4.1. Brownian motion[[Student version, December 8, 2002]] 105
250500750100012501500200 300 400 500 600 700100200300400500600700jj〈(
xj(^2) )
〉
(xj
(^2) )
ab
100 200 300 400 500 600 700 100
Figure 4.5:(Mathematical functions.) (a)Squared deviation (xj)^2 for a single, one-dimensional random walk of
700 steps. Each step is one unit long. The solid line showsjitself; the graph shows that (xj)^2 is not at all the same
asj.(b)As(a), but this time the dots represent theaverage〈(xj)^2 〉overthirty such walks. Again the solid line
showsj.This time〈(xj)^2 〉does resemble the idealized diffusion law (Equation 4.4).
Returning to the physics of Brownian motion, our result means that even if we cannot see
the elementary steps in our microscope, we can nevertheless confirm Idea 4.5a and measureD
experimentally: Simply note the initial position of a colloidal particle, wait a timet,note the final
position, and calculatex^2 / 2 t.Repeat the observation many times; the average ofx^2 / 2 tgivesD.
The content of Idea 4.5a is that the value ofDthus found will not depend on the elapsed timet.
Once we measureD,Idea 4.5b lets us relate it to the microscopic, unobservable parameters
Land ∆t. Unfortunately, we cannot solve one equation for two unknowns: Just measuringDis
not enough to find specific values for either one of these parameters. We need a second relation
relatingLand ∆tto some observation, so that we can solve two equations for the two unknowns.
Section 4.1.4 will find the required additional relation.
Wecan extend all these ideas to two or more dimensions (Figure 4.2). For a walk on a two-
dimensional checkerboard with squares of sideL,westill defineD=L^2 /2∆t.Now,however,each
step is a diagonal, and so has lengthL
√
- Also, the positionrNis a vector, with two components
xNandyN.Thus〈(rN)^2 〉=〈(xN)^2 〉+〈(yN)^2 〉=4Dtis twice as great as before, since each term
on the right separately obeys Idea 4.5a. Similarly, in three dimensions we find
〈(rN)^2 〉=6Dt. diffusion in three dimensions (4.6)It may seem confusing to keep track of all these different cases. But the important features about
the diffusion law are simple: In any number of dimensions, mean-square displacement increases
linearly with time, so the constant of proportionalityDhas dimensionsL^2 T−^1. Remember this,
and many other formulas will be easy to remember.
4.1.3 The diffusion law is model-independent
Our mathematical treatment of the random walk made some drastic simplifying assumptions. One
may well worry that our simple result, Idea 4.5, may not survive in a more realistic model. This
subsection will show that on the contrary,the diffusion law is universal—it’s independent of the
model, as long as we have some distribution of random, independent steps.