\equiv \sum_{N_{\Omega}=0}^{\infty}
\mathcal{Z}^{N_{\Omega},Q,E,\vec  P}(V,T,\mu)$. However when taking the ratio (\[MCEprob\]) auxiliary parameters chemical potential and temperature drop out: $$\label{MCEprob2}
P_{mce}(N_{\Omega})~ \equiv~ \frac{Z(V,N_{\Omega},Q,E,\vec P)}{Z(V,Q,E,\vec
  P)} ~=~ \frac{\mathcal{Z}^{N_{\Omega},Q,E,\vec
    P}(V,T,\mu)}{\mathcal{Z}^{Q,E,\vec P}(V,T,\mu)}~.$$ The main difference between the two versions of partition functions is that for $Z(V,N_{\Omega},Q,E,\vec P)$ one is confronted with a heavily oscillating (or even irregular) integrant, while for $\mathcal{Z}^{N_{\Omega},Q,E,\vec
  P}(V,T,\mu)$ the integrant becomes ($T$,$\mu$ correctly chosen) very smooth. Thus, introduction of $T$ and $\mu$ allows to derive (and use) the asymptotic solution of Ref.[@clt].

We have a total number of 6 conserved ‘charges‘, and hence we need to solve the 6-dimensional Fourier integral for the numerator in Eq.(\[MCEprob2\])[^1]: $$\begin{aligned}
\label{MCEInt}
\mathcal{Z}^{N_{\Omega},Q,E, \vec P}&=&  
\int \limits_{-\pi}^{\pi} \frac{d \phi_{N_{\Omega}}}{2\pi}  
\int \limits_{-\pi}^{\pi} \frac{d \phi_Q}{2\pi}  
\int \limits_{-\infty}^{\infty} \frac{d \phi_E}{2\pi}  
\int \limits_{-\infty}^{\infty} \frac{d \phi_{P_x}}{2\pi}  
\int \limits_{-\infty}^{\infty} \frac{d \phi_{P_y}}{2\pi}  
\int \limits_{-\infty}^{\infty} \frac{d \phi_{P_z}}{2\pi} \nonumber \\ 
&\times& e^{-iN_{\Omega} \phi_{N_{\Omega}}}~ e^{-iQ\phi_Q} ~e^{- iE \phi_E} 
~e^{-iP_x \phi_{P_x}}~ e^{-iP_y \phi_{P_y}}~ e^{-iP_z \phi_{P_z}}   \nonumber \\ 
&\times& \exp \left[  V \sum_k \psi_k \left( \phi_{N_{\Omega}}\phi_Q, \phi_E,
\phi_{P_x},\phi_{P_y},\phi_{P_z} \right) \right].\end{aligned}$$ The summation in (\[MCEInt\]) should be taken over the single particle partitions $V \psi_k=z_k$ of all considered particle species $k$. The Wick-rotated fugacities $\phi_{Q}$, etc. are related to the individual conservation laws. The distinction between the 