118 Chapter 4. Random walks, friction, and diffusion[[Student version, December 8, 2002]]
find:^8
dc
dt
=D
d^2 c
dx^2. diffusion equation (4.19)
In more advanced texts, you will see the diffusion equation written as∂c∂t=D∂(^2) c
∂x^2. The curly
symbols are just a stylized way of writing the letter “d,” and they refer to derivatives, as always.
The∂notation simply emphasizes that there is more than one variable in play, and the derivatives
are to be taken by wiggling one variable while holding the others fixed. This book will use the more
familiar “d” notation.
T 2 Section 4.4.2′on page 133 casts the diffusion equation in vector notation.
4.4.3 Precise statistical prediction of random processes
Something magical seems to have happened. Section 4.4.1 started with the hypothesis of random
molecular motion. But the diffusion equation (Equation 4.19) isdeterministic;that is, given the
initial profile of concentrationc(x,0), we can solve the equation andpredict the futureprofilec(x, t).
Did we get something from nothing? Almost—but it’s not magic. Section 4.4.2 started from the
assumption that the number of random-walking particles, and in particular the number in any one
slice, was huge. Thus we have a large collection of independent random events, each of which can
take either of two equally probable options, just like a sequence of coin flips. Figure 4.3 illustrates
how in this limit the fraction taking one of the two options will be very nearly equal to 1/2, as
assumed in the derivation of Equation 4.19.
Equivalently, we can consider a smaller number of particles, but imagine repeating an observation
on them many times and finding the average of the flux over the many trials. Our derivation can be
seen as giving this average flux,〈j(x)〉,interms of the average concentration,c(x)=〈N(x)〉/(LY Z).
The resulting equation forc(x)(the diffusion equation) is deterministic. Similarly, a deterministic
formula for the squared displacement (the diffusion law, Equation 4.5 on page 104) emerged from
averaging many individual random walks (see Figure 4.5).
When we don’t deal with the ideal world of infinitely repeated observations, we should expect
some deviation of actual results from their predicted average values. Thus for example the peak
in the coin-flipping histogram in Figure 4.3c is narrow, but not infinitely narrow. This deviation
from the average is calledstatistical fluctuation.For a more interesting example, we’ll see that
the diffusion equation predicts that a uniformly mixed solution of ink in water won’t spontaneously
assemble itself into a series of stripes. Certainly thiscouldhappen spontaneously, as a statistical
deviation from the behavior predicted by the diffusion equation. But for the huge number of
molecules in a drop of ink spontaneous unmixing is so unlikely that we can forget about the
possibility. (Section 6.4 on page 182 will give a quantitative estimate.) Nevertheless, in a box
containing justtenink molecules, there’s a reasonable chance of finding all of them on the left-hand
side, a big nonuniformity of density. The chance is 2−^10 ,or≈ 0 .1%. In such a situation, the average
behavior predicted by the diffusion equation won’t be very useful in predicting what we’d see: The
statistical fluctuations will be significant, and the system’s evolution really will appear random, not
deterministic.
So we need to take fluctuations seriously in the nanoworld of single molecules. Still, there are
many cases in which we study large enough collections of molecules for the average behavior to be
agoodguide to what we’ll actually see.
(^8) Some authors call Equation 4.19 “Fick’s second law.”