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Brownian Motion

Let's model a random motion happening around 0,with equal probability to go left and right with same amount each step.

Xi=+ΔxP(Xi)=12Xi=-ΔxnP(Xi)=12X=X1+X2++Xn
X=X1++XnXi=12×(-Δx)+12×(Δx)=0X=0

So average is zero

On the other hand variance is not zero, since

var{Xi}=X2i-Xi2=(12)×(Δx)2+12×(-Δx)2var{x1}=var{x2}==var{Xn}=(Δx)2

Since each step statistically independent The sum of variance is

X2=ne0=1var{Xi}=nΔx2

Since total duration of walk is

t=nΔtn=tΔtX2=(Δx2Δt)t

So Brownian motion happens round 0 and with time it diffuses. That is the variance increases with number of steps. The term in front of time is equal to the twice of diffusion coefficient

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