D is a constant, say k, then we do get the equation

${\frac {\partial \phi }{\partial t}}=k\nabla ^{2}\phi .$

The latter equation is treated at heat equation.

In fact, the case where D is constant (or at least independent of φ) is very common. So it might be better to redirect diffusion equation to heat equation and move this article to nonlinear diffusion equation. -- Jitse Niesen (talk) 20:39, 20 November 2005 (UTC)

Missing \cdot

Dear Sir

In the first equation after: "The equation is usually written as: ....

is missing a dot after the nabla: "...= nabla . ( D( ..."

"Nabla dot" is the divergency.

The italian version for "Diffusion equation" is correct. It has the "dot".

150.163.46.38 23:25, 25 May 2007 (UTC) Ivan J.Kantor

You are completely right. Thanks for bringing this to our notice. I now fixed it. -- Jitse Niesen (talk) 05:10, 27 May 2007 (UTC)

Merge with Fick's second law

As far as I can see Fick's second law and the diffusion equation are the same equation, therefore shouldn't the articles be merged? Eraserhead1 15:27, 12 July 2007 (UTC)

References and citations

I need to use this page for scholarly purposes. The article, as is, is naked. There are no substantial references and citations. Were one to arrive on the page and one was not already an expert then one would despair :-) I propose to try to make the article more accessible.--Михал Орела (talk) 08:34, 4 January 2009 (UTC)

All the papers I'm currently reading seem to cite this one particular source: "[Ish78] ISHIMARU A.: Wave Propagation and Scattering in Random Media. Academic Press, 1978. 1, 2" -Krackpipe (talk) 21:37, 11 April 2010 (UTC)

what was not known prior to einstein?

which part is einstein's work and which is old knowledge by Fick, Brown, Maxwell, Sutherland? — Preceding unsigned comment added by 134.7.248.206 (talk) 03:47, 6 May 2018 (UTC)

About anisotropic diffusion

In the Statement section, I understand the first and third equations but not the middle one

I fail to understand why one might need to represent the diffusion coefficient as a matrix. What would such a matrix (and its components) represent specifically? When does it arise, in contrast with the usual form of the equation?

I checked out the anisotropic diffusion article: the coefficient seems to be a scalar and there is no mention of a need of a matrix. — Preceding unsigned comment added by 70.91.187.165 (talk) 07:04, 26 June 2018 (UTC)

History section

The history section was largely material from, and in some part direct copy, of this article: Okino, Takahisa (2015). "Mathematical Physics in Diffusion Problems". Journal of Modern Physics. 06 (14): 2109–2144. doi:10.4236/jmp.2015.614217. ISSN 2153-1196.{{cite journal}}: CS1 maint: unflagged free DOI (link) It describs the author's own research and I can find no independent citations of that paper. I've removed the section, reverting it to the previous version from Feb 2016. StarryGrandma (talk) 21:40, 14 February 2019 (UTC)

Textbook reference for "The thermodynamic view"

The recently added "The thermodynamic view" looks like a textbook content and it needs a textbook reference. Johnjbarton (talk) 02:34, 10 April 2024 (UTC)