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An equation of motion for open quantum systems incorporating memory effectsand initial correlations with the environment is presented in terms of aneffective Liouville operator that solely acts on states of the system. Theenvironment can induce memory effects via the frequency dependence of theeffective Liouville and initial correlations can be mapped to a shiftedfrequency dependent initial state within the system. The equation of motiongeneralizes the well known semi-group dynamic equations. In generic systems theeffective Liouville has a non-degenerate zero mode. By probability conservationone can demonstrate that a generic open system reaches, in the long time limit,a stationary state, which is independent of any initial condition.

Equation of Motion for Open Quantum Systems incorporating Memory and Initial Correlations