Equality statements for entropy change in open systems

The entropy change of a (non-equilibrium) Markovian ensemble is calculated from (1) the ensemble phase density $p(t)$ evolved as iterative map, $p(t) = \mathbb{M}(t) p(t- \Delta t)$ under detail balanced transition matrix $\mathbb{M}(t)$, and (2) the invariant phase density $\pi(t) = \mathbb{M}(t)^{\infty} \pi(t) $. A virtual measurement protocol is employed, where variational entropy is zero, generating exact expressions for irreversible entropy change in terms of the Jeffreys measure, $\mathcal{J}(t) = \sum_{\Gamma} [p(t) - \pi(t)] \ln \bfrac{p(t)}{\pi(t)}$, and for reversible entropy change in terms of the Kullbach-Leibler measure, $\mathcal{D}_{KL}(t) = \sum_{\Gamma} \pi(0) \ln \bfrac{\pi(0)}{\pi(t)}$. Five properties of $\mathcal{J}$ are discussed, and Clausius' theorem is derived.