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For edge current $J_{ij}=w_{ij}{P}_i-w_{ji}{P}_j$, the total entropy production rate isIt vanishes precisely when every occupied edge satisfies 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.

A directed transition count is its conditional jump rate multiplied by the occupation probability of its departure state. Net current is the forward count minus the reverse count, multiplied by the transported charge. On a stationary finite tree each edge has zero net current, although both directed counts can be nonzero. Different edges can have different traffic.

A steady state does not by itself give every fluctuating observable an entropy production symmetry. If the dimensionless total entropy production ${\Sigma }_t$ has the same-law finite relation and a large deviation principle with rate function $I$ for ${\Sigma }_t/t$, then, on its common domain,The first expression denotes leading exponential scaling; the second follows by taking the scaled logarithm of the probability ratio. A finite-state, autonomous Markov process with time-even states, positive bidirectional rates and a positive stationary distribution supplies a standard setting for a finite total-entropy production relation. The stochastic system entropy is then bounded, so its endpoint contribution does not change the long-time rate function of bath entropy production. ^[ref]

Unbounded state spaces can have endpoint fluctuations on the scale relevant to a large deviation principle. Dropping system entropy, changing the measured observable to heat, or changing the reverse preparation therefore requires a fresh argument. The exact finite integral fluctuation theorem, a same-law finite relation, and an asymptotic rate function symmetry are distinct claims.

Stochastic thermodynamics assigns work, heat, and entropy production to individual fluctuating trajectories while recovering ordinary thermodynamics after ensemble averaging.