If you are looking at diffusion as descried by the Langevin equation, then the probability distribution function for displacements is Gaussian at all times, and you can characterize it with the mean-square displacement. In physical systems, the Langevin equation is replaced with the Mori equation, and life can readily become much more complicated.

To understand odd results for the diffusion coefficients it is also helpful to have look at the Green-Kubo relation for D. This relates the coefficient with the autocorrelation function of the particle velocity. Time dependency enters the diffusion constant on time scales shorter or equal to the correlation time of the particle motion. The structure of the correlation function will then strongly influence the precise time dependency. And in that case negative outcomes could be possible (anticorrelations in the motion: oscillations?). One could discuss whether or not you can still call this diffusion.

Thank you for your kind replies, Prof. Philies and Prof. Huinink. However, in addition to your answers, i will like to know what feature in a restricted geometry gives rise to D. I was introduced to restricted geometries/porous media years back with articles in which analyses were done with D(t). Meanwhile, in an attempt to recover the models via the Bloch-Torrey (diffusion NMR) equation, i could not get time-dependence of the diffusion coefficient. I wonder what may be responsible for this.

In the process of building the diffusion model, i get negative D; i have used several methods to get rid of the negative but it keeps staring in my face. This is what prompted my question: maybe if the origin of D in restricted geometries is known to me (i have searched literature but got nothing), possibly there may be a logical explanation to why negative appears or my models are wrong!

What NMR experiment are you doing? Normally if you work "pulse-field gradient" type of experiments or stimulated echo's in case of a static gradient, the thing you actually measure is the displacement squared via the phase angle squared. Via the Einstein-relation you then obtain a value for D. It is pretty hard to make something negative out of this. Can you clarify what type of experiment you have in mind?

Even in case of geometrical restrictions I would not expect a sign change.

Actually i am working on computational models based on PFG (pulsed field gradient) experiments. I am actually required to solve the Bloch-Torrey equation analytically in which the solutions can work very easily with PFG.

The problem is that the negative D leads to a very scary implication; a complex velocity (D=ivK, where K is a constant, v is the velocity and i is the complex notation). Is this even possible at all?

Michael, I can advise you to read the chapter on dynamics from Chandler:

http://www.amazon.com/Introduction-Modern-Statistical-Mechanics-Chandler/dp/0195042778

I teach it my students and it is very instructive to understand what should really end up in a diffusion coefficient. It connects the Langevin equation (see comments of George), the Einstein result for diffusion (R^2=6Dt) and the Green-Kubo relations (see my own remark). It also clarifies the comment of Robin: D~ and not given that v is the particle velocity with respect to an average motion induced by an external field.

Thank you all for your contributions. They have all been very helpful to me. Thanks.

Prof. Huinink, thanks for the link

In simple case of diffusion, negative D is the same as time inversion. So it may have something to do with time arrow and entropy production. Or any circumstance under which system is not "diffusive" but behaves other way around (anti-diffusion). It doesn't apply in your case if you are looking for analytic solution of those equations - but it's one interpretation of negative D.

Thank you Prof Kuklik. Can you kindly explain your response please. Does anti-diffusion exist in real systems (like biological system)?

Please have a look at paper "Negative diffusion coefficient in a two-dimensional lattice-gas system with attractive nearest-neighbor interactions" as example of this concept. In general, negative diffusion coefficient would denote process of "concentration" as opposed to diffusion. Thus, entropy is decreasing. That means, it's not a random walk problem but additional forces are acting opposing to diffusion process resulting in concentration. I think in biological systems it would be more correct to talk about transport phenomena rather than anti-diffusion.

Michael, Idon't understand why are you trying to derive time dependent diffusion coefficient in your original question. For a system with fixed configuration and temperature, the diffusion coefficient should be constant.

The statement that PFGNMR measures the mean-square displacement, which goes back to Tanner and co-author whose name will doubtless come to mind as soon as I finish typing, corresponds exactly to the statement that the particle displacement is a Gaussian random Markoff process and, thanks to Doob's theorem, that the relaxation is a simple exponential. If the relaxation is not a simple exponential, then PFGNMR is not giving you the mean-square displacement, perhaps by orders of magnitude. I have a paper on this now being reviewed at Phys Rev E. The figures are a bit hefty, so I have not put it up at Arxiv.org yet, but I did extensive computer simulations on Mori-Zwanzig diffusion.

Dr. Wang, I am actually looking at the influence of changes of fluid flow in tissues around the boundaries between normal and diseased conditions. In this case, the diffusion coefficient is likely to be D(x). As you rightly pointed out, the problem we are trying to solve has complex configuration and this has been the challenge!

Thank you Prof. Philies. I guess my challenge could be the attempt to force exponential relaxation behavior on the system. Your response is quite enlightening. I would really love to have your article when it is published!

I got the following assignment, Kindly guide me in this regard.

Solve the following differential equation for transport of f(x,y,z,t) by MS Excel

∂f/∂t+Ux ∂f/∂x+Uy ∂f/∂z+Uz ∂f/∂z=K((∂^2 f)/(∂x^2 )+(∂^2 f)/(∂y^2 )+(∂^2 f)/(∂z^2 ))+ σf+g(x,y,z,t)

1. You don’t have to consider all the terms of above equation.

Example 1. Considering only diffusion & Production terms, and setting g(x,y,z,t)= C(constant), then

(∂^2 f)/(∂x^2 )+(∂^2 f)/(∂y^2 )+(∂^2 f)/(∂z^2 )=C/k (Poisson Equation)

Example 2. Considering only time derivative and diffusion terms, then

∂f/∂t =k ((∂^2 f)/(∂x^2 )+(∂^2 f)/(∂y^2 )+(∂^2 f)/(∂z^2 )) (Diffusion Equation)

Of course you can consider all the terms. And also you may solve more than two equations.

2. You can use any numerical schemes

3. You can set initial and boundary conditions freely

4. You have to consider the physical meaning of the equation and condition you solve. (Ex. Diffusion of substances, thermal conduction etc.)

5. You have to do presentation about:

Solved equation, initial & boundary condition with the physical meaning you considered

Used numerical schemes( if possible, comparison of numerical schemes)

Calculation results and discussion

Negative diffusion is always associated with attraction or adhesion of particles. Several biological models are relevant to negative diffusion However, a proper derivation requires the correction of the 4th order spatial derivative, that stabilizes the equation. This in physics corresponds to the infamous Cahn-Hilliard equation https://en.wikipedia.org/wiki/Cahn–Hilliard_equation

As you will be able to observe, for certain parameter constellation, the Cahn-Hilliard equation produces aggregation/clustering patterns. and this is exactly the effect of negative diffusion.

Dear brothers,

I have also in a recent work come a cross a negative diffusion constant and couldn't give it a meaning, that is when the electric field in certain medium diffuses with a diffusion constant= 1/(conductivity time permeability)....Could any body help me what is happening over there...