Computational Physics

12.2 The variational Monte Carlo method 381

probability 1− 2 α. This is clearly a Markov process as described inSection 10.3.

We are interested in the probabilityρ(x,t)to find the walker at sitex=naat

timet=mh, wherenandmare both integer. This probability satisfies the master

equation of the Markov process:

ρ(x,t+h)−ρ(x,t)=α[ρ(x+a,t)+ρ(x−a,t)− 2 ρ(x,t)]≈αa^2

∂^2 ρ(x,t)

∂x^2

.

(12.22)

For smallh, the left hand side can be written ash(∂ρ/∂t), and definingγ=a^2 α/h,

we can write the continuum form of the master equation (for smalla)as

∂ρ(x,t)

∂t

=γ

∂^2 ρ(x,t)

∂x^2

. (12.23)

This equation is called thediffusion equation: it describes how the probability

distribution of a walker evolves in time. It may equivalently be interpreted as the

density distribution for a large collection of independent walkers.

Consider the following function:

G(x,y;t)=

1

√

4 πγt

e−(x−y)

(^2) /( 4 γt)

. (12.24)

This function has the following properties:

• Considered as a function ofyandt, keepingxfixed, it is a solution of the
  diffusion equation fort>0.


• Fort→0,Greduces to a delta-function:

G(x,y;t)→δ(x−y)fort→0. (12.25)

Gis called theGreen’s functionof the diffusion equation. This function can be

used to write the time evolution of any initial distributionρ(x,0)of this equation

in integral form:

ρ(y,t)=

∫

dxG(x,y;t)ρ(x,0), (12.26)

which can easily be checked using the properties ofG. Inspection of the Green’s

function shows that it is normalised, that is,

∫

dyG(x,y;t)=1, independent ofx

andt.

The Green’s function can be interpreted as the probability distribution of a single

walker which starts off at positionxatt=0. We can useGto construct a new

Markov process corresponding to the diffusion equation. We discretise the time in

steps t. We start with a walker localised atxatt=0. Then we move this walker

to a new positionyat time twith probability distributionG(x,y; t). From this,