114 Chapter 4. Random walks, friction, and diffusion[[Student version, December 8, 2002]]
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1020304050frequencymonthly return, %Figure 4.9:(Experimental data.) Ubiquity of random walks. The distribution of monthly returns for a 100-security
portfolio, January 1945–June 1970. [Data from (Malkiel, 1996).]
contains no useful information that will enable an investor consistently to beat other investors.
If this idea is correct, then some of our results from random-walk theory should show up in
financial data. Figure 4.9 shows the distribution of step sizes taken by the market value of a stock
portfolio. The value was sampled at one-month intervals, over 306 consecutive periods. The graph
indeed bears a strong resemblance to Figure 4.3. In fact, Section 4.6.5 below will argue that the
distribution of step sizes in a random walk should be a Gaussian, approximately as seen in the
figure.
4.4 More about diffusion
4.4.1 Diffusion rules the subcellular world
Cells are full of localized structures; “factory” sites must transport their products to distant “cus-
tomers.” For example, mitochondria synthesize ATP, which then gets used throughout the cell. We
may speculate that thermal motion, which we have found is a big effect in the nanoworld, somehow
causes molecular transport. It’s time to put this speculation on a firmer footing.
Suppose we look at one colloidal particle—perhaps a visible pollen grain—every 1/30 second, the
rate at which an ordinary video camera takes pictures. An enormous number of collisions happen
in this time, and they lead to some net displacement. Each such displacement is independent of the
preceding ones, just like the successive tosses of a coin, because the surrounding fluid is in random
motion. It’s true that the steps won’t be all the same length, but we saw in Section 4.1.3 that
correcting this oversimplification complicates the math but doesn’t change the physics.
With enough patience, one can watch a single particle for, say, one minute, note its displacement
squared, then repeat the process enough times to get the mean. If we start over, this time using
two-minute runs, the diffusion law says that we should get a value of〈(xN)^2 〉twice as great as
before, and we do. The actual value of the diffusion constantDneeded to fit the observations the
diffusion law (Idea 4.6) will depend on the size of the particle and the nature of the surrounding