In the large volume limit energy, charge, and particle density in the MCE will correspond to GCE values. This is required by the thermodynamic equivalence of ensembles for mean quantities. MCE and CE partition functions are generally obtained from their GCE counterpart by multiplication with delta-functions, which pick out a set of micro states consistent with a particular conservation law. Here it will be of considerable advantage to use Fourier representations of delta-functions, similar to the treatment in Section \[GCECE\]. This method could be considered to be a Fourier spectral analysis of the generalized GCE partition function [@clt].

The normalized conditional probability distribution of multiplicity $N_{\Omega}$ can be defined by the ratio of the values of two partition functions: $$\label{MCEprob}
P_{mce}(N_{\Omega})~\equiv~ \frac{\textrm{number of all states with $N_{\Omega}$,
    $Q$, $E$, and $\vec P = \vec 0$}}{\textrm{number of all states with $Q$,
    $E$, and  $\vec P= \vec 0$} }~.$$ The real MCE partition function and our modified version are connected as $Z(V,N_{\Omega},Q,E,\vec P)  \equiv  \mathcal{Z}^{N_{\Omega},Q,E,\vec
  P}(V,T,\mu) e^{+E/T} e^{-Q\mu/T}$. In either case the normalization in Eq.(\[MCEprob\]) is given by the partition functions with fixed values of $Q,E,\vec P$, but arbitrary particle number $N_{\Omega}$, hence $Z(V,Q,E,\vec
P) \equiv \sum_{N_{\Omega}=0}^{\infty} Z(V,N_{\Omega},Q,E,\vec P)$, or $\mathcal{Z}^{Q,E,\vec P}(V,T,\mu) 