Landau criterion ($v_L={\inf }_{p>0}ϵ(p)/p$)

The Landau criterion asks whether creating an excitation can lower a moving superfluid's energy. For a weakly coupled, heavy obstacle moving through a uniform liquid initially at zero temperature, let $ϵ(p)>0$ be the excitation energy in the fluid rest frame, with momentum magnitude $p$. Transferring momentum $\mathbf{p}$ from an obstacle moving at velocity $\mathbf{v}$ supplies, to leading order in its negligible recoil, energy $\mathbf{v}\cdot \mathbf{p}$. Emission requiresFor an isotropic dispersion relation, the easiest direction is parallel to $\mathbf{v}$, giving the necessary kinematic conditionAn infimum includes cases in which the limiting slope occurs only as $p\to 0$. A finite-mass obstacle of mass $M$ instead loses $\mathbf{v}\cdot \mathbf{p}-p^2/(2M)$, so the corresponding emission inequality includes the extra positive recoil term $p^2/(2M)$ on the left. This is an energy and momentum balance, not a calculation of an emission rate.

The Landau criterion excludes spontaneous emission into the specified spectrum below its threshold. Above it, kinematic permission alone does not guarantee strong coupling or rapid decay. It is not a universal observed flow threshold: walls, channel geometry and quantum vortex nucleation can enable other decay routes, and finite temperature permits scattering from existing excitations. Using only bulk phonons and rotons leaves those processes outside the model.

Landau's 1941 energy and momentum argument used a roton approximation different from the finite-momentum minimum illustrated below.^[ref]

Landau received his Nobel Prize in Physics in 1962 while hospitalized in Moscow after a car accident.^[ref]

For a deliberately simple Bose-Einstein condensation comparison, stipulate an ideal free-particle excitation law $ϵ=p^2/(2m)$. Then $ϵ/p=p/(2m)$ has infimum zero. Macroscopic one-particle occupation therefore supplies no positive emission threshold by this argument alone. By contrast a linear phonon law $ϵ=cp$ yields $v_L=c$. The dispersion relation and the collective response contain information that the occupation criterion alone does not supply; Bose-Einstein condensation and thermodynamic superfluidity must be tested separately.

To check the picture algebraically, on the stipulated roton branch,Thus tangency occurs at $q_L=\sqrt{4.3}$, slightly above the roton energy minimum. At an energy minimum $e^{\prime }=0$, but a tangent through the origin needs $e^{\prime }=e/q>0$. Minimizing energy and minimizing the energy-to-momentum ratio are different operations. This geometric construction makes the criterion useful even when the full dispersion relation is supplied numerically rather than by a simple formula.

In a parabolic condensate in the Thomas–Fermi approximation for a condensate, the local sound speed is $c(D)={\omega }_r\sqrt{R^2-D^2}/\sqrt{2}$. Equating it to the obstacle speed $\Omega D$ gives the displayed threshold. This is the stipulated local phonon estimate; a finite strongly depleting obstacle can modify flow and nucleate vortices, so the inequality is not a universal exact onset law for every stirring experiment.

For a two-dimensional Thomas–Fermi condensate at fixed coupling, $N=\pi n_0{R}^2/2\propto {\mu }^2$, so $R\propto N^{1/4}$. Slow exponential atom loss shrinks the cloud while a fixed stirring radius becomes a larger fraction of its size. At $\Omega ={\omega }_r/\sqrt{2}$ and initial $D/R=1/2$, the local-sound-speed threshold is reached at $t_c=2\tau \ln 2$, before the shrinking cloud edge reaches the beam.

The excitation-based kinematic velocity bound in the Landau criterion. For a heavy weak obstacle and an isotropic excitation spectrum, emitting momentum parallel to the obstacle's motion costs $E(p)-vp$; requiring this to remain positive for every excitation gives the displayed infimum. Vortex nucleation or other dynamical processes can make a measured critical velocity lower than this bound.

The contact-gas spectrum $E_k=\sqrt{{ϵ}_k({ϵ}_k+2g{n}_0)}$ gives $\min E_k/(\hslash k)=\sqrt{g{n}_0/m}$. This energetic homogeneous-flow bound need not equal a measured critical speed when vortices or boundaries enable other dissipation.

For a repulsive homogeneous condensate with Bogoliubov energy $E(p)=\sqrt{(p^2/2m)(p^2/2m+2\mu )}$, the infimum of $E(p)/p$ is the speed of sound $\sqrt{\mu /m}$. This is energetic stability against the given bulk modes, not a universal measured dissipation threshold.

A roton is an excitation near a local minimum of a superfluid's dispersion relation at nonzero momentum $p_0$. The displayed local approximation defines a positive energy gap $\Delta$ and positive curvature effective mass $m_r$; it is not a free helium atom or a macroscopic quantum vortex.^[ref] In the stipulated plot, these correspond to $p_0=2p_{\ast }$, $\Delta =0.6E_{\ast }$ and $m_r=p_{\ast }^2/(4E_{\ast })$. The plot illustrates how a low energy at appreciable momentum can lower the Landau criterion below the speed of sound, without assigning any measured parameter to real helium.

Landau's 1947 revision joined the low-momentum linear part to a spectrum with a finite-momentum minimum, retaining the macroscopic superfluid theory while changing its excitation model.^[ref]