= Solution
Treat $p_n(t)$ as the <probability> of the complete coarse-grained state $n$ at time $t$, with $p_n\geq0$ and $\sum_np_n=1$. In the column-vector convention of the <master equation>, $W_{nm}$ for $n\ne m$ is the transition rate \b[from $m$ to $n$]. The short-time transition law is
$$
P(n,t+dt\mid m,t)=\delta_{nm}+W_{nm}dt+o(dt).
$$
Nonnegative transition <probabilities> and conservation of total <probability> require
$$
\boxed{W_{nm}\geq0\ (n\ne m),\qquad
\sum_nW_{nm}=0,\qquad W_{mm}=-\sum_{n\ne m}W_{nm}\leq0.}
$$
For a <finite state space> these conditions define a <continuous-time Markov chain>. The <Markov property> means that, conditional on the present state, no further history affects the transition <probabilities>. A time-independent $W$ additionally assumes time-homogeneous rates; the waiting time in a state is exponential, and multiple jumps in an infinitesimal interval contribute beyond first order. A reduced description that omits relevant slow variables can retain memory and need not obey this closed equation. <Quantum coherences> are likewise absent from this classical population model. For infinite state spaces one also needs conditions preventing infinitely many jumps in finite time. Neither equal forward/backward rates nor <detailed balance> is required merely to conserve <probability>.
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