128 Chapter 4. Random walks, friction, and diffusion[[Student version, December 8, 2002]]
times minus the gradient of temperature.^12 Section 5.2.1′on page 166 will give another important
example.
T 2 Section 4.6.4′on page 134 mentions other points about electrical conduction.
4.6.5 Diffusion from a point gives a spreading, Gaussian profile
Let’s return to one dimension, and to the question of time-dependent diffusion processes. Sec-
tion 4.4.2 on page 115 posed the question of finding the full distribution function of particle posi-
tions after an initial density profilec(x,0) has spread out for timet.Certainly if we release many
particles all at one place, we expect the resulting distribution to get broader with time. We might
therefore guess that the solution we seek is a Gaussian; perhapsc(x, t)=? Be−x
(^2) /(2At)
whereA
andBare some constants. This profile has the desired property that its varianceσ^2 =Atindeed
grows with time. But substituting it into the diffusion equation, we find that it is not a solution,
regardless of what we choose forAandB.
Before abandoning our guess, notice that it has a more basic defect: It’s not properly normalized
(see Section 3.1.1 on page 66). The integral
∫∞
−∞dxc(x, t)isthe total number of particles and cannot
change in time. The proposed solution doesn’t have that property.
Your Turn 4g
a. Show that. Then show that the profilec(x, t)=const√t e−x^2 /^4 Dtdoes always maintain the same
normalization. Find the constant, assumingNparticles are present. [Hint: Use the change of
variables trick from the Example on page 68.]
b. Substitute your expression from (a) into the one-dimensional diffusion equation, take the
derivatives, and show that with this correction we do get a solution.
c. Verify that〈x^2 〉=2Dtfor this distribution: It obeys the fundamental diffusion law (Equa-
tion 4.5 on page 104).
The solution you just found is the function shown in Figure 4.12. You can now find the inflection
points, where the concentration switches from increasing to decreasing, and verify that they move
outward in time, as described in Section 4.5.2 and Figure 4.11 on page 119.
The result of Your Turn 4g pertains to one-dimensional walks, but we can promote it to three
dimensions. Letr=(x, y, z). Because each diffusing particle moves independently in all three
dimensions, we can use the multiplication rule for probabilities: The concentrationc(r)isthe
product of three factors of the above form. Thus
c(r,t)=N
(4πDt)^3 /^2
e−r(^2) / 4 Dt
. fundamental pulse solution (4.27)
In this formula the symbolr^2 refers to the length-squared of the vectorr,that is,x^2 +y^2 +z^2.
Equation 4.27 has been normalized to makeN the total number of particles released att=0.
Applying your result from Your Turn 4g(c) tox, y,andzseparately and adding the results recovers
the three-dimensional diffusion law, Equation 4.6.
Weget another important application of Equation 4.27 when we recall the discussion of polymers.
Section 4.3.1 argued that, while a polymer in solution is constantly changing its shape, still its mean-
square end-to-end length is a constant times its length. We can now sharpen that statement to say
that thedistributionof end-to-end vectorsrwill be Gaussian.
(^12) Various versions of this law are sometimes called “Newton’s law of cooling,” or “Fourier’s law of conduction.”