116 Chapter 4. Random walks, friction, and diffusion[[Student version, December 8, 2002]]
x−^32 Lx−^12 Lx+^12 LLXYZxyzLabFigure 4.10:(Schematic.) Simultaneous diffusion of many particles in three dimensions. For simplicity we consider
adistribution uniform inyandzbut nonuniform inx,and subdivide space into imaginary bins centered atx−L,
x,x+L,...The planes labeled “a,” “b” represent the (imaginary) boundaries between these bins. X,Y,andZ
denote the overall size of the system.
der, log its position at various timesti,then repeat the experiment and compute the probability
distributionP(x, t)dxusing its definition (Equation 3.3 on page 67). But we have already seen in
Section 4.4.1 that an alternative approach is much easier in practice. If we simply release atrillion
random walkers in some initial distributionP(x,0), then monitor theirdensity,we’ll find the later
distributionP(x, t), automatically averaged for us over those trillion independent random walks,
all in one step.
Imagine, then, that we begin with a three-dimensional distribution that is everywhere uniform in
they, zdirections but nonuniform inx(Figure 4.10). We again simplify the problem by supposing
that, on every time step ∆t,every suspended particle moves a distanceLeither to the right or to
the left, at random (see Section 4.1.2). Thus about half of a given bin’s population hops to the left,
and half to the right. And more will hop from the slot centered onx−Lto the one centered onx
than will hop in the other direction, simply because there were more atx−Lto begin with.
LetN(x)bethe total number of particles in the slot centered atx,andY, Zthe widths of the
box inthey, zdirections. Thenetnumber of particles crossing the bin boundary “a” from left to
right is then the difference betweenNevaluated at two nearby points, or^12
(
N(x−L)−N(x))
;we
count the particles crossing the other way with a minus sign.
Wenow come to a crucial step: The bins have been imaginary all along, so we can, if we choose,
imagine them to be very narrow. But the difference between a function, likeN(x), at two nearby
points isLtimes the derivative ofN:
N(x−L)−N(x)→−L
d〈N(x)〉
dx. (4.17)
The point of this step is that we can now simplify our formulas by eliminatingLaltogether, as
follows.
The number density of particles,c(x), is just the numberN(x)inaslot, divided by the volume
LY Zof the slot. Clearly, the future development of the density won’t depend on how big the box