Remarks

Phase space methods are powerful tools for solving Master equations. The resulting PDEs, however, are often non-trivial and cannot be solved exactly. This is particularly true if more than one degree of freedom is involved and one has to solve systems of PDEs.

Systems of partial differential equations are really complicated beasts: in contrast to systems of ordinary differential equations, they are not equivalent to a single PDE of higher order, cf. the discussion in Courant/Hilbert `Methoden der Mathematischen Physik'.

Related problems occur in the theory of the Laser, where one has to deal with PDEs containing derivatives up to infinite order. This is discussed in the book by Scully/Lamb. Another, very recent challenge are systems of Master equations with non-linear couplings between bosonic and electronic degrees of freedom in nano-electromechanical systems.