Computational Physics - Department of Physics

12.2 Diffusion Equation and Random Walks 385

12.2.2Random Walks

Consider now a random walker in one dimension, with probabilityRof moving to the right

andLfor moving to the left. Att= 0 we place the walker atx= 0 , as indicated in Fig. 12.2. The

walker can then jump, with the above probabilities, either to the left or to the right for each

time step. Note that in principle we could also have the possibility that the walker remains

in the same position. This is not implemented in this example. Every step has length∆x=l.

Time is discretized and we have a jump either to the left or to the right at every time step.

Let us now assume that we have equal probabilities for jumping to the left or to the right, i.e.,

• • • • • • • •

.. − 3 l − 2 −l x= 0 l 2 l 3 l ..

Fig. 12.2One-dimensional walker which can jump either to the left or to the right. Every step has length

∆x=l.

L=R= 1 / 2. The average displacement afterntime steps is

〈x(n)〉=

n

∑

i

∆xi= 0 ∆xi=±l,

since we have an equal probability of jumping either to the left or to right. The value of〈x(n)^2 〉

is

〈x(n)^2 〉=

(

n

∑

i

∆xi

)(

n

∑

j

∆xj

)

=

n

∑

i

∆x^2 i+

n

∑

i 6 =j

∆xi∆xj=l^2 n.

For many enough steps the non-diagonal contribution is

N

∑

i 6 =j

∆xi∆xj= 0 ,

since∆xi,j=±l. The variance is then

〈x(n)^2 〉−〈x(n)〉^2 =l^2 n. (12.3)

It is also rather straightforward to compute the variance forL 6 =R. The result is

〈x(n)^2 〉−〈x(n)〉^2 = 4 LRl^2 n.

In Eq. (12.3) the variablenrepresents the number of time steps. If we definen=t/∆t, we can

then couple the variance result from a random walk in one dimension with the variance from

the diffusion equation of Eq. (12.2) by defining the diffusion constant as

D=

l^2

∆t

.

In the next section we show in detail that this is the case.