Computational Physics - Department of Physics

388 12 Random walks and the Metropolis algorithm

0

20

40

60

80

100

0 20 40 60 80 100

σ^2

Time stepst

Fig. 12.3Time development ofσ^2 for a random walker. 100000 Monte Carlo samples were used with the
function ran1 and a seed set to− 1.

-0.04

-0.02

0

0.02

0.04

0 20 40 60 80 100

〈x(t)〉

Time stepst

Fig. 12.4Time development of〈x(t)〉for a random walker. 100000 Monte Carlo samples were used with the
function ran1 and a seed set to− 1.

∂w(x,t)

∂t

≈

w(i,n+ 1 )−w(i,n)

∆t

,

whereas the gradient is approximated as

D

∂^2 w(x,t)

∂x^2

≈D

w(i+ 1 ,n)+w(i− 1 ,n)− 2 w(i,n)

(∆x)^2

,

resulting in the discretized diffusion equation