 Solution

Here $p_n(t)$ is the probability of occupying state $n$, and $W_{nm}$ for $n\ne m$ is the transition rate from $m$ to $n$. More precisely, over $dt>0$ that transition has probability $W_{nm}dt+o(dt)$. This is a time-homogeneous continuous-time Markov chain: the present state contains all information needed for the next transition, the rates do not depend on time, and finite rates make multiple jumps have probability $o(dt)$. In a physical rate description, untracked coherent dynamics or memory must be negligible on the timescale being modeled.

The master equation separates incoming and outgoing terms:Columns, rather than rows, sum to zero with this indexing. These conditions preserve normalization and nonnegative probabilities. For infinitely many states one also needs a well-defined nonexplosive process. Detailed balance is an additional equilibrium condition, not a general requirement on a master equation.