Information, entropy, and heat far from equilibrium

For equilibrium states, the Shannon entropy, $H =- \int d\Gamma \,p(\Gamma) \ln p(\Gamma) $, coincides with the thermodynamic entropy. It has lately been recognized that for systems in nonequilibrium steady states a thermodynamic description based on $H$ becomes feasible, as well. In the present work we derive various generalizations of the second law for nonequilibrium processes with additional information supplied from an external or an internal memory. We will show that the irreversible entropy production is bounded from below by the information transferred from the memory to the system.