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.

In an equilibrium stochastic process with detailed balance, a sample trajectory and its physical time reversal have equal full probabilities when their equilibrium starting probabilities are included. If $P[\gamma \mid i]$ is the conditional weight for a path from $i$ to $j$, thenFor coarse equilibrium states with free energies $F_i$ and $p_i^{\mathrm{eq}}\propto e^{-F_i/(k_{B}T)}$, this givesThese are conditional path weights, rather than probabilities of a path after conditioning additionally on both endpoint events. Time reversal also reverses time-odd quantities such as momentum; a time-even order parameter is only read in reverse order. Magnetic fields and prescribed driving must be reversed in the comparison when required by the physical dynamics.^[ref]