The Master Equation

The master equation is the most direct probabilistic description of a Markov process on a set of states. Instead of following one noisy trajectory, it governs the entire probability distribution $P_n(t)$ , evolving it forward through transition rates. The name "master" reflects its authority: once the rates are fixed, this single first-order equation controls every statistical observable — means, variances, correlation functions, relaxation times — that the process can produce. Radioactive decay, chemical reactions, queues, epidemics, Glauber spin dynamics, and the coarse-grained hopping of a Brownian particle are all master equations with different rates. Physically it is bookkeeping for a fluctuating system. Probability flows into state $n$ from every state $m$ at rate $W_{m\to n}P_m$ and flows out of $n$ at rate $W_{n\to m}P_n$ ; the net imbalance is $\dot P_n$ . This gain-loss structure is exact for any continuous-time Markov chain and conserves total probability by construction. Written as a matrix equation $\dot{\mathbf P}=\mathsf W\mathbf P$ , the entire dynamics is encoded in the spectrum of the rate matrix $\mathsf W$ — and that spectrum has a rigid structure this article makes precise: a single zero eigenvalue whose eigenvector is the stationary state, and a left-eigenvector $(1,1,\dots)$ that enforces probability conservation, with every other eigenvalue carrying a non-positive real part that sets a relaxation rate. Two further threads run through the subject. Detailed balance selects time-reversible dynamics: when each microscopic transition exactly balances its reverse, the stationary state is a true equilibrium with no probability currents, and an H-theorem guarantees monotone relaxation to the Boltzmann distribution — irreversibility emerging from reversible-looking rates. And coarse-graining a continuous master equation, through the Kramers-Moyal expansion, produces the Fokker-Planck equation, making the master equation the discrete parent of every continuum stochastic description. This is the article the stochastic-methods chapter later recalls when it revisits the master equation in a Markov-chain and Monte-Carlo setting.

Key takeaways

The master equation is a first-order linear ODE for the probability $P_n(t)$ of state $n$ , expressing $\dot P_n$ as the net probability current: gain from other states minus loss to them.
It is built from transition rates $W_{n\to m}\ge0$ that, under the Markov assumption, depend only on the present state; total probability is conserved because the rate-matrix columns sum to zero.
In matrix form $\dot{\mathbf P}=\mathsf W\mathbf P$ : the rate matrix has one zero eigenvalue (the stationary state) with left-eigenvector $(1,1,\dots)$ enforcing conservation, and all other eigenvalues have non-positive real part (relaxation rates).
Detailed balance $W_{m\to n}P_n^{\rm eq}=W_{n\to m}P_m^{\rm eq}$ makes the dynamics reversible, gives a real spectrum, and drives an H-theorem: monotone relaxation to the Boltzmann distribution.
Coarse-graining a continuous master equation via the Kramers-Moyal expansion yields the Fokker-Planck equation — the master equation is the discrete parent of continuum diffusions.
Key idea: probability is a conserved fluid on a network of states; $\dot P_n$ is the net flux into $n$ , each link carrying a one-way current $W_{m\to n}P_m$ . Stationarity, equilibrium, relaxation rates, and the Fokker-Planck limit all follow from the spectrum of the rate matrix.

The core insight

Probability is a conserved fluid on the state space: $\dot P_n$ is the net flux into $n$ , each link carrying a one-way current $W_{m\to n}P_m$ . Everything — stationarity, equilibrium, relaxation rates, the Fokker-Planck limit — follows from the structure of the rate matrix.

Where this lives today

The master equation is the exact substrate of stochastic simulation. In systems biology and chemical kinetics, the chemical master equation governs the probability distribution over molecule-number states; Gillespie's stochastic simulation algorithm generates exact sample trajectories of precisely this equation, and it is the workhorse for modeling gene expression, signaling networks, and any reaction system where molecule numbers are small enough that intrinsic noise matters. Whether a deterministic rate-equation description suffices or a full stochastic treatment is needed is decided by the system-size scaling that the Kramers-Moyal expansion makes precise. In statistical physics the master equation supplies relaxational dynamics for equilibrium models. Glauber and Metropolis single-spin-flip rules are transition rates chosen to satisfy detailed balance with respect to the Boltzmann weight, so their stationary state is the equilibrium ensemble; the slowest eigenvalue of the resulting rate matrix sets the relaxation time and diverges as critical slowing down near a phase transition. These same rules are the basis of Markov-chain Monte Carlo, the topic the stochastic-methods chapter develops when it recalls the master equation in a simulation setting.

The physical picture

Imagine a network of boxes (the states) joined by pipes (the transitions). Each pipe carries a one-way current of probability proportional to how full the upstream box is and to the pipe's conductance — the rate. The master equation says the level in each box changes at exactly the net flow rate: water in minus water out. That is all it is, and its power comes from that simplicity being exact for any Markov process. Stationarity — constant levels — can happen in two distinct ways, and telling them apart is central. Either every pipe individually carries zero net current (detailed balance, a true equilibrium), or net currents circulate around closed loops while each box stays balanced (a non-equilibrium steady state with persistent probability flux). Global balance, the statement $\dot P_n=0$ , is strictly weaker than detailed balance; the difference is the difference between equilibrium and a driven steady state. Because each transition is Markovian, the system has no memory: the future depends only on which box you are in now, not on how you arrived. This is exactly what makes the dynamics a closed first-order ODE rather than an integro-differential equation with memory kernels. It is also what lets the whole evolution be read off from a single matrix. That matrix — the rate matrix $\mathsf W$ — has a rigid spectral skeleton, shown in the figure. One eigenvalue is pinned exactly at zero: its right-eigenvector is the stationary distribution, and its left-eigenvector is $(1,1,\dots)$ , the constant vector that sums probability. Every other eigenvalue lies in the left half-plane, its negative real part a relaxation rate; the slowest of these sets the longest timescale of the system. When jumps become small and frequent, the discrete pipe network blurs into a smooth flow, the gain-loss balance becomes a continuity equation with drift and diffusion, and the Fokker-Planck description emerges.

Figure

The probability-network picture: one-way currents on links, in minus out sets the rate of change

Figure

Punchline: the rate-matrix spectrum — one eigenvalue pinned at 0 (stationary), the rest relaxation rates

The core equations

The master equation is a gain-loss balance for the probability of each discrete state. Probability enters $n$ from every state $m$ and leaves $n$ toward every state $m$ , each transition weighted by its rate and the current occupancy of the source state.

\frac{\mathrm{d}P_n(t)}{\mathrm{d}t} = \sum_{m\neq n}\big[\,W_{m\to n}\,P_m(t) - W_{n\to m}\,P_n(t)\,\big] \qquad (1)

where: $P_n(t)$ — probability of being in state $n$ ; $W_{m\to n}$ — transition rate from $m$ to $n$ (probability per unit time, $\ge0$ ); the sum runs over all other accessible states. Read the two terms as gain (into $n$ ) and loss (out of $n$ ); their difference is the rate of change of $P_n$ . Collecting the coefficients into a matrix turns the balance into a linear system. The off-diagonal entries are the rates into a state; the diagonal is minus the total escape rate, which guarantees probability conservation through vanishing column sums.

\frac{\mathrm{d}P_n}{\mathrm{d}t} = \sum_m \mathsf{W}_{nm}\,P_m,\qquad \mathsf{W}_{nm} = W_{m\to n} - \delta_{nm}\sum_{k}W_{n\to k} \qquad (2)

where: $\mathsf{W}_{nm}$ — rate-matrix element (off-diagonal $\ge0$ ); the column sums $\sum_n\mathsf{W}_{nm}=0$ enforce $\partial_t\sum_n P_n=0$ . This is the object whose spectrum governs the whole dynamics: $\mathsf W$ is generically non-symmetric, but its structure fixes one eigenvalue at zero and pushes the rest into the left half-plane. The stationary state is defined by vanishing time derivative — global balance — which requires only that total inflow equal total outflow for each state. This is weaker than requiring each link to balance.

\sum_{m\neq n} W_{m\to n} P_m^{\mathrm{eq}} = \sum_{m\neq n} W_{n\to m} P_n^{\mathrm{eq}} \qquad (3)

where: $P_n^{\mathrm{eq}}$ — stationary distribution, $\dot P_n=0$ . A stationary state satisfying only this may still carry circulating currents around loops: a non-equilibrium steady state. Detailed balance is the stronger, equilibrium condition: each individual transition balances its reverse. It fixes every rate ratio to a Boltzmann factor and is what a system in true thermal equilibrium obeys.

W_{m\to n}\,P_n^{\mathrm{eq}} = W_{n\to m}\,P_m^{\mathrm{eq}}\qquad\Longrightarrow\qquad \frac{W_{n\to m}}{W_{m\to n}} = e^{-(E_m-E_n)/k_BT} \qquad (4)

where: $P_n^{\mathrm{eq}}\propto e^{-E_n/k_BT}$ is the Boltzmann weight; transitions upward in energy are suppressed exponentially. Detailed balance makes $\mathsf W$ symmetrizable (real spectrum) and, as shown below, drives an H-theorem. Two canonical examples anchor the abstraction. Radioactive decay is a pure-death process; the birth-death process generalizes it with up and down rates and models populations, queues, and reaction counts.

\frac{\mathrm{d}P_n}{\mathrm{d}t} = b_{n-1}P_{n-1} + d_{n+1}P_{n+1} - (b_n+d_n)P_n \qquad (5)

where: $b_n$ — birth (up) rate from state $n$ ; $d_n$ — death (down) rate. For pure decay, $b_n=0$ and $d_n=n\lambda$ , giving the mean $\langle n\rangle=Ne^{-\lambda t}$ .

Derivation AThe rate-matrix spectrum: a zero eigenvalue, conservation, and relaxation

The goal is to establish the spectral skeleton that governs every master equation: probability conservation, the existence of a stationary state as a zero eigenvalue, and the sign of the relaxation eigenvalues. The strategy is to read structure off the rate matrix directly — the column-sum property gives conservation and pins one eigenvalue at zero, the left-eigenvector $(1,1,\dots)$ makes that pinning explicit, and (with detailed balance) a similarity transform to a symmetric matrix proves the spectrum is real and non-positive. Watch the two-sided eigenvector structure: the zero eigenvalue has right-eigenvector $P^{\rm eq}$ but left-eigenvector $(1,1,\dots)$ , and it is the left one that carries conservation.

Step 1. Prove conservation from the column sums. Summing the matrix equation over the final index, $\tfrac{\mathrm d}{\mathrm dt}\sum_n P_n=\sum_m\big(\sum_n\mathsf W_{nm}\big)P_m$ , and $\sum_n\mathsf W_{nm}=\sum_{n\neq m}W_{m\to n}-\sum_k W_{m\to k}=0$ . Why: the gain terms into all states from $m$ exactly equal the total loss out of $m$ , so no probability is created or destroyed — this is built into the matrix, not imposed by hand.

Step 2. Identify the zero eigenvalue. Vanishing column sums mean the constant row vector $\mathbf 1=(1,1,\dots)$ is a left-eigenvector with eigenvalue $0$ : $\mathbf 1\,\mathsf W=0$ . A left-eigenvector with eigenvalue $0$ forces a right-eigenvector with eigenvalue $0$ as well, and that right-eigenvector is the stationary distribution $P^{\rm eq}$ (it satisfies $\mathsf W P^{\rm eq}=0$ ). Why: the left null-vector is the conservation law; the matching right null-vector is the state the system relaxes to — the two are different vectors for the same eigenvalue because $\mathsf W$ is non-symmetric.

Step 3. Bound the other eigenvalues. By the Gershgorin circle theorem each eigenvalue lies in a disc centred at the (negative) diagonal $-\sum_k W_{n\to k}$ with radius $\sum_{m\neq n}W_{n\to m}$ equal in magnitude to the centre; every such disc touches the origin from the left, so $\mathrm{Re}\,\lambda\le0$ for all eigenvalues. Why: the non-negative off-diagonal rates and the conservation-fixed diagonal keep the whole spectrum in the closed left half-plane, guaranteeing that perturbations decay rather than grow.

Step 4. With detailed balance, prove the spectrum is real. Define $\tilde{\mathsf W}_{nm}=\mathsf W_{nm}\sqrt{P_m^{\rm eq}/P_n^{\rm eq}}$ ; detailed balance $W_{m\to n}P_m^{\rm eq}=W_{n\to m}P_n^{\rm eq}$ makes $\tilde{\mathsf W}$ symmetric, so it has a real spectrum, and $\tilde{\mathsf W}$ is similar to $\mathsf W$ . Why: a symmetrizable matrix has real eigenvalues, so a detailed-balance master equation relaxes without oscillation — pure exponential decay of each mode, the slowest setting the relaxation time. *Verified: verify_aq-sm-master-equation.py diagonalizes a two-state and a three-state detailed-balance rate matrix, confirming the zero eigenvalue, the left null-vector $(1,1,1)$ , the right null-vector proportional to $P^{\rm eq}$ , the real spectrum, and $\mathrm{Re}\,\lambda\le0$ .*

Go deeper · Derivation B · 1 minThe H-theorem: irreversibility from reversible rates

Derivation BThe H-theorem: irreversibility from reversible rates

The goal is to prove that a master equation obeying detailed balance relaxes monotonically to equilibrium — that a relative-entropy functional cannot increase — thereby producing macroscopic irreversibility from microscopically reversible rates. The strategy is to define an H-functional measuring the distance from the stationary distribution, differentiate it along the dynamics, symmetrize the double sum over pairs, and invoke an elementary sign inequality. The point to appreciate is where the arrow of time enters: not in the rates (which are reversible under detailed balance) but in the convexity inequality applied to a distribution that starts away from equilibrium.

Step 1. Define $H=\sum_n P_n\ln(P_n/P_n^{\rm eq})$ and differentiate: $\dot H=\sum_n\dot P_n\big(\ln(P_n/P_n^{\rm eq})+1\big)$ , where the $+1$ term drops because $\sum_n\dot P_n=0$ by conservation. Why: $H$ is the relative entropy (Kullback-Leibler divergence) from equilibrium, non-negative by Gibbs' inequality and zero only at $P=P^{\rm eq}$ , so it is the natural Lyapunov candidate.

Step 2. Insert the master equation: $\dot H=\sum_{n,m}\big(W_{m\to n}P_m-W_{n\to m}P_n\big)\ln(P_n/P_n^{\rm eq})$ . Why: substituting the gain-loss form exposes the pairwise current on each link, which is what the symmetrization will act on.

Step 3. Symmetrize over the pair by relabelling $n\leftrightarrow m$ and averaging, then use detailed balance $W_{m\to n}P_m^{\rm eq}=W_{n\to m}P_n^{\rm eq}$ : $\dot H=-\tfrac12\sum_{n,m}W_{n\to m}P_n^{\rm eq}\,(x-y)\ln(x/y)$ with $x=P_m/P_m^{\rm eq}$ , $y=P_n/P_n^{\rm eq}$ . Why: detailed balance is what lets the two directions of each link be written with a common non-negative prefactor $W_{n\to m}P_n^{\rm eq}$ , isolating the sign in the factor $(x-y)\ln(x/y)$ .

Step 4. Apply the elementary inequality $(x-y)\ln(x/y)\ge0$ for all positive $x,y$ , so every summand is non-positive and $\dot H\le0$ , with equality only when $P_n/P_n^{\rm eq}$ is constant, i.e. $P=P^{\rm eq}$ . Why: $\ln$ is increasing, so $x>y\Rightarrow\ln(x/y)>0$ and $x<y\Rightarrow\ln(x/y)<0$ ; the product with $(x-y)$ is always non-negative — this convexity fact is the entire source of the arrow of time here. *Verified: verify_aq-sm-master-equation.py confirms $(x-y)\ln(x/y)\ge0$ over a random sample of positive pairs and that $H=\sum P\ln(P/P^{\rm eq})\ge0$ for a test distribution (Gibbs inequality).*

Limiting cases

The master equation specializes and reduces in several important regimes:

Case	Result	Meaning
Detailed balance	$W_{m\to n}P_n^{\rm eq}=W_{n\to m}P_m^{\rm eq}$	Each transition balances its reverse; the steady state is a true equilibrium with no probability currents, real relaxation spectrum, and an H-theorem.
Small-jump (Fokker-Planck) limit	$\partial_t P=-\partial_x[A_1 P]+\tfrac12\partial_x^2[A_2 P]$	Keeping the first two Kramers-Moyal moments turns the master equation into a drift-diffusion equation.
Symmetric nearest-neighbour hopping	$\partial_t P=D\,\partial_x^2 P$	Equal left/right rates $w$ on a lattice of spacing $a$ give the diffusion equation with $D=wa^2$ .
Two-state (telegraph) process	$\dot P_1=-k_{12}P_1+k_{21}P_2$	The minimal master equation; relaxes exponentially at rate $k_{12}+k_{21}$ to $P_1^{\rm eq}=k_{21}/(k_{12}+k_{21})$ .

Worked examples

Example 1 — The two-state master equation: exact relaxation and the spectrum. This solves the minimal master equation completely and reads off the rate-matrix eigenvalues.

Step 1. Take states $1,2$ with rates $k_{12}$ (from 1 to 2) and $k_{21}$ (from 2 to 1). Using $P_2=1-P_1$ , the dynamics closes: $\dot P_1=k_{21}-(k_{12}+k_{21})P_1$ .

Step 2. The stationary state sets $\dot P_1=0$ : $P_1^{\rm eq}=k_{21}/(k_{12}+k_{21})$ . For $k_{12}=2,\ k_{21}=0.5$ this is $P_1^{\rm eq}=0.5/2.5=\mathbf{0.2}$ .

Step 3. Integrate the linear ODE: $P_1(t)=P_1^{\rm eq}+[P_1(0)-P_1^{\rm eq}]e^{-(k_{12}+k_{21})t}$ , a single exponential relaxation at rate $k_{12}+k_{21}=\mathbf{2.5}$ .

Step 4. Read the spectrum. The $2\times2$ rate matrix $\mathsf W=\begin{pmatrix}-k_{12}&k_{21}\\k_{12}&-k_{21}\end{pmatrix}$ has eigenvalues $\{0,\,-(k_{12}+k_{21})\}=\{\mathbf{0},\,\mathbf{-2.5}\}$ . The zero eigenvalue is the stationary mode; the nonzero one is the relaxation rate — exactly the exponent in $P_1(t)$ . The slowest nonzero eigenvalue magnitude is the relaxation rate.

Example 2 — Detailed balance fixes the rate ratio to a Boltzmann factor. This shows how equilibrium constrains the rates themselves.

Step 1. Suppose states $1,2$ have energies $E_1<E_2$ with gap $\Delta E=E_2-E_1$ at temperature $T$ . Equilibrium demands $P_n^{\rm eq}\propto e^{-E_n/k_BT}$ , so $P_1^{\rm eq}/P_2^{\rm eq}=e^{+\Delta E/k_BT}$ .

Step 2. Detailed balance $W_{1\to2}P_1^{\rm eq}=W_{2\to1}P_2^{\rm eq}$ then forces the rate ratio $W_{2\to1}/W_{1\to2}=P_1^{\rm eq}/P_2^{\rm eq}=e^{+\Delta E/k_BT}$ : downhill (higher rate) beats uphill.

Step 3. For $\Delta E=k_BT$ (gap equal to thermal energy), the rate ratio is $e^{1}\approx\mathbf{2.72}$ — downhill transitions are about 2.72 times faster than uphill. For $\Delta E=3k_BT$ it is $e^{3}\approx\mathbf{20.1}$ .

Step 4. This is the microscopic content of "systems fall toward low energy": the rates are not free; equilibrium ties their ratio to the Boltzmann factor. Detailed balance is what makes the stationary state the equilibrium Boltzmann distribution rather than an arbitrary steady state.

Origins · 1 minmaster equation and how this was discovered

master equation

term coined by A. Nordsieck, W. E. Lamb & G. E. Uhlenbeck (1940); framework by N. G. van Kampen · The master equation as a named object entered physics through a 1940 paper by Arnold Nordsieck, Willis Lamb, and George Uhlenbeck on the statistics of cosmic-ray showers, who called the governing gain-loss equation the "master equation" because it controls all the moments of the distribution. Its systematic development as the foundation of non-equilibrium statistical mechanics is owed above all to Nicolaas Godfried van Kampen, whose treatise Stochastic Processes in Physics and Chemistry organized the master equation, the system-size expansion, and the passage to the Fokker-Planck equation into the standard modern framework. The H-theorem for master dynamics under detailed balance formalizes how irreversible macroscopic relaxation emerges from microscopically reversible transition rates.

History

1940

— Nordsieck, Lamb, and Uhlenbeck coin the term "master equation" in a study of cosmic-ray shower statistics

1943

— Chandrasekhar's review consolidates master and Fokker-Planck equations for stochastic problems in physics

1954

— Glauber-type single-spin-flip dynamics recast Ising relaxation as a detailed-balance master equation

1976

— Gillespie's stochastic simulation algorithm samples exact trajectories of the chemical master equation

1981

— van Kampen's treatise establishes the master equation and system-size expansion as the standard framework

Going further

References. N. G. van Kampen — Stochastic Processes in Physics and Chemistry, 3rd ed. (Elsevier, 2007), Ch. V–VIII · C. W. Gardiner — Handbook of Stochastic Methods, 3rd ed. (Springer, 2004), Ch. 7 · M. Kardar — Statistical Physics of Particles (Cambridge, 2007), Ch. 3 · R. K. Pathria & P. D. Beale — Statistical Mechanics, 3rd ed. (Elsevier, 2011), Ch. 15 · F. Reif — Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965), Ch. 15 · J. P. Sethna — Statistical Mechanics: Entropy, Order Parameters, and Complexity (Oxford, 2006), Ch. 8 · H. Risken — The Fokker-Planck Equation, 2nd ed. (Springer, 1989).

Provenance. Equations and derivations follow van Kampen (Ch. V, VIII, X) and Gardiner (Ch. 7). A verifier recomputes: (1) probability conservation from vanishing column sums; (2) the rate-matrix zero eigenvalue with left null-vector $(1,1,\dots)$ and right null-vector $\propto P^{\rm eq}$ ; (3) $\mathrm{Re}\,\lambda\le0$ and, under detailed balance, a real spectrum; (4) the two-state eigenvalues $\{0,-(k_{12}+k_{21})\}$ and steady state $P_1^{\rm eq}=k_{21}/(k_{12}+k_{21})$ ; (5) the H-theorem sign via $(x-y)\ln(x/y)\ge0$ ; (6) the diffusion limit $D=wa^2$ ; (7) the decay mean $\langle n\rangle=Ne^{-\lambda t}$ . Cross-checked against Kardar Ch. 3, Pathria & Beale Ch. 15, Reif Ch. 15, Sethna Ch. 8. No numerical constants are asserted beyond these standard structural results.

Builds on: Brownian Motion · The Diffusion Equation · Wiener Process and White Noise Leads to: The Fokker-Planck Equation · The Kramers-Moyal Expansion · Kramers' Escape Rate Theory · The Fluctuation-Dissipation Theorem

What is the difference between the master equation and the Fokker-Planck equation?

The master equation is the general gain-loss balance for a Markov process, valid for discrete states or arbitrary jumps. The Fokker-Planck equation is its small-jump limit: keep only the first two terms of the Kramers-Moyal expansion of a continuous master equation and you get a drift-diffusion PDE. So Fokker-Planck is a special case — the master equation is the parent, the Fokker-Planck equation is its $n=2$ continuum child, exact when jumps are small (or the system large).

How can reversible rates produce irreversible relaxation?

Detailed balance makes the rates time-symmetric, but the H-theorem shows that the relative entropy from equilibrium, $H=\sum P\ln(P/P^{\rm eq})$ , can only decrease along the dynamics — because of the convexity inequality $(x-y)\ln(x/y)\ge0$ . The arrow of time is not in the rates; it is in the fact that the system starts in a special (low-entropy, out-of-equilibrium) distribution and convexity forbids it from moving away from equilibrium on average. Reversible microscopic rates plus a special initial condition give irreversible macroscopic relaxation.

Why must the rate matrix have a zero eigenvalue?

Because probability is conserved. Conservation means the constant vector $(1,1,\dots)$ is a left-eigenvector with eigenvalue zero: summing the equation over all final states gives zero. A left null-vector forces a right null-vector at the same eigenvalue, and that right-eigenvector is the stationary distribution. So the existence of a steady state is not an extra assumption — it is guaranteed by conservation, encoded as the zero eigenvalue of $\mathsf W$ .

Can a master equation have oscillating relaxation?

For a generic rate matrix, yes — eigenvalues can be complex (with non-positive real part), giving damped oscillations. But if detailed balance holds, the rate matrix is similar to a symmetric matrix, so its spectrum is entirely real and relaxation is pure exponential decay with no oscillation. Complex eigenvalues are therefore a signature of a non-equilibrium (detailed-balance-violating) steady state with circulating currents.

Why do the transition rates have units of inverse time rather than being probabilities?

Because the master equation is continuous in time. $W_{n\to m}\,\mathrm dt$ is the probability of the transition in an infinitesimal interval $\mathrm dt$ , so $W_{n\to m}$ itself is a probability per unit time — a rate. It can be larger than 1 (a fast transition) and carries units of $1/$ time. This is the crucial distinction from a discrete-time Markov chain, whose transition matrix entries are genuine probabilities in $[0,1]$ with rows summing to 1.

How does the diffusion equation come out of a hopping master equation?

Take symmetric nearest-neighbour hops at rate $w$ on a lattice of spacing $a$ . The gain-loss terms form a discrete second difference, $w[P(x-a)-2P(x)+P(x+a)]$ . Taylor-expanding to second order gives $wa^2\,\partial_x^2 P$ , i.e. the diffusion equation $\partial_t P=D\,\partial_x^2 P$ with $D=wa^2$ . This is precisely the leading (second-order) Kramers-Moyal term, and it is why lattice random walks converge to continuum diffusion.