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.
تحميل البحث