Past exam of the thermal and statistical physics course of the University of Cambridge / 2021 / Section B / 6

Let each step have length $a$, and let $K$ be the number of right steps among $N$ independent steps. Then $K$ has a binomial distribution and the physical displacement is $x=a[K-(N-K)]=a(2K-N)$. Therefore $K=(N+x/a)/2$, and the position probability mass isThis is zero unless $x/a$ is an integer, $|x/a|\le N$, and $N+x/a$ is even. The parity condition matters because the walk alternates between the two sublattices. If distance is measured in lattice spacings, replace $x/a$ by $x$.

Normalization follows from the binomial theorem. Its first two moments provide useful checks:These are exact discrete-time statements; a smooth continuum probability density requires coarse-graining over the lattice and its parity oscillation.

The one-step master equation for the biased random walk isExpand in a Taylor series and retain the first time derivative and the first two space derivatives. Since $w_{+}+w_{-}=1$ and $w_{-}-w_{+}=2ϵ$,Thus the usual continuum truncation gives the requested Fokker-Planck equation,The diffusion coefficient has units $\mathrm{lengt}{\mathrm{h}}^2/time$, while $C$ has units of velocity. The physical drift velocity is $v=-C=a(w_{+}-w_{-})/\tau$, because the standard advection term is $-v{\partial }_{x}P$. A rightward bias therefore means $ϵ<0$, $C<0$, and $v>0$.

The approximation must be stated carefully. The truncation is a controlled diffusion limit of a weakly biased random walk when $a\to 0$, $\tau \propto a^2$, and $ϵ\propto a$, 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:One obtains the same correction by retaining $\frac{1}{2}{\tau }^2{\partial }_t^{2}P$ in the time expansion and using the leading advection equation to replace ${\partial }_t^{2}P$ by $v^2{\partial }_x^{2}P$. The result is $D_{\mathrm{fixed}}=D_0-v^2\tau /2$. 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.

For this radial random walk on a regular tree, the drawing is an infinite regular tree of degree $z$, not a planar lattice with closed loops. The radial coordinate is the number of edges in the unique path to the root. A non-root vertex has one neighbour inward and $z-1$ neighbours outward, so equal probabilities for its $z$ edges giveLet $r$ be radial distance measured in edge units. For $r>1$, probability arriving in shell $r$ comes from shell $r-1$ by an outward step or from shell $r+1$ by an inward step:This has precisely the same bulk master equation as the one-dimensional biased random walk, withThe outward bias is geometric: beyond the first shell, the number of vertices is $M_r=z(z-1{)}^{r-1}$, so there are more outward than inward choices even though no edge direction is dynamically preferred. $P(r,N)$ is the probability summed over an entire shell, not the probability at one specified vertex; no additional shell-size factor belongs in the boxed recurrence.

The equality concerns the evolution equation away from the root, not equality to the unrestricted full-line binomial distribution at all positions. The root imposes a different boundary rule.

At the root all $z$ neighbours are in shell one, so the probability of leaving the root for that shell is one. Conversely, a walker anywhere in shell one chooses its unique rootward edge with probability $1/z$. The two boundary recurrences are thereforeTogether with $P(0,0)=1$ and $P(r>0,0)=0$, these equations determine the distribution. In particular $P(1,1)=1$ and $P(0,2)=1/z$, which check the special root coefficient. For the original indexing, replace $N+1$ by $N$ and $N$ by $N-1$ on the right-hand sides. Summing these equations and the bulk recurrence conserves ${\sum }_{r\ge 0}P(r,N)=1$.

Farther than a few edge lengths from the root, replace the radial coordinate $r$ by a physical coordinate $x=ar$ and smooth the master equation. It has the same constant-coefficient Fokker-Planck equation as in (b). Under the conventional first-time-derivative truncation,The radial drift $v=-C=a(z-2)/(z\tau )$ is outward. For $z=2$ the tree is the infinite line, its radial distance is an unbiased reflected walk, and $C=0$.

The root cannot absorb probability or allow negative radial distance. Write the equation as a continuity equation, using the Fokker-Planck probability currentThe appropriate continuum boundary condition is reflecting, or zero flux:It is not generally ${\partial }_{x}P(0,t)=0$: that simpler condition is correct only when $C=0$. With the finite-bias fixed-step Gaussian convention from (b), use $D=D_{\mathrm{fixed}}=2a^2(z-1)/(z^2\tau )$ instead; the drift sign and zero-current boundary condition are unchanged. A continuum boundary condition represents the microscopic root region after coarse-graining, rather than reproducing each discrete root probability at individual step times.

For the infinite system with outward drift, total probability remains one, but probability at a fixed distance from the root eventually decreases as the walker moves away. The question concerns this local decay, not a loss of total probability.

The independent dimensions of the continuum coefficients are $[D]=L^2/t$ and $[C]=L/t$. The drift-diffusion decay time and its associated length are, up to numerical factors:Equivalently, equating the drift distance $|C|t$ to the diffusive width of order $\sqrt{Dt}$ gives this crossover scale. Rescaling $x=(D/|C|)y$ and $t=(D/C^2)s$ removes the dimensional coefficients from the equation.

With the conventional coefficients used in the requested continuum derivation,where the last expression suppresses the numerical factor as requested. The elementary drifted Gaussian distribution on the full line hasAt fixed $x$, its exponential decay factor is $e^{-C^{2}t/(4D)}$, so an exponential e-folding time would conventionally be called $4D/C^2$, consistent with the same dimensional scale. The reflecting half-line changes the spatial form and algebraic prefactor. The substitution $P=e^{-Cx/(2D)}\psi$ gives ${\partial }_t\psi =D{\partial }_x^2\psi -(C^2/4D)\psi$, displaying the same exponential scale for outward drift.

For $z=2$, $C=0$ and no finite drift-induced exponential decay time exists: the probability near the root decays only algebraically. For the finite-bias, deterministic-step Gaussian envelope discussed in (b), the corresponding dimensional expression is instead $D_{\mathrm{fixed}}/C^2=2\tau (z-1)/(z-2{)}^2$. These continuum estimates are not exact return-probability decay rates for the lattice walk at arbitrary $z$; in particular, a diffusion approximation need not describe far-tail return events accurately at strong bias.