Question on Drift-Diffusion Equation

[Geometry_dude]

Hi, there! I was pondering on the drift-diffusion equation lately and there are some things I don't understand. I hope that some of you are more knowledgeable on the topic than me and maybe can point me to some literature.

The situation: Fick's first law of diffusion is given by
$$ \vec j _{diff} = -D \, \nabla \rho \, ,$$
where $D$ is the diffusion constant, $\rho$ is the probability density of finding the point mass in a certain region of space and $\vec j _{diff}$ is the probability current density for diffusion. If there is also a drift (e.g. due to a force or because the observer is moving at nonzero constant velocity with respect to the point mass), then the probability current density for that is
$$ \vec j _{drift} =\rho \, \vec v \, ,$$
where $\vec v$ is the drift velocity. Adding the two current densities gives the total current density $\vec j$.
Now, because we want probability conservation, we assume the continuity equation
$$\frac{\partial \rho}{\partial t}+ \nabla \cdot \vec j = 0$$
holds.
Plugging everything into it yields the drift-diffusion equation. So far, so good.

My questions are:
1) In principle one can define a second velocity $\vec u$ via
$$\vec j = \rho \, \vec u \, . $$
I've seen this being done in the literature. Now, is $\vec u$ a velocity that one can measure? What does it mean?

2) What if I don't know $\vec v$? Is there something like Newton's second law that let's me determine it? For instance, what do I do if an external electromagnetic field acts on the mass?

 

[NFuller]

1) In principle one can define a second velocity ⃗uu→\vec u via

⃗j=ρ⃗u.j→=ρu→.​

\vec j = \rho \, \vec u \, .
I've seen this being done in the literature. Now, is ⃗uu→\vec u a velocity that one can measure? What does it mean?

The only way I can see this being valid is if the concentration field $\rho$ is constant. In that case $\vec{j}$ is just the flux density which is equal to the flux from drift $\vec{j}_{drift}$

2) What if I don't know ⃗vv→\vec v? Is there something like Newton's second law that let's me determine it? For instance, what do I do if an external electromagnetic field acts on the mass?

$\vec{v}$ is generally the velocity profile of the fluid carrying the particles along. Finding $v$ requires one to solve the governing equations for the fluid motion; this could be the Navier-Stokes eq., Stokes eq., Eulers eqs., etc.

 

[Geometry_dude]

The issue is, the fluid interpretation is not really what I'm looking for here. In an applied math course I took we derived the 1-dimensional diffusion equation from a statistical model and then generalized it to the drift-diffusion equation (well, I did the generalizing...). We assumed the mass moves on a 1-dimensional lattice and has equal probability of going left or right in one time step. Then we played around a bit and found a way to argue what happens in the continuum model where the lattice spacing and time steps go to zero (depending on each other). So I'm really looking for the probabilistic interpretation, where $\rho$ is the probability density for finding the mass in a certain region of space.

I understand that the drift velocity $\vec v$ really does model an overall drift of the ensemble, but I don't quite get how it all relates to the positions and velocities of the individual masses. Also $\vec u$ does seem to make sense in the general case. Mathematically, it is directly related to the expectation value of the position of the point mass, but I woul like to have a more direct understanding in terms of said quantities.

$\vec{v}$ is generally the velocity profile of the fluid carrying the particles along. Finding $v$ requires one to solve the governing equations for the fluid motion; this could be the Navier-Stokes eq., Stokes eq., Eulers eqs., etc.

Yes, these equations are special cases of Newton's second law in the continuum case! So why are these equations in $\vec v$ and not in $\vec u$?

EDIT: Do you know what the meaning of $\nabla \rho / \rho$ is? I've read that (modulo a constant factor) it is referred to as the osmotic velocity.

 

[Chestermiller]

I think what you want to write is $$\vec{j}=\rho \vec{u}-D\nabla{\rho}$$ where $\vec{j}$ is the total flux resulting from mean convective transport $\vec{u}$ and diffusion.

 

[Lord Jestocost]

In an applied math course I took we derived the 1-dimensional diffusion equation from a statistical model and then generalized it to the drift-diffusion equation (well, I did the generalizing...). We assumed the mass moves on a 1-dimensional lattice and has equal probability of going left or right in one time step. Then we played around a bit and found a way to argue what happens in the continuum model where the lattice spacing and time steps go to zero (depending on each other). So I'm really looking for the probabilistic interpretation, where ρ\rho is the probability density for finding the mass in a certain region of space.

Based upon your writing, I suppose that you are looking for something like "biased random walk" models. See, for example: [PDF]Random walks - University of Lethbridge

 

[Geometry_dude]

Yes, I think you are right. Do you have a good reference where these things are treated rigorously from the point of view of stochastic equations?

I think what you want to write is $$\vec{j}=\rho \vec{u}-D\nabla{\rho}$$ where $\vec{j}$ is the total flux resulting from mean convective transport $\vec{u}$ and diffusion.

Well, yes, I said this above (your $\vec u$ is my $\vec v$), but did not write it out in full:
$$\rho \vec u = \rho \vec v - D \nabla \rho $$
or, equivalently (excluding zeros of $\rho$)
$$\vec u = \vec v - D \frac{\nabla \rho}{\rho} $$

 

[Geometry_dude]

I think I have partially understood the correct interpretation of these quantities and I have also found some references.

The interpretation: Since the velocities of the individual particles are random, $\vec u$, $\vec v$ and $\vec w := -D \nabla \rho / \rho$ do not refer to individual velocities in general, but must be computed from the positions and velocities of all of the particles in the ensemble. $\vec w$ is a quantity that results from the Brownian motion (diffusion), i.e. it indicates a net motion of the particles in the ensemble simply due to the fact that there's more particles in one place than another and they move into each direction with equal probability. One can compare this with the motion of a gas (at rest with respect to the observer) into a region of lower concentration. $\vec v$ is the so called drift velocity and it roughly represents a bias of the overall motion of the particles into a particular direction - e.g. due to a (pseudo-)force acting on each of the particles in the ensemble in the same way (excluding the diffusion). This is also the reason why the force equation must be formulated in terms of $\vec v$.
$\vec u$ is the total velocity of the ensemble, if one combines the two effects. I'm not sure whether it is really the pointwise mean velocity of the particles, but it does govern the time evolution of the probability density via its flow (which is probability preserving).

References for a rigorous treatment of Brownian motion and an introduction to the topic are given in the book "Probability Theory" by Klenke.
He recommends "Brownian Motion" by Mörters and Peres, "Continuous Martingales and Brownian Motion" by Revuz and Yor, as well as "Brownian Motion" by Schilling and Partzsch.