134 Chapter 4. Random walks, friction, and diffusion[[Student version, December 8, 2002]]
4.6.2′ Actually, a wide range of organisms have basal metabolic rates scaling with a power of
body size that is less than three. All that matters for the structure of our argument is that this
“allometric scaling exponent” is bigger than 1.
4.6.4′
- Section 3.2.5 on page 81 mentioned that frictional drag must generateheat. Indeed it’s well
known that electrical resistance creates heat, for example, in your toaster. Using the First Law,
wecan calculate the heat: Each ion passed between the plates falls down a potential hill, losing
potential energyq×∆V.The total number of ions per time making the trip isI/q,sothe power
(energy per time) expended by the external battery is ∆V×I.Using Ohm’s law gives the familiar
formula: power =I^2 R. - The conduction of electricity through a copper wire is also a diffusive transport process, and also
obeys Ohm’s law. But the charge carriers are electrons, not ions, and the nature of the collisions is
quite different from salt solution. In fact, the electrons could pass perfectly freely through a perfect
single crystal of copper; they only bounce off imperfections (or thermally induced distortions) in
the lattice. Figuring out this story required the invention of quantum theory. Luckily your body
doesn’t contain any copper wires; the picture developed above is adequate for our purposes.
4.6.5′
1.Gilbert says: Something is bothering me about Equation 4.27. For simplicity let’s work in just
one dimension. Recall the setup (Section 4.1.2): At timet=0,Irelease some random-walkers at
the origin,x=0.Ashort timetlater the walkers have takenNsteps of lengthL,whereN=t/∆t.
Thennoneof the walkers can be found farther away thanxmax=±NL=tL/∆t. And yet, the
solution Equation 4.27 says that the densityc(x, t)ofwalkers is nonzero for anyx,nomatter how
large! Did we make some error or approximation when solving the diffusion equation?
Sullivan: No, Your Turn 4g showed that it was an exact solution. But let’s look more closely
at the derivation of the diffusion equation itself—maybe what we’ve got is an exact solution to an
approximateequation. Indeed it’s suspicious that we don’t see the step sizeL,nor the time step
∆tanywhere in Equation 4.19.
Gilbert: Now that you mention it, I see that Equation 4.17 replaced the discrete difference of the
populationsNin adjacent bins by aderivative,remarking that this was legitimate in the limit of
smallL.
Sullivan: That’s right. But we took this limitholdingDfixed,whereD=L^2 /(2∆t). So we’re also
taking ∆t→0aswell. At any fixed timet,then, we’re taking a limit where the number of steps is
becoming infinite. So the diffusion equation is an approximate, limiting representation of a discrete
random walk. In this limit, the maximum distancexmax=2Dt/Lreally does become infinite, as
implied by Equation 4.27.
Gilbert: Should we trust this approximation?
Let’s help Gilbert out by comparing the exact, discrete probabilities for a walk ofNsteps to
Equation 4.27, and seeing how fast they converge with increasingN.Weseek the probability that
arandom walker will end up at a positionxafter a fixed amount of timet.Wewant to explore
walks of various step sizes, while holding fixed the macroscopically observable quantityD.
Suppose thatNis even. AnN-step random walk can end up at one of the points (−N),(−N+
2),...,N.Extending the Example on page 101 shows that the probability to take (N+j)/ 2 steps