= Diffusion limit of a weakly biased random walk
{title2=$V=a(\alpha-\beta)/\tau,\ D=a^2/(2\tau)$}
For independent lattice steps of length $a$ at intervals $\tau$, with probabilities $\alpha+\beta=1$, the diffusive scaling $a^2/\tau\to2D$ and $(\alpha-\beta)a/\tau\to V$ requires a bias of order $a$. Expanding the <master equation> gives the <advection-diffusion equation> $p_t+Vp_x=Dp_{xx}$. The finite-step variance coefficient is $2a^2\alpha\beta/\tau$; it has the same limit because the squared bias vanishes. Fixed nonzero bias does not give finite drift under this diffusive scaling.
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