Master equation

Forward equation for continuous-time Markov jump processes; describes probability flow between states via transition rates.

Master equation

The master equation is the Kolmogorov forward equation for Markov jump processes. It is the standard starting point for stochastic models of relaxation, driven systems, and coarse-grained dynamics.

Prerequisites: probability measure , Markov semigroup .

Setup: rates and generator

Let $S$ be a countable state space. Specify jump rates $q_{ij}\ge 0$ for $i\ne j$ . Define the generator matrix $Q$ by

$Q_{ij}=q_{ij}$ for $i\ne j$ ,
$Q_{ii}=-\sum_{j\ne i} q_{ij}$ (so each row sums to $0$ ).

Let $p_i(t)=\mathbb{P}(X_t=i)$ and write $p(t)$ as a row vector.

Master equation (component form)

For each state $i$ ,

\frac{d}{dt}p_i(t)=\sum_{j\ne i}\Big(p_j(t)\,q_{ji}-p_i(t)\,q_{ij}\Big).

The first term is inflow from other states into $i$ , and the second is outflow from $i$ .

Master equation (matrix form) and semigroup solution

In row-vector convention,

\frac{d}{dt}p(t)=p(t)\,Q, \qquad p(t)=p(0)\,e^{tQ}.

The family $P_t=e^{tQ}$ is the transition semigroup from Markov semigroups .

Stationary distribution

A distribution $\pi$ is stationary if

\pi Q=0,

equivalently $\pi P_t=\pi$ for all $t\ge 0$ .

Detailed balance (equilibrium vs nonequilibrium)

A stationary $\pi$ satisfies detailed balance when

\pi_i q_{ij}=\pi_j q_{ji}\qquad (i\ne j).

If detailed balance fails, the stationary state may still exist, but it typically carries nonzero steady currents (a hallmark of nonequilibrium steady states, compare nonequilibrium steady state ).

Relative entropy decay (common Lyapunov function under detailed balance)

When detailed balance holds, relative entropy $D(p(t)\|\pi)$ is typically nonincreasing in time; this expresses relaxation toward equilibrium in an information-theoretic sense.