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For independent lattice steps of length $a$ at intervals $\tau$, with probabilities $\alpha +\beta =1$, the diffusive scaling $a^2/\tau \to 2D$ and $(\alpha -\beta )a/\tau \to V$ requires a bias of order $a$. Expanding the master equation gives the advection-diffusion equation $p_t+V{p}_x=D{p}_{xx}$. 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.

For transverse coupling with equal mode amplitudes $\lambda$, Fermi's golden rule and the oscillator creation operator and annihilation operator givewhere the first index is the final spin and $g$ is the density of states per unit angular frequency. Decay creates a quantum and carries the factor $n_{\Omega }+1$; excitation annihilates one and carries $n_{\Omega }$. For nonuniform resonant occupations, $n_{\Omega }$ is their density-weighted mean.

With a thermal Bose-Einstein distribution, $n_{\Omega }/(n_{\Omega }+1)=e^{-\beta \hslash \Omega }$. The two-state master equation therefore has the Boltzmann distribution as its unique stationary populations when ${\Gamma }_0>0$. The population deviation relaxes at rate ${\Gamma }_0(2n_{\Omega }+1)$. If there is no resonant coupling, both rates vanish and stationarity does not determine the populations.

The thermal oscillator occupation is the Bose-Einstein distribution. With the rate notation ${\Gamma }_{\eta \sigma }$ for a transition from $\sigma$ to $\eta$, the master equation for the up population isAt thermal equilibrium, assuming ${\Gamma }_0>0$, vanishing time derivative is detailed balance between the two spin states. HenceorTogether with normalization, this gives $p_{\uparrow }=1/(1+e^{\beta \hslash \Omega })$ and $p_{\downarrow }=e^{\beta \hslash \Omega }/(1+e^{\beta \hslash \Omega })$, the Boltzmann distribution for the energy eigenvalues $\pm \hslash \Omega /2$. At zero temperature, absorption vanishes and emission relaxes the spin into its down ground state.

The nonzero-rate condition is necessary: if $\lambda =0$ or $g(\Omega )=0$, both rates vanish, so every initial pair of populations is stationary within this approximation. In that case stationarity alone cannot imply a thermal distribution. The equilibrium conclusion therefore presumes coupling to resonant oscillator modes.

The pair-birth rate here is the constant immigration rate $\lambda$, while deaths occur at total rate $\beta n$. This interpretation is fixed by the requested mean formula. With $P_n(t)=\mathrm{Pr}(N_t=n)$ and $P_j=0$ for $j<0$, the master equation isThe Markov jump-process generator acts as $Lf(n)=\lambda [f(n+2)-f(n)]+\beta n[f(n-1)-f(n)]$. Taking $f(n)=n$ gives $m^{\prime }=2\lambda -\beta m$, henceFor the second moment $s=\mathbb{E}{N}_t^2$, the two jumps give $s^{\prime }=4\lambda m+4\lambda -2\beta s+\beta m$. Differentiating $V=s-m^2$ therefore yields $V^{\prime }=4\lambda +\beta m-2\beta V$. Substitution of the mean and solution of this scalar linear equation giveThus $V(t)\to 3\lambda /\beta$, assuming $\beta >0$ and finite initial second moment. This is pair immigration with linear deaths; births at rate $\lambda n$ would be a different model and would not give the printed mean.

Let $p_j^n$ be the probability of occupying site $ja$ after $n$ steps, with $p_j^0=1_{\{j=0\}}$ and $\alpha +\beta =1$. The master equation isWrite $p_j^n\approx ap(ja,n\tau )$ so that the limiting $p$ is a probability density. A Taylor expansion of this master equation givesThe weakly biased diffusion limit of a weakly biased random walk keeps $a^2/\tau$ and $(\alpha -\beta )a/\tau$ finite as $a,\tau \to 0$, so $\alpha -\beta =O(a)$. The limiting advection-diffusion equation isIts initial condition is the Dirac delta function at the origin. At finite spacing, the exact variance gained per step is $a^2[1-(\alpha -\beta {)}^2]=4a^2\alpha \beta$. Thus a finite-step central limit theorem gives the variance-matched coefficient $2a^2\alpha \beta /\tau$; its difference from the displayed $D$ vanishes in the stated weak-bias diffusion limit of a weakly biased random walk. Keeping a fixed nonzero bias while taking diffusive space-time scaling would instead make $V$ diverge.

Multiply the master equation by $s^n$ and sum over $n\ge 0$. For the immigration terms,For the death terms, shifting the first sum givesTherefore the probability generating function obeys

The additional transition is $n\to n+k$ at rate $ϵ(n)$. Its gain term at state $n$ comes from state $n-k$, so the new master equation iswhere the gain term is zero when $n<k$. In the generating-function equation the new contribution isFor a general state-dependent $ϵ$, this is not closed in $\varphi$. Even when it is closed, $s^k-1$ is not proportional to $s-1$ for $k>1$, so the Poisson ansatz used in part (a) is not preserved.

Write $E_n(t)=P(E,n)$ and $C_n(t)=P(ES,n)$. The master equation balances the transitions $E_n\to C_n$ at rate $k_1$, $C_n\to E_n$ at rate $k_1^{\prime }$, and $C_n\to E_{n+1}$ at rate $k_2$:for $n\ge 0$, with $C_{-1}=0$.

The distribution is the geometric distribution on $\{1,2,\dots \}$ with success parameter $q=e^{-\lambda t}$. Its expected value is thereforeEquivalently, multiplying the master equation by $n$ and summing gives $m^{\prime }(t)=\lambda m(t)$ with $m(0)=1$.

This is a Yule process, a pure birth process with state-dependent birth rate ${\lambda }_n=\lambda n$. Probability flows into state $n$ from $n-1$ and leaves it for $n+1$, so the master equation is

Let $w_{ij}$ be the transition rate from $i$ to $j$ and define the probability current $J_{ij}=w_{ij}{P}_i-w_{ji}{P}_j=-J_{ji}$. The master equation is ${\dot{P}}_i=-{\sum }_j{J}_{ij}$. Differentiating the Shannon entropy $S_{\mathrm{sys}}=-{\sum }_i{P}_i\log P_i$ and symmetrizing givesAdding the entropy flow to the environment produces the nonnegative entropy production rate of a Markov chainThe paper's displayed $\sum P\log P$ is the negative of the thermodynamic system entropy, so its derivative has the opposite sign before the environmental contribution is added.

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.

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.