User:Brian P. Josey/Notebook/2010/11/02

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Diffusion with Drift
Today I am searching for some information on diffusion with drift. Normally, when you have particles suspended in a solution, they will move around the solute in a random pattern. This is due to Brownian motion, but forces complicate things. When an additional force is added to the system, the particles will have a higher likelihood to move in the direction of the force. For example, if you leave a sample of fine mud in water on a table, the mud will still spread out, but have a gradient of density with a higher density near the bottom of the container.

Back in July, I had spent some time reading through Biological Physics by Philip Nelson, to get acquainted with biophysics as whole, and to look into the things that other people think about. Decent chapter on diffusion and Brownian motion, which I actually took notes on and posted to my notebook, here. In my notes from the reading, there is a very useful equation to describe the motion of a diffusive particle in three dimensions. This equation is:

$$ \langle (\vec {r}_N)^2 \rangle = 6 D t $$

This equation describes the average expected distance traveled as a function of time. The $$ \vec {r} $$ represents the distance traveled, and the subscript N is to denote the number of discrete steps taken. The derivation of this assumes the particle moves in a discrete number of steps of equal size. The t is the elapsed time, and D is a diffusion constant. This constant is found my measuring the step size, L, squared over twice the time over multiple readings. So mathematically:

$$ D= \frac {L^2} {2 \Delta t} $$

This can then be used to measure the concentration of the particles being measured, c, as a function of both space and time. This is just the simple diffusion partial differential equation:

$$ \frac {\partial c} {\partial t} = D \frac {\partial ^2 c}{\partial x ^2} $$

While this is all fairly basic, and commonly talked about, my real interest lie where there are forces working in the system. Normally, Nelson writes up clear descriptions of physical processes, he leaves the diffusion with drift issue out. However, he does have an example that I could potentially use to build up my own derivation. It is the Nernst relation, section 4.6.3, and I will pick it up Thursday.


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