= Infrared qualification of condensate fluctuation counting
A small noncondensed fraction provides a useful global expansion parameter for a <weakly interacting Bose gas>, but does not bound every long-wavelength mode individually. In <Bogoliubov theory>, mode amplitudes can grow as momentum approaches zero, while their integrated contribution remains finite in some dimensions and regimes.
A fluctuation integral includes both the small-momentum behavior and the momentum-space measure. For example, an integrand proportional to $1/k$ contributes $\int_0 dk\,k^{d-2}$ in $d$ spatial dimensions, which converges at zero only for $d>1$. Thermal factors can make the infrared behavior more singular. Thus <condensate fluctuation power counting> must be supplemented by checking the relevant integrals, system size and <temperature>. The uniform phase mode and fixed <particle number> constraint also require their own treatment.
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