A self-consistent master equation and a new kind of cumulants

Generalized master equations correspond to different kinds of cumulants. Here, we discuss the recently introduced non-crossing cumulants from a physical point of view and propose the corresponding integro-differential master equation as a new type of equation for a self-consistent treatment of memory effects. We prove the cluster property of the non-crossing cumulants, and show that the Gaussian approximation of our equation is given by a random matrix process. As an instructive example for our expansion formula we treat the random frequency-modulated oscillator.

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