= Solution
The one-step <master equation> for the <biased random walk> is
$$
P(x,t+\tau)=w_+P(x-a,t)+w_-P(x+a,t),\qquad t=N\tau.
$$
Expand in a <Taylor series> and retain the first <time> <derivative> and the first two space <derivatives>. Since $w_++w_-=1$ and $w_--w_+=2\epsilon$,
$$
\tau\partial_tP=2a\epsilon\,\partial_xP+\frac{a^2}{2}\partial_x^2P+\cdots.
$$
Thus the usual continuum truncation gives the requested <Fokker-Planck equation>,
$$
\partial_tP=D_0\partial_x^2P+C\partial_xP,\qquad
\boxed{D_0=\frac{a^2}{2\tau},\qquad C=\frac{2a\epsilon}{\tau}.}
$$
The <diffusion coefficient> has units $\mathrm{length^2/time}$, while $C$ has units of <velocity>. The physical <drift velocity> is $v=-C=a(w_+-w_-)/\tau$, because the standard advection term is $-v\partial_xP$. A rightward bias therefore means $\epsilon<0$, $C<0$, and $v>0$.
The approximation must be stated carefully. The truncation is a controlled <diffusion limit of a weakly biased random walk> when $a\to0$, $\tau\propto a^2$, and $\epsilon\propto a$, keeping $D_0$ and $C$ finite. It is also the spatial <diffusion> approximation for a continuous-time jump process with exponentially distributed waiting times of mean $\tau$.
If the original fixed-step-time walk has a finite bias, its exact <variance> from (a) instead gives the <diffusion coefficient> of the Gaussian envelope about the drifting mean:
$$
\boxed{D_{\rm fixed}=\frac{a^2}{2\tau}(1-4\epsilon^2)=\frac{2a^2w_+w_-}{\tau}.}
$$
One obtains the same correction by retaining $\tfrac12\tau^2\partial_t^2P$ in the <time> expansion and using the leading advection equation to replace $\partial_t^2P$ by $v^2\partial_x^2P$. The result is $D_{\rm fixed}=D_0-v^2\tau/2$. In particular, a deterministic one-way walk has zero <variance>, as this expression requires. Thus $D_0$ is the usual intended first-time-derivative approximation, not an exact variance-growth coefficient at arbitrary bias. The later continuum estimates use that convention, with the fixed-time alternative identified where relevant.
Back to article page
