⟨x²⟩ = 0 = Δx² · steps
Steps: 0 | D = Δx²/2Δt = 0.5
KS p-val vs Gaussian: —
Central Limit Theorem → Heat Equation
A 1D random walk takes steps ±Δx each time Δt. After n steps, position X_n = Σ ξ_i where ξ_i = ±Δx with probability 1/2.
As Δx,Δt→0 with D = Δx²/(2Δt) fixed, the distribution P(x,t) converges to the diffusion equation:
∂P/∂t = D ∂²P/∂x²
with solution P(x,t) = e^{-x²/4Dt}/√(4πDt).
This is the CLT in PDE form. The variance grows linearly: ⟨x²⟩ = 2Dt. Larger Δx → same continuum limit if D is preserved.
random walk
diffusion equation
CLT
Brownian motion