4.6. Track 2[[Student version, December 8, 2002]] 135
-2 0 2 40.10.20.30.4P(x)xa
-2 0 2 40.10.20.30.4P(x)xb
Figure 4.15:(Mathematical functions.) The discrete binomial distribution forNsteps(bars),versus the corre-
sponding solution to the diffusion equation(curve).Ineach case the random walk under consideration had 2Dt=1
in the arbitrary units used to expressx;thusthe curve is given by (2π)−^1 /^2 e−x^2 /^2 .The discrete distribution has
been rescaled so that the area under the bars equals one, for easier comparison to the curves. (a)N=4. (b)N=14.
to the right (and hence (N−j)/ 2 steps left), ending upjsteps from the origin, is
Pj=N!
2 N
(N+j
2)
!
(N−j
2)
!
. (4.28)
Such a walk ends up at positionx=jL.Weset the step sizeLbyrequiring a fixed, givenD:
Noting that ∆t=t/NandD=L^2 /2∆tgives thatL=
√
2 Dt/N.Thus, if we plot a bar of width
2 Land heightPj/(2L), centered onx=jL,then the area of the bar represents the probability
that a walker will end up atx.Repeating for all even integersjbetween−Nand +Ngives a bar
chart to be compared to Equation 4.27. Figure 4.15 shows that the approximate solution is quite
accurate even for small values ofN.
Strictly speaking, Gilbert is right to note that the true probability must be zero beyondxmax,
whereas the approximate solution (Equation 4.27) instead equals (4πNdiffust)−^1 /^2 e−(xmax)
(^2) /(4Dt)
.
But the ratio of this error to the peak value ofP,(4πNdif f ust)−^1 /^2 ,ise−N/^2 ,which is already
less than 1% whenN=10.
Similar remarks apply to polymers: TheGaussian modelof a polymer mentioned at the end of
Section 4.6.5 gives an excellent account of many polymer properties. We do need to be cautious,
however, about using it to study any property that depends sensitively on the part of the distribution
representing highly extended molecular conformations.
Your Turn 4h
Instead of graphing the explicit formula, use Stirling’s approximation (see the Example on
page 101) to find the limiting behavior of the logarithm of Equation 4.28 whenN→∞,holding
x, t,andDfixed. Express your answer as a probability distributionP(x, t)dx,and compare to
the diffusion solution.
- Once we’ve found one solution to the diffusion equation, we can manufacture others. For ex-
ample, ifc 1 (x, t)isone solution, then so isc 2 (x, t)=dc 1 /dt,aswesee by differentiating both