106 Chapter 4. Random walks, friction, and diffusion[[Student version, December 8, 2002]]
Forsimplicity we’ll continue to work in one dimension. (Besides being mathematically simpler
than three dimensions, the 1d case will be of great interest in Section 10.4.4.) Suppose that our
checker makes steps of various lengths. We are given a set of numbersPk,the probabilities of taking
steps of lengthkL,wherekis an integer. The lengthkjof stepjcan be positive or negative, for
forward or backward steps. We assume that the relative probabilities of the various step sizes are
all the same for each step (that is, each value ofj). Letubethe average value ofkj:
u=〈kj〉=∑
kkPk. (4.7)uis a form of average drift motion superimposed on the random walk. The analysis of the preceding
subsection corresponds to the special caseP± 1 =^12 ,with all the otherPk=0.Inthis caseu=0.
The mean position of the walker is now:
〈xN〉=〈xN− 1 〉+L〈kN〉=〈xN− 1 〉+uL=NuL. (4.8)Toget the last equality, we noticed that a walk ofNsteps can be built one step at a time; after
each step the mean displacement grows byuL.
The mean displacement is not the whole story: We know from our earlier experience that
diffusion concerns thefluctuationsabout the mean. Accordingly, let us now compute the variance
(or mean-square deviation, Equation 3.11) of the actual position about its mean. Repeating the
analysis leading to Equation 4.3 gives that
variance(xN) ≡〈(xN−〈xN〉)^2 〉=〈(xN− 1 +kNL−NuL)^2 〉
= 〈(
(xN− 1 −u(N−1)L)+(kNL−uL)) 2
〉
= 〈(xN− 1 −u(N−1)L)^2 〉+2〈(xN− 1 −u(N−1)L)(kNL−uL)〉+L^2 〈(kN−u)^2 〉.
(4.9)Just as before, we now recall thatkL,the length of theNthstep, was assumed to be a random
variable, statistically independent of all the previous steps. Thus the middle term of the last formula
becomes 2L〈xN− 1 −u(N−1)L〉〈kN−u〉,which is zero by the definition ofu(Equation 4.7). Thus
Equation 4.9 says that the variance ofxNincreases by a fixed amount on every step, or
variance(xN)=〈(xN− 1 −〈xN− 1 〉)^2 〉+L^2 〈(kN−〈kN〉)^2 〉
=variance(xN− 1 )+L^2 ×variance(k).AfterNsteps the variance is thenNL^2 ×variance(k). Supposing the steps to come every ∆t,so
thatN=t/∆t,then gives
variance(xN)=2Dt. (4.10)
In this formula,D= L
2
2∆t×variance(k). In the special case whereu=0(no drift), Equation 4.10
just reduces to our earlier result, Idea 4.5a!
Thus the diffusion law (Idea 4.5a) is model-independent. Only the detailed formula for the
diffusion constant (Idea 4.5b) depends on the microscopic details of the model.^3 Such universality,
whenever we find it, gives a result great power and wide applicability.
(^3) T 2 Section 9.2.2′on page 340 will show that similarly, the structure of the three-dimensional diffusion law
(Equation 4.6) does not change if we replace our simple model (diagonal steps on a cubic lattice) by something more
realistic (steps in any direction).