We review and further develop a Keldysh function integration method for the study of Lindblad evolution of many-body driven dissipative quantum systems. A systematic and pedagogical explanation of general boson and fermion Lindbladian dynamics is presented. Of particular interest are the inherent properties of stationary distribution functions determined by the Lyapunov equation. This framework is applied to study examples of Lindblad dynamics in the context of band theory, disorder, collisionless collective modes, and mean field theory.

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