Equality statements for entropy change in open systems

Abstract

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, $\mathcal{J}(t) = \sum_Γ [p(t) - π(t)] \ln \bfrac{p(t)}{π(t)}$, and for reversible entropy change in terms of the Kullbach-Leibler measure, $\mathcal{D}_{KL}(t) = \sum_Γ π(0) \ln \bfrac{π(0)}{π(t)}$. Five properties of $\mathcal{J}$ are discussed, and Clausius' theorem is derived.

12 pages