4.5. Functions, derivatives, and snakes under the rug[[Student version, December 8, 2002]] 119
c c cxx xabcdc/dt<^0
dc/dt> 0∗∗
0 AFigure 4.11:(Mathematical functions.) (a)Auniform (well-mixed) solution has constant concentrationc(x)of
solute. The graph of a constant functionc(x)isahorizontal line, and so has dc/dx=0and d^2 c/dx^2 =0. (b)The
graph of a linear function,c(x)=ax+b,isastraight line and so has d^2 c/dx^2 =0.Ifthe slope, dc/dx,isnot zero,
then this function represents a uniform concentrationgradient.The dashed lines denote two fixed locations; see the
text. (c)Alump of dissolved solute centered onx=0.The curvature, d^2 c/dx^2 ,isnownegative near the bump, zero
at the points labeled∗,and positive beyond those points. The ensuing flux of particles will be directed outward. This
flux will deplete the concentration in the region between the points labeled with stars, while increasing it elsewhere,
for example at the point labeledA.These fluxes change the distribution from the solid curve at one instant of time
to the dashed curve at a later instant.
T 2 Section 4.4.3′on page 133 mentions a conceptual parallel to quantum mechanics.
4.5 Functions, derivatives, and snakes under the rug
4.5.1 Functions describe the details of quantitative relationships
Before solving the diffusion equation, it’s important to get an intuitive feeling for what the symbols
are saying. Even if you already have the technical skills to handle equations of this sort, take some
time to see how Equation 4.19 summarizes everyday experience in one terse package.
The simplest possible situation, Figure 4.11a, is a suspension of particles that already has
uniform density at timet=0.Becausec(x)isaconstant, Fick’s law says there’s zero net flux.
The diffusion equation says thatcdoesn’t change: A uniform distribution stays that way. In the
language of this book, we can say that it stays uniform because any nonuniformity would increase
its order, and order doesn’t increase spontaneously.
The next simplest situation, Figure 4.11b, is a uniform concentration gradient. The first deriva-
tive dc/dxis the slope of the curve shown, which is a constant. Fick’s law then says there’s a
constant fluxjto the right. The second derivative d^2 c/dx^2 is thecurvatureof the graph, which is
zero for the straight line shown. Thus, the diffusion equation says that once againcis unchanging
in time: Diffusion maintains the profile shown. This conclusion may be surprising at first, but it
makes sense: Every second, the net number of particles entering the region bounded by dashed
lines in Figure 4.11b from the left is just equal to the net numberleavingto the right, socdoesn’t
change.
Figure 4.11c shows a more interesting situation: abumpin the initial concentration at 0. For
example, at the moment when a synaptic vesicle fuses, it suddenly releases a large concentration