12.2 Diffusion Equation and Random Walks 383
whereDis the so-called diffusion constant, with dimensionality length^2 per time. If the num-
ber of particles is conserved, we have the continuity equation
∂j(x,t)
∂x=−
∂w(x,t)
∂t,
which leads to
∂w(x,t)
∂t
=D
∂^2 w(x,t)
∂x^2, (12.1)
which is the diffusion equation in one dimension.
With the probability distribution functionw(x,t)dxwe can use the results from the previous
chapter to compute expectation values such as the mean distance
〈x(t)〉=∫∞
−∞xw(x,t)dx,or
〈x^2 (t)〉=
∫∞
−∞x^2 w(x,t)dx,which allows for the computation of the varianceσ^2 =〈x^2 (t)〉−〈x(t)〉^2. Note well that these
expectation values are time-dependent. In a similar way we can also define expectation values
of functionsf(x,t)as
〈f(x,t)〉=∫∞
−∞f(x,t)w(x,t)dx.Sincew(x,t)is now treated as a PDF, it needs to obey the same criteria as discussed in the
previous chapter. However, the normalization condition
∫∞
−∞
w(x,t)dx= 1
imposes significant constraints onw(x,t). These are
w(x=±∞,t) = 0 ∂nw(x,t)
∂xn
|x=±∞= 0 ,implying that when we study the time-derivative∂〈x(t)〉/∂t, we obtain after integration by
parts and using Eq. (12.1)
∂〈x〉
∂t =∫∞
−∞
x
∂w(x,t)
∂t dx=D∫∞
−∞
x
∂^2 w(x,t)
∂x^2 dx,leading to
∂〈x〉
∂t
=Dx
∂w(x,t)
∂x
|x=±∞−D
∫∞
−∞∂w(x,t)
∂x
dx,implying that
∂〈x〉
∂t
= 0.
This means in turn that〈x〉is independent of time. If we choose the initial positionx(t= 0 ) = 0 ,
the average displacement〈x〉= 0. If we link this discussion to a random walk in one dimension
with equal probability of jumping to the left or right and with an initial positionx= 0 , then
our probability distribution remains centered around〈x〉= 0 as function of time. However, the
variance is not necessarily 0. Consider first
∂〈x^2 〉
∂t
=Dx^2 ∂w(x,t)
∂x
|x=±∞− 2 D∫∞
−∞x∂w(x,t)
∂x
dx,