12.3 Microscopic Derivation of the Diffusion Equation 391
Using the binomial formula
n
∑
k= 0
(
n
k)
aˆkbˆn−k= (a+b)n,ee we have that the transition matrix afterntime steps can be written as
Wˆn(nε)) =^1
2 nn
∑
k= 0(
n
k)
RˆkLˆn−k,or
Wˆn(nε)) =^1
2 nn
∑
k= 0(
n
k)
Lˆn−^2 k=^1
2 nn
∑
k= 0(
n
k)
Rˆ^2 k−n,and usingRmi j=δi,(j+m)andLmi j=δ(i+m),jwe arrive at
W(il−jl,nε) =
1
2 n(
n
1
2 (n+i−j))
|i−j|≤n
0 else, (12.7)
andn+i−jhas to be an even number. We note that the transition matrix for a Markov process
has three important properties:
- It depends only on the difference in spacei−j, it is thus homogenous in space.
- It is also isotropic in space since it is unchanged when we gofrom(i,j)to(−i,−j).
- It is homogenous in time since it depends only the difference between the initial time
and final time.
If we place the walker atx= 0 att= 0 we can represent the initial PDF withwi( 0 ) =δi, 0.
Using Eq. (12.4) we have
wi(nε) =∑
j(Wn(ε))i jwj( 0 ) =∑
j1
2 n(
n
1
2 (n+i−j))
δj, 0 ,resulting in
wi(nε) =1
2 n(
n
1
2 (n+i))
|i|≤n.We can then use the recursion relation for the binomials
(
n+ 1
1
2 (n+^1 +i)
)
=
(
n
1
2 (n+i+^1 ))
+
(
n
1
2 (n+i−^1 ))
(12.8)to obtain the discretized diffusion equation. In order to achieve this, we definex=il, wherel
andiare integers, andt=nε. We can then rewrite the probability distribution as
w(x,t) =w(il,nε) =wi(nε) =1
2 n(
n
1
2 (n+i))
|i|≤n,and rewrite Eq. (12.8) as
w(x,t+ε) =^1
2
w(x+l,t)+^1
2
w(x−l,t).Adding and subtractingw(x,t)and multiplying both sides withl^2 /εwe have