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) = M(t)p(t−∆t) under detail balanced transition matrix M(t), and (2) the invariant phase density π(t) = M(t)∞π(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, J (t) = ∑ Γ [p(t) − π(t)] ln [p(t)/π(t)], and for reversible entropy change in terms of the Kullbach-Leibler measure, DKL(t) = ∑ Γ π(0) ln [π(0)/π(t)]. Five properties of J are discussed, and Clausius’ theorem is derived. ∗Electronic address: [email protected]