5.1. INTUITIVE INTRODUCTION TO DIFFUSIONS 159
Sincevk,nis the probability of findingTn·δbetweenk·δand (k+2)·δ, and
since this interval has length 2·δwe can say that the ratiovk,n/(2δ) measures
locally the probability per unit length, that is the probability density. The
last relation above implies that the ratiovk,n/ntends to
v(t,x) = ((2·δ)/(√
2 ·πDt)) exp(−[x−ct]^2 /(2Dt))It follows by the definition of integration as the sums of quantities repre-
senting densities times geometric lengths or areas, that sums of probabilities
vk,ncan be approximated by integrals and the result may be restated as
P[a < Tnδ < b]→(1/(√
2 πDt))∫baexp(−(x−ct)^2 /(Dt))The integral on the right may be expressed in terms of the standard normal
distribution function.
Note that we derived the limiting approximation of the binomial distri-
bution
vk,n∼((2δ)/(
√
2 πDt)) exp(−[x−ct]^2 /(2Dt))by applying the general form of the Central Limit Theorem. However, it
is possible to derive this limit directly through careful analysis. The direct
derivation is known as the DeMoivre-Laplace Theorem and it is the most
basic form of the Central Limit Theorem.
Differential Equation Solution of the Limit Question
Another method is to start from the difference equations governing the ran-
dom walk, and then pass to the differential equation in the limit. We can
then obviously generalize the differential equations, and find out that the
differential equations govern well-defined stochastic processes depending on
continuous time. Since differential equations have a well-developed theory
and many tools to manipulate, transform and solve them, this method turns
out to be useful.
Consider the position of the walker in the random walk at thenth and
(n+ 1)st trial. Through a first step analysis the probabilitiesvk,nsatisfy the
difference equations:
vk,n+1=p·vk− 1 ,n+q·vk+1,n