# User:Brian P. Josey/Notebook/2010/07/09

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## Ch 4 Random Walks, Friction, and DiffusionContinuing from my notes yesterday. ## Random Walks and Diffusion
Where: -
**x**is the displacement of the marker after_{i}**i**number of steps -
**L**is the length of each step, assumed to be constant for both forward and backwards and each iteration -
**N**is the total number of steps -
**k**is either +1 or -1, representing a movement in the positive or negative direction depending on how the coin lands, notice the last term becomes just L_{N}^{2}in the above equation
This can then be generalized to:
Then if we wait a time - The mean-square displacement increases linearly in time for a one-dimensional random walk given by:
- where the constant
**D**equals L^{2}/(2Δt)
But it is very important to remember that the brackets indicate an average, and any particular example will not conform exactly to the diffusion law, instead the average over a large number of examples will conform to the law. This law then can be generalized for higher dimensions. To do this, x is swapped out with the position vector, and using the fact that the step size in a 2 dimensional diagonal is actually , the diffusion law in two-dimensions is:
and similarly, in three-dimensions it is:
This diffusion law, however, is fairly simplistic, and in the real world random walks don't take discrete steps of uniform size, there is a continuum of step sizes. Fortunately, the
This value,
Then by measuring the variance in the motion, omitted to save space and time, the variance in the mean displacement is found by:
where
If the drift, ## FrictionDiffusion is essentially the result of random fluctuations, and it can be argued that friction is also the result of the same interactions. As a simple example, imagine a body, mass
Since each collision is independent of all other collisions, then each step,
where
This ζ = 6πη and plugging in the relationship between the drift velocity, force and zeta I get:
which has made more than its fair share of appearances in my notebook. Of course,
it then comes down to Einstein's relation: ζ Which all assembled together allows for the mass of molecules to be estimated.
- When I assumed that the collisions wiped away all memory of all older collisions, I was assuming the initial momentum of the particle was comparable to the momentum of the molecules striking it. A bullet fired from a gun will not come to a sudden stop if fired into water simply because one fast moving water molecule hit it in the sweet spot. It will however slow down as many particles hit it.
## More on Random WalksPolymers are interesting, in that they move around and change their shapes depending on the forces acting on them from whatever medium they are suspended in. Like an individual particle moving around in water, a protein, which is a polymer of amino acids gets knocked around by the water it is in, and changes its shape slightly from one moment to the next. A really simplified polymer, one with N number of units connected so that there can be any angle between them, like chains of paperclips, and a constant distance between adjoining units. Here the distance between any two can be thought of as the vector sum of moving from one monomer to the last one in question hitting each of the intervening monomers in the process. The coil size of the polymer then increases proportionally as the square root of the molar mass. Of course, this doesn't work with polymers composed of monomers that are strongly self-attractive, which results in globular proteins and follows a non-random walk pattern. Polymers can be classified as "compact" or "extended" depending on their properties. ## More on DiffusionDiffusion is a vital process on the cellular level for it allows proteins, nutrients and everything else to move through out the cell to supply whatever is needed to different parts of the cell for important life processes. Of course, it is easier for a prokaryote to survive on diffusion than eukaryote like us, which is why we study kinesin in this lab. And diffusion is very simple, and is governed by a single equation. The equation is called "Fick's Law" and is:
Where: -
*j*is measures the net number of particles moving from one side to another, -
*D*is the same factor as above, -
*x*is measure of length, and -
*c*is the number density of particles as a function of distance.
It's important to understand that the particles are not being pushed away by each other, in fact the book assumes that all movement is independent of the location of other particles, but by probability. Simply put, if each particle is allowed to move to the left or right equally, and there are more on the left than the right, it would make more sense that there would be more particles moving from the higher concentration on the left to the right side than there would be going in the opposite direction. The particles don't "want" to be alone, they just end up that way. Extending Fick's law into three dimensions and using vectors you get:
and the concentration as a vector function changes also:
It should be noted that it's not just the number of particles that gets dissipated through the diffusion process. |