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.
 * &mdash; 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 &phi;) 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: 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)


 * This new section appears to be a direct cut and paste of the cite It was added by 01:35, 10 April 2024‎ User:Mohammad alsalamat (talk | contribs) and so is a self-cite (which, in this case, I think is fine cause this is basic stuff.) The blog-post itself points to several textbooks on pharma. Adding them here seems ... wrong; citing pharma textbooks for physics articles goes in the wrong direction. 67.198.37.16 (talk) 19:21, 18 May 2024 (UTC)
 * Thanks for that information. However I don't see how it solves this issue. The content is self-generated, not independently verifiable. Johnjbarton (talk) 21:13, 18 May 2024 (UTC)
 * Well, if you take the very first eqn $$\mathbf{\mu}={\mu_0}- RT\ln(c)$$, as a given, the rest of the section follows easily by conventional algebra (yes, I just checked). And doing conventional textbook-level algebra in wikipedia pages is not OR or SYNTH, its just "explaining things" (and this section is not a bad explanation, its pretty decent). The only sticky part would be the very first eqn, which cannot be found verbatim in the article on chemical potential, nor can it be found in grand canonical ensemble, written as it currently is. However, I believe it is entirely correct (just take the log of the grand canonical ensemble, and divide by N to get the concentration.). I can't look it up, my chemistry books (just like yours) disappeared more than a few decades ago. Perhaps it would be better to skip the first two eqns, and just quote the third? That would make it fit into the conventional differential paradigm used in thermo. On the other hand, it is exactly integrable, so no need to start with the differential form. Actually, I'm not sure. Certainly, things change as a flame front or other chemical reaction pass through. So, yeah, ha ha (nervous laughter) actually having an actual reference would be better. Surely this stuff is generally available on-line. Don't force me to do search-engine work. 67.198.37.16 (talk) 23:11, 18 May 2024 (UTC).
 * Ah, whatever. Looks like this brand-new content is almost a verbatim copy of what is already in Fick's laws of diffusion. That content dates back to before 2012 (which is as far as I looked). I'm not going to fine-tooth-comb it, to make sure these two articles are fully consistent. I will remove the ref, as it was clearly created by synthesizing existing WP articles. 67.198.37.16 (talk) 23:46, 18 May 2024 (UTC)
 * So Circular_reporting and we are back to where we were. Johnjbarton (talk) 00:27, 19 May 2024 (UTC)

stochastic diffusion eqn
While reading spinodal decomposition, I find an equation which I think is called the "stochastic diffusion equation", written as


 * $$\partial_t\phi=\nabla ( m\nabla\mu + \xi(x) )\;,$$


 * where $$m$$ is the diffusive mobility, $$\xi(x)$$ is some random noise such that $$\langle\xi(x)\rangle=0$$, and the chemical potential $$\mu$$ is derived from the Landau free-energy...

I believe that this is entirely correct. The main issue is that diffusive mobility is a red link, which makes the relation to the eqn here harder than it should be. The additional noise term is also a "convetional modification", to create a stochastic differential equation out of the usual PDE of this article; I'd like to see the details. 67.198.37.16 (talk) 00:21, 19 May 2024 (UTC)