= Density susceptibility
{title2=$\chi_\rho(q)=\delta\rho_q/\delta\mu_q$}
The static <density susceptibility> is the <linear response> of <number density> to a spatially varying <chemical potential>:
$$
\chi_\rho(q)=\frac{\delta\rho_{\mathbf q}}{\delta\mu_{\mathbf q}}.
$$
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:
$$
\delta E=\int d^3x\left[
\frac g2(\delta\rho)^2+
\frac{\hbar^2}{8m\rho_0}|\nabla\delta\rho|^2
-\delta\mu\,\delta\rho\right],
$$
with repulsive <contact interaction> $g>0$, background number density $\rho_0$ and particle <mass> $m$. Stationarity gives
$$
g\,\delta\rho-\frac{\hbar^2}{4m\rho_0}\nabla^2\delta\rho=\delta\mu.
$$
Thus
$$
\chi_\rho(q)=\left[g+\frac{\hbar^2q^2}{4m\rho_0}\right]^{-1}.
$$
The expression without $\hbar^2$ uses units $\hbar=1$. 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>:
$$
\kappa_T=\rho_0^{-2}\chi_\rho(0)=\frac1{g\rho_0^2}.
$$
Thermal and interaction corrections beyond the stated zero-temperature approximation can change this response.
Back to article page
