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## Main Question or Discussion Point

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 lets me determine it? For instance, what do I do if an external electromagnetic field acts on the mass?

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 lets me determine it? For instance, what do I do if an external electromagnetic field acts on the mass?