If we consider the thermal energy as kinetic energy, then we can write the equation

where the mean square velocity is proportional to the temperature and inversely
proportional to the mass of the particle. When we imagine a bunch of these
particles whizzing around, it is clear that they will be colliding with other
particles quite regularly. With each collision, the velocity and direction of
motion will change. With 10^{23} particles the situation is complicated enough that
we can essentially consider each change in motion of a single particle to be
uncorrelated with all previous collisions. This situation can be modelled as a
random walk. A single particle is started at some point and has equal
probability of moving one step to the left or one step to the right.

Fig. 5.1.1

We’ll call the step length d and the time interval
t (these will be treated as
constants, although they will differ for different particles and different
temperatures). So if we let our particle "walk" for 5 steps, its path might be
RRLRL which would leave it one step to the right of its starting point. This
procedure is isomorphic to tossing a coin, so we can learn something about it by
looking at that situation. For *N* tosses of a fair coin, the probability of *x*
heads appearing is given by the binomial distribution.

Fig. 5.1.2

Naturally the highest probability is given to the situation in which there are an equal number of heads and tails, but there is a spread in the distribution.

Going back to the random walk of a particle, we can set up an equation that tells
us the displacement of the particle after *n* time intervals

If we were to perform a random walk with *N* particles, then the average
displacement would be given by the sum of each individual displacement
divided by *N*

Substituting our earlier expression, we get

The plus/minus term becomes zero, so on average the particle stays where it is. This just shows the obvious fact that the mean displacement is equal to zero (due to the symmetry of equal probability for movement to the right and left). To see how the particles will spread out with time, we can look at the root-mean-square displacement (since the square of a negative number is positive, this will show us how the particles move away from their starting point). First we square the displacement

Then we take the mean

Again the plus/minus term goes to zero, so the mean square displacement is given by

Since *x _{i}*(0) = 0, it follows that <

Since d and t are considered constants, we can replace them with a single constant

This is the diffusion equation, where *D* is the diffusion constant (the reason
for the 2 will become apparent). If we look at the root-mean-square displacement,
we get

So we can see that the spreading of the particle is proportional to the square root of time. This means that to cover twice as much distance, four times as much time will be required.

If we go back to the velocity due to thermal motion

We can substitute this into

to get

Now using the equipartition theorem again, we can see that

Thus allowing one to determine the mean time between collisions by measuring macroscopic quantities. This is essentially what Einstein did to prove the atomic hypothesis and show how we could measure the size of atomic particles. Although we rarely see past the celebrity of relativity and quantum theory, this was surely Einstein’s most fruitful contribution to the physics that people actually do.

Fig. 5.1.3

At time *t* + t, half of the particles at *x*
will move one step to the right, and half of the particles at *x* +
d will move to the left. The net number of
particles crossing the boundary will be

The net flux is the number of particles moving divided by the cross-sectional area per unit time

We can multiply the right side by
d^{2}/d^{2}

We saw before that d^{2}/2t
is the diffusion constant. The number of particles
at a point divided by the area times the length of each point is simply the
number of particles per unit volume. We usually call this the
concentration.

Now, as d becomes very small, the limit of the above equation as d approaches zero is simply the definition of a partial derivative, so we can write it as

This is Fick’s first law and it simply states that the net flux at a point is proportional to the concentration gradient. It is now clear why the factor of 2 was introduced into the definition of the diffusion constant, to make it compatible with Fick’s equation. We can generalize this result to include time as well by assuming that particles are conserved. If we consider a thin box, the concentration change in some time period will be a function of the fluxes at either side of the box.

Fig. 5.1.4

So the number of particles accumulating in the box is given by

On the right side the t’s and the *A*’s cancel and in
the limit in which both t
and d go to zero we’re left with

Substituting Fick’s first law leaves us with Fick’s second law

This is often referred to as the diffusion equation and is perhaps the most significant equation in all of biology. Although it can be written out with just a few symbols, when it is generalized to three dimensions, solutions can be very hard to come by (although the equation itself looks quite elegant--it won't hurt to take a quick peek).

In fact, the best way to obtain solutions is to look in a book called "The Mathematics of Diffusion" by John Crank and hope that he solved it for the situation which you are studying.

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Document last updated Oct. 14, 1998.

Copyright © 1997, Ken Muldrew.