  For the multi-mode Dicke model in a transport setting that exhibits
collective boson transmissions, we construct the equation of motion for the
cumulant generating function. Approximating the exact system of equations at
the level of cumulant generating function and system operators at lowest order,
allows us to recover master equation results of the Full Counting Statistics
for certain parameter regimes at very low cost of computation. The
thermodynamic limit, that is not accessible with the master equation approach,
can be derived analytically for different approximations.
