 Entropy production

Entropy production $\Delta S_{\mathrm{tot}}$ is the entropy change of the system plus that of its environment. For a discrete-state stochastic process, the stochastic system entropy at the occupied state is $s(t)=-k_B\log p_{i(t)}(t)$. A single thermal reservoir contributes $\Delta S_{\mathrm{bath}}=Q_{\mathrm{bath}}/T$, with positive heat into the reservoir. Including the actual endpoint probabilities is essential even if the final ensemble has not reached thermal equilibrium.^[ref]

Under the hypotheses of the integral fluctuation theorem,Applying Jensen's inequality to the convex exponential function gives $e^{-⟨\Delta S_{\mathrm{tot}}⟩/k_B}\le 1$, hence $⟨\Delta S_{\mathrm{tot}}⟩\ge 0$. This constrains the mean, while a short individual sample trajectory may have negative entropy production. Equating a medium-only path-weight ratio to total entropy production would omit the stochastic system term.