  We investigate the number fluctuations in small cells of quantum gases
pointing out important deviations from the thermodynamic limit fixed by the
isothermal compressibility. Both quantum and thermal fluctuations in weakly as
well as highly compressible fluids are considered. For the two-dimensional (2D)
superfluid Bose gas we find a significant quenching of fluctuations with
respect to the thermodynamic limit, in agreement with recent experimental
findings. An enhancement of the thermal fluctuations is instead predicted for
the 2D dipolar superfluid Bose gas, which becomes dramatic when the size of the
sample cell is of the order of the wavelength of the rotonic excitation induced
by the interaction.

Quantum Gases