For spectral actions consisting of the average number of particles and
arising from open systems made of general free $q$-particles (including Bose,
Fermi and classical ones corresponding to $q=\pm 1$ and $0$, respectively) in
thermal equilibrium, we compute the asymptotic expansion with respect to the
natural cut-off. We treat both relevant situations relative to massless and non
relativistic massive particles, where the natural cut-off is $1/\beta=k_{\rm
B}T$ and $1/\sqrt{\beta}$, respectively. We show that the massless situation
enjoys less regularity properties than the massive one. We also treat in some
detail the relativistic massive case for which the natural cut-off is again
$1/\beta$. We then consider the passage to the continuum describing infinitely
extended open systems in thermal equilibrium, by also discussing the appearance
of condensation phenomena occurring for Bose-like $q$-particles, $q\in(0,1]$.
We then compare the arising results for the finite volume situation (discrete
spectrum) with the corresponding infinite volume one (continuous spectrum).



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