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=+Δx→P(Xi)=12Xi=-Δxn→P(Xi)=12X=X1+X2+…+Xn
⟨X⟩=⟨X1⟩+…+⟨Xn⟩⟨Xi⟩=12×(-Δx)+12×(Δx)=0⟨X⟩=0
So average is zero
On the other hand variance is not zero, since
var{Xi}=⟨X2i⟩-⟨Xi⟩2=(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=nΔt⇒n=tΔt⟨X2⟩=(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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