 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 $T$,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.^[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.