Local detailed balance

Local detailed balance states that the logarithm of the probability ratio between a trajectory and its time reverse equals the entropy delivered to the environment in units of $k_B$. For one heat bath, $\log (P_F/P_B)=\Delta Q/(k_{B}T)$ when positive $\Delta Q$ flows into the bath.

For an equilibrium process, a path and its physically time-reversed path have equal probability after including their equilibrium starting weights. Conditional on their endpoints, this meansThis is the path version of detailed balance. A time-even order parameter is read backward without changing sign, while a transport flux changes sign. The Onsager--Machlup path probability implements this reversal in a coarse-grained stochastic description.