Kronecker delta-function (limits of integration $\left[-\pi,\pi \right]$) for discrete quantities and the Dirac delta-function (limits of integration $\left[-\infty,\infty \right]$) for continuous quantities is important here, however for deriving an asymptotic solution it will not be. To simplify (\[MCEInt\]) we change to shorthand notation for $\phi_j =
(\phi_{N_{\Omega}}\phi_Q, \phi_E, \vec \phi_P)$ and the conserved ‘charge‘ vector $ Q^j = (N_{\Omega},Q,E,\vec P)$. We again split the single particle partition functions in two parts. The first part counts the number of momentum states observable to our detector, while the second part counts momentum states invisible to our detector: $$\begin{aligned}
\psi_{k} \left( \phi_j\right)  &=&  \frac{g_k}{\left( 2\pi\right)^3} \int
\limits_{ \Omega} d^3 p ~ e^{-\frac{\varepsilon_k - q_k \mu}{T}} ~e^{i q_{k,
    \Omega}^j \phi_j}~ +~ \frac{g_k}{\left( 2\pi\right)^3} 
\int \limits_{ \bar \Omega} d^3 p~ e^{-\frac{\varepsilon_k-q_k \mu}{T}} ~ e^{i
  q_{k,\bar \Omega}^j \phi_j}~.\end{aligned}$$ For the ‘charge‘ vector of all measured particle species $k$ we write $q^j_{k,\Omega} = (1,q_k,\varepsilon_k,\vec p_k)$ for momenta inside $\Omega$, and $q^j_{k, \bar \Omega} = (0,q_k,\varepsilon_k,\vec p_k)$ for momenta outside of $\Omega$. For all unobserved particle species we write $q^j_{k,\Omega}=q^j_{k, \bar \Omega} =(0,q_k,\varepsilon_k,\vec p_k) $. Here $q_k$ is the electrical charge of particle species $k$, and $\varepsilon_k$ and $\vec p_k$ are its energy and momentum vector. In Ref.[@clt], 