Continuum Limit: Random Walk → Diffusion Equation

A 1D lattice random walk with step size a and time step τ: in the continuum limit a→0, τ→0 with D = a²/2τ fixed, the master equation converges to the diffusion PDE ∂P/∂t = D∂²P/∂x². Compare histogram of N walkers against the exact Gaussian solution P(x,t) = (4πDt)^(−½) exp(−x²/4Dt).

N walkers 1000

Step size a 1.0

D = a²/2 0.50

Bias p(+) 0.50

Fick's law: J = −D ∇P
D = a²/2τ (symmetric walk)
Drift v = a(2p−1)/τ
⟨x²⟩ = 2Dt (+ v²t² with drift)

MSD slope: —