= Local detailed balance
<Local detailed balance> constrains forward and reverse transition rates by the <entropy> delivered to the environment. For a <continuous-time Markov chain> whose resolved states have no hidden internal <entropy>, write $w_{ij}$ for the rate from $i$ to $j$. For one <thermal reservoir> at <temperature> $T$,
$$
\log\frac{w_{ij}}{w_{ji}}=\frac{q_{ij}}{k_BT},
$$
where $q_{ij}>0$ is <heat> released into the reservoir. Internally degenerate coarse states require their internal <entropy> contribution as well. Reservoir particle exchange or several reservoirs similarly require the appropriate chemical and thermal terms.<Stochastic thermodynamics, fluctuation theorems, and molecular machines by Udo Seifert>[§6.2, local detailed balance for discrete-state dynamics]{ref}
For a static collection of nondegenerate <energy levels> without external <work>, $q_{ij}=E_i-E_j$; the rates then have the equilibrium <Boltzmann factor> ratio. For a whole <sample trajectory>, the corresponding ratio is initially a ratio of conditional forward/reverse path weights under matched reversed driving. Including the initial/final ensemble <probabilities> adds the system <entropy> change and gives total <entropy production>. This distinction prevents identifying reservoir <entropy> alone with a ratio of full path <probabilities>.
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