# 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 ${\displaystyle dX=\sigma (x,t)dW}$ immediately follows the Fokker-Planck equation ${\displaystyle {\frac {\partial }{\partial t}}\phi (x,t)={\frac {\partial ^{2}}{\partial x^{2}}}\left(D(x,t)\phi (x,t)\right)}$, where ${\displaystyle D(x,t)=\sigma (x,t)^{2}/2}$. — 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

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

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:
${\displaystyle {\frac {\partial \phi }{\partial t}}=\nabla \cdot D(\phi )\nabla \phi ({\vec {r}},t)}$
just the same as this:
${\displaystyle {\frac {\partial \phi }{\partial t}}=D(\phi )\nabla ^{2}\phi ({\vec {r}},t)}$
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
${\displaystyle \nabla \cdot D(\phi )\nabla \phi ({\vec {r}},t)=(\nabla D(\phi ))\cdot (\nabla \phi ({\vec {r}},t))+D(\phi )\nabla ^{2}\phi ({\vec {r}},t).}$
However, if the diffusion coefficient D is a constant, say k, then we do get the equation
${\displaystyle {\frac {\partial \phi }{\partial t}}=k\nabla ^{2}\phi .}$
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)