 Density susceptibility

The static density susceptibility is the linear response of number density to a spatially varying chemical potential:Here $\rho$ denotes particle number per volume, not mass density, and $\mathbf{q}$ is a wavevector.

For a uniform weakly interacting Bose-Einstein condensate at zero temperature, the Gross–Pitaevskii equation gives a quadratic static energy for a small density perturbation:with repulsive contact interaction $g>0$, background number density ${\rho }_0$ and particle mass $m$. Stationarity givesThusThe expression without ${\hslash }^2$ uses units $\hslash =1$. The gradient term suppresses response at short wavelengths; the inverse susceptibility is the density stiffness.

At $q=0$, $(\partial \rho /\partial \mu {)}_T=1/g$ within this mean-field model. The isothermal compressibility additionally contains the density normalization:Thermal and interaction corrections beyond the stated zero-temperature approximation can change this response.