$-e({\mathbf{E}}+{\mathbf{v}}\times\mathbf{B})$ is the Lorentz force with $\mathbf{B}$ being the magnetic flux density, $\gamma$ is a damping constant, $g[n]$ is an energy functional of the fluid, and the term $\nabla\left(\frac{\delta g[n]}{\delta n}\right)$ denotes the quantum pressure. Complementary to Eq. \[eq:hd\], the dynamics of the free-electron density is given by $$\partial_t n+\nabla\cdot(n{\mathbf{v}})=0,$$ which is the well-known continuity relation that relates the velocity ${\mathbf{v}}$ and the density $n$.

The hydrodynamic dynamics described by Eq. (\[eq:hd\]) is obviously nonlinear in ${\bf v}$, but in the following we only consider the linear response of the electron gas on external fields. One can write a perturbation expansion ${\bf v} \simeq {\bf v}_{0} + {\bf v}_{1}$ and similarly for the electric and magnetic fields and for the density. Since in the absence of an external field ${\bf v} = {\bf v}_{0} = {\bf 0}$, both the nonlinear term ${\mathbf{v}}\cdot\nabla{\mathbf{v}}$ and the magnetic induction field $\mathbf{B}$ disappear due to the linearization [@ToscanoEtal2015]. If we furthermore assume the energy functional to be of the Thomas-Fermi form, then we obtain for the linearized quantum pressure $$\label{eq:qpressure}
-\nabla\left(\frac{\delta g[n]}{\delta n}\right)= -m_e\beta^2\frac{1}{n_0}\nabla n,$$ where $\beta^2=\frac{3}{5}v_F^2$ with $v_F$ being the Fermi velocity. The zero-order (i.e. equilibrium) density $n_0$ is constant within the plasmonic 