where only multiplicity distributions in the full momentum space were considered, the general ‘charge‘ vector took the form $q^j_{k,4\pi} =
(n_k,q_k,\varepsilon_k,\vec p_k)$, where $n_k$ is the multiplicity of this particle. For stable particles $n_k=1$ in case they are observed, and $n_k=0$ if they are not measured, while for unstable particles $n_k$ could also denote the number of measurable decay products.

For large system volume the main contribution to the integral (\[MCEInt\]) comes from a small region around the origin [@VolDep]. Thus we proceed by Taylor expansion of the integrant of (\[MCEInt\]) around $\phi_j=\vec
0$. In this context $ \Psi \left( \phi_j\right) = \sum_k \psi_k \left(
  \phi_j\right) $ would be called the cumulant generating function (CGF). Cumulants (expansion terms) are defined by differentiation of the CGF at the origin: $$\label{kappa_n}
\kappa_n^{j_1,j_2,\dots,j_n } ~\equiv~ \left(-i\right)^n\frac{\partial^n
\Psi \left( \phi_j \right) }{\partial \phi_{j_1}
  \partial \phi_{j_2} \dots \partial \phi_{j_n} }
\Bigg|_{\phi_j = \vec 0}~.$$ Generally are cumulants tensors of rank $n$ and dimension defined by the number of conserved quantities. Here $\kappa_1$ is a 6 component vector, while $\kappa_2$ is a $6 \times 6$ matrix, etc.

The parts of the integrant related to discrete quantities, i.e. $N_{\Omega}$ and $Q$, are now not $2\pi$ periodic anymore 