Entropy production
= Entropy production
{title2=$\Delta S_{\rm tot}$}
{wiki}
<Entropy production> $\Delta S_{\rm tot}$ 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 $\Delta S_{\rm bath}=Q_{\rm bath}/T$, with positive <heat> into the reservoir. Including the actual endpoint <probabilities> is essential even if the final ensemble has not reached <thermal equilibrium>.<Entropy production along a stochastic trajectory and an integral fluctuation theorem by Udo Seifert>[Equations 14–18 and 30: endpoint weights, total entropy and integral theorem]{ref}
Under the hypotheses of the <integral fluctuation theorem>,
$$
\left\langle e^{-\Delta S_{\rm tot}/k_B}\right\rangle=1.
$$
Applying <Jensen's inequality> to the convex <exponential function> gives $e^{-\langle\Delta S_{\rm tot}\rangle/k_B}\leq1$, hence $\langle\Delta S_{\rm tot}\rangle\geq0$. 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.