Clas Blomberg - Physics of life-Elsevier Science (2007)

224 Part V. Stochastic dynamics
This means that the distribution only depends on the distance to some point. This is the case
if the distribution at the onset is spherical symmetric. It may start out from a particular
point or a spherical region and then spread in all directions. It should also mean that the
system itself is spherical symmetric, for instance a sphere round an original starting point.
When the distribution function only depends on the distance r to a certain centre, the
diffusion equation becomes:
n
⎛ 2 n
D
2n
⎞
t r r r2
⎝⎜
⎠⎟
(22.8)
The normal distribution function in three dimensions, above is a solution and it can be
confirmed by direct calculation.
22B
Diffusion-controlled reactions
We can here go to a situation encountered in the step equation form in the previous chapter,
and which describes a diffusion-controlled reaction, where the system is determined by the
following steps. (A similar model based on discrete steps is considered in Section 20G):
1. The distribution n(r, t) is confined to a spherical region r R. At the boundary, we have
a reflecting condition, which is expressed by the condition: n/r 0 when r R.
2. There is an absorbing region at r r 0 , which represents a strong binding site. As in the
step equations, this corresponds to a condition: n(r 0 , t) 0.
The solution for this situation is similar to that of the discrete equations—Chapter 20.
A spherical symmetric solution of the equation that fulfils the absorbing condition at r r 0
is given by the following expression:
n (,) r t n
k
0
sin( kr ( r))
r
0
Dk2
e
t
(22.9)
Again, this can be confirmed by direct calculation. The sine function is such that n(r 0 ) 0 for
all times, which should be the case at an absorbing boundary. n 0 is a constant that determines
the total density.
This shall fulfil the reflecting condition at the outer boundary, which yields a relation for k;
nr (,) t k⋅ cos( k( R
r0)) sin( kR ( r0))
0
rR
r
R
R2
(22.10)
This can be written as:
tan( kR ( r))
kR
0
(22.11)