Biological Physics: Energy, Information, Life

136 Chapter 4. Random walks, friction, and diffusion[[Student version, December 8, 2002]]

 2

 1

0

1 1

2

3

t

0

1

2

c

 1

0

1

x

Figure 4.16:(Mathematical functions.) Diffusion from an initial concentration step. Time increases as we move
diagonally downward. The sharp step gradually smooths out, starting from its edges.

sides of the diffusion equation. Similarly, the antiderivativec 2 (x, t)=

∫x
dx′c 1 (x′,t)yields a so-
lution. The latter procedure, applied to the fundamental pulse solution in Your Turn 4g, gives a
new solution describing the gradual smoothing-out of a sharp concentration step; see Figure 4.16.
Mathematicians give the function√^2 π

∫x

0 dx

′e−(x′)^2 the name Erf(x), theerror function.