Geodesic
Exact master equations describing reduced dynamics of the Wigner function
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 5, pp. 203-219.

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Master equations of different types describe the evolution (reduced dynamics) of a subsystem of a larger system generated by the dynamic of the latter system. Since in some cases the (exact) master equations are relatively complicated, there exist numerous approximations for such equations which are also called master equations. In the paper, one develops an exact master equation describing the reduced dynamics of the Wigner function for quantum systems obtained by a quantization of a Hamiltonian system with a quadratic Hamilton function. First, one considers an exact master equation for first integrals of ordinary differential equations in infinite-dimensional, locally convex spaces. After this, one applies the obtained results to develop an exact master equation corresponding to a Liouville type equation (which is the equations for first integrals of the (system of) Hamilton equation(s)); the latter master equation is called a master Liouville equation, it is a linear first-order differential equation with respect to a function of real variable taking values in a space of functions on the phase space. If the Hamilton equation generating the Liouville equation is linear, then the vector fields that define the first-order linear differential operators in the master Liouville equations are also linear, which in turn implies that for a Gaussian reference state the Fourier transform of a solution of the master Liouville equation also satisfies a linear differential equation. 
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J. Kupsch; O. G. Smolyanov. Exact master equations describing reduced dynamics of the Wigner function. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 5, pp. 203-219. http://geodesic.mathdoc.fr/item/FPM_2006_12_5_a15/

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