  We present a method to describe driven-dissipative multi-mode systems by
considering a truncated hierarchy of equations for the correlation functions.
We consider two hierarchy truncation schemes with a global cutoff on the
correlation order, which is the sum of the exponents of the operators involved
in the correlation functions: a 'hard' cutoff corresponding to an expansion
around the vacuum, which applies to a regime where the number of excitations
per site is small; a 'soft' cutoff which corresponds to an expansion around
coherent states, which can be applied for large excitation numbers per site.
This approach is applied to describe the bunching-antibunching transition in
the driven-dissipative Bose-Hubbard model for photonic systems. The results
have been successfully benchmarked by comparison with calculations based on the
corner-space renormalization method in 1D and 2D systems. The regime of
validity and strengths of the present truncation methods are critically
discussed.
