394 12 Random walks and the Metropolis algorithm
12.3.2Continuous Equations
Hitherto we have considered discretized versions of all equations. Our initial probability dis-
tribution function was then given by
wi( 0 ) =δi, 0 ,
and its time-development after a given time step∆t=εis
wi(t) =∑
jW(j→i)wj(t= 0 ).The continuous analog towi( 0 )is
w(x)→δ(x), (12.10)
where we now have generalized the one-dimensional positionxto a generic-dimensional vec-
torx. The Kroeneckerδfunction is replaced by theδdistribution functionδ(x)att= 0.
The transition from a statejto a stateiis now replaced by a transition to a state with
positionyfrom a state with positionx. The discrete sum of transition probabilities can then
be replaced by an integral and we obtain the new distributionat a timet+∆tas
w(y,t+∆t) =∫
W(y,x,∆t)w(x,t)dx,and aftermtime steps we have
w(y,t+m∆t) =∫
W(y,x,m∆t)w(x,t)dx.When equilibrium is reached we have
w(y) =∫
W(y,x,t)w(x)dx.We can solve the equation forw(y,t)by making a Fourier transform to momentum space. The
PDFw(x,t)is related to its Fourier transformw ̃(k,t)through
w(x,t) =∫∞
−∞
dkexp(ikx)w ̃(k,t), (12.11)and using the definition of theδ-function
δ(x) =1
2 π∫∞
−∞dkexp(ikx),we see that
w ̃(k, 0 ) = 1 / 2 π.
We can then use the Fourier-transformed diffusion equation
∂w ̃(k,t)
∂t
=−Dk^2 w ̃(k,t), (12.12)with the obvious solution
w ̃(k,t) =w ̃(k, 0 )exp[
−(Dk^2 t))
=
1
2 π
exp[
−(Dk^2 t)]
.
Using Eq. (12.11) we obtain