I have been reading about self-diffusion of a particle here: http://www-hsc.usc.edu/~rfarley/Diffusion-SI-2003.pdf and on page 9 it gives Einstein's relation as

$(\Delta x)^2 = 2 D t$

and I can estimate roughly how far the particle will have diffused after time $t$.

Since this is self-diffusion, the diffusion velocity should be constant, but from the above equation

$\frac{dx}{dt} = \sqrt\frac{D}{2t}$

so the particle diffuses with infinite speed at $t=0$, and the velocity decreases with time.

Clearly my understanding is wrong. Why is it not possible to differentiate the Einstein equation to get the diffusion velocity like this?


"Einstein's relation" is derived from the random walk problem. In the random walk problem in one dimension, we consider an ensemble of particles which have equal probability of moving left or right at any given instant in time. Because of this, as your link says, the average displacement of the particles is zero since approximately equal number of particles will move to the left as to the right. Also, keep in mind that a given particle may move left, then right, then left, then right, and ultimately never go anywhere. However, the mean square displacement will not be zero. Basically, this means that while the mean position of particle ensemble will remain zero, the particles will in fact spread out on either side of x=0 where they begin (as in the picture in your link). Think of this an analogy to mean vs standard deviation of a normal distribution. As you let the clock run, more and more particles will end up getting farther away from x=0 with equal probability on either side, increasing the standard deviation of the distribution but not changing the mean. In any case, this is why you can't simply take the square root and differentiate this equation. This equation describes a statistical property of an ensemble of particles, not the position of a particle as a function of time. Thus, while this equation can be used to estimate how long a particle may take to diffuse a given distance, we can't use it to estimate the velocity of a particle. If we observe a given particle in the process of self-diffusion, it's velocity will actually be changing a lot. It may move in one direction, bump into some other particle, reverse direction, bump into another particle, go back in the same direction, etc.. This random motion is called Brownian Motion. The particle isn't moving with some easily described velocity, but is actually moving randomly and sometimes even stopping instantaneously. I hope that makes sense.


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