= 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:
$$
\dot p_n=\sum_{m\ne n}(W_{nm}p_m-W_{mn}p_n),\qquad\boxed{W_{nm}\ge0\ (n\ne m),\quad W_{mm}=-\sum_{n\ne m}W_{nm}.}
$$
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>.
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