Superfluid

A superfluid exhibits superfluidity, with phase stiffness and a phase-gradient current. In an appropriate ordered or quasi-ordered regime it supports low-speed flow without viscous dissipation, and its long-wavelength Goldstone boson has a sound-like dispersion. True thermodynamic phase selection and Bose-Einstein condensation need separate infrared conditions.

The normal component of a superfluid describes the momentum and entropy carried by excitations rather than the phase-gradient condensate current. Its velocity need not equal the superfluid velocity. In phenomenological condensate equations, prescribing its motion supplies a reference flow for dissipation; a full two-component fluid theory would also evolve its momentum and energy.

These are two responses of one liquid, not separable kinds of atoms. Their fractions need not equal the Bose-Einstein condensation fraction.^[ref]

The two-fluid model of superfluidity gives the normal component of a superfluid and the superfluid independent velocities.

An elementary uniform-flow example makes the distinction operational. Write the mass flux per unit area and the ideal entropy flux asHere $s$ is specific entropy, in $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$, and ${\rho }_s,{\rho }_n$ are response mass densities. The mass density ${\rho }_s$ used here gives the three-dimensional phase stiffness coefficient ${\rho }_s(\hslash /m{)}^2$ when ${\mathbf{v}}_s=(\hslash /m)\nabla \theta$ is substituted into the kinetic energy density ${\rho }_s{v}_s^2/2$. For a stipulated ${\rho }_s=3\rho /4$, ${\rho }_n=\rho /4$ and $v_n=3u$ along a channel, zero net mass flux requires $v_s=-u$. Yet $j_S=3\rho su$ is positive. Opposing velocities can therefore transport entropy while transporting no net mass.

This example explains why excellent apparent heat transport need not be ordinary diffusion of heat through a stationary liquid. In the ideal zero-net-mass-flux limit, the heat flux associated with this entropy transport is $T{j}_S$. Viscous losses, quantum vortices and external heat exchange require additional terms. The model also supports second sound.

Closely spaced disks undergo torsional oscillations in He-II. Viscous entrainment of the normal component contributes a temperature-dependent moment of inertia, while vortex-free superfluid need not rotate with the disks. The interpretation requires the normal component to follow the drive under the apparatus's spacing and frequency conditions. The experiment is associated with Elephter Andronikashvili.

Second sound is an entropy and temperature wave. At negligible thermal expansion, it has almost zero net mass flux. Lev Davidovich Landau predicted two sound modes in 1941.^[ref]

The thermomechanical effect couples a temperature difference to a pressure difference across a superleak. For the closed-chamber model in the apparatus figure, consider neighboring equilibrium states with small velocities, negligible height difference and pure helium-4. Let ${\mu }_m$ be chemical potential per unit mass, in $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$. The Gibbs free energy differential per unit mass givesThe superleak allows the superfluid to redistribute until the two sides have equal ${\mu }_m$. Setting $d{\mu }_m=0$ therefore givesThe sign matters: heating one side at initially equal pressure lowers its ${\mu }_m$; inflow raises its pressure until chemical balance is restored. The hotter side has the higher pressure. The integral follows the constant-${\mu }_m$ path; it need not be evaluated at fixed pressure when material properties vary appreciably.

For an original local example, stipulate constant $\rho =150\mathrm{kg}{\mathrm{m}}^{-3}$, $s=100\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$ and $T_h-T_c=0.020\mathrm{K}$. The predicted pressure difference is $300\mathrm{Pa}$. These are teaching assumptions, not helium measurements. This static thermodynamics relation predicts neither the required heater power nor the filling time: those require heat leakage, channel conductance and chamber heat capacity. An outlet would introduce kinetic energy and a flowing balance, so the static $300\mathrm{Pa}$ is not a predicted fountain height.

A superleak is a narrow or porous connection that strongly suppresses the viscous normal component of a superfluid while allowing appreciable superfluid flow. Treating ${\mathbf{v}}_n\simeq 0$ is a regime assumption, not a perfect material filter for particular atoms. The thermomechanical effect uses that distinction to maintain a temperature difference during chemical equilibration. The idealization fails when normal leakage or dissipative quantum vortex motion becomes appreciable.

For a condensate $\psi =\sqrt{n}{e}^{iS}$ with kinetic operator $-{\hslash }^2{\nabla }^2/(2m)$, its particle current is $n\mathbf{u}$ with the displayed velocity. This follows by subtracting the Gross–Pitaevskii equation from its complex conjugate. More generally, in units where the kinetic operator is $-D{\nabla }^2$, the current velocity is $2D\nabla S$. The complex argument gradient and current velocity coincide only when $D=1/2$.

Stationary one-dimensional flow has constant particle current $j=\rho v$. A lower-density segment must flow faster at the same current and also has smaller local speed of sound $\sqrt{g\rho /m}$. It can therefore lose local phonon energetic stability before a denser segment.

Combining zero-current bulk densities with quantized circulation gives an approximate winding bound. If equal halves have densities ${\rho }_0(1-a)$ and ${\rho }_0(1+a)$, $0<a<1$, the more restrictive local condition is $n^2<(m{R}^2{\mu }_0/{\hslash }^2)(1-a)/(1+a{)}^2$. A self-consistent finite-current calculation must also allow kinetic pressure to change those densities.

For a complex superfluid order parameter $\psi =\sqrt{n}{e}^{i\theta }$, the nonrelativistic Landau-Ginzburg kinetic term contains $-n\dot{\theta }$. Thus number density and phase are canonically conjugate, so a phase-selected state necessarily fluctuates in particle number.

In a first-order condensate action, density and phase are canonically paired variables. Linear density and phase oscillations are therefore coupled components of a single Bogoliubov branch, rather than two independent positive-frequency particles. For $i\dot{\psi }=-{\psi }^{\prime \prime }+\beta (|\psi {|}^2-1)\psi$, writing $\psi \simeq 1+\rho /2+i\theta$ gives $\dot{\rho }=-2{\theta }^{\prime \prime }$ and $\dot{\theta }={\rho }^{\prime \prime }/2-\beta \rho$, hence ${\omega }^2=k^4+2\beta k^2$.

Number-phase conjugacy is the canonical relation between a conserved $U(1)$ charge and the phase of its order parameter. A state of sharp number cannot also have a sharply selected broken-symmetry phase.

A thermal phase fluctuation is a finite-temperature fluctuation of the phase of a complex order parameter. For a linearly dispersing phase mode, its infrared variance is proportional to $T\int d^{d}k/k^2$, which prevents true finite-temperature long-range order when $d\le 2$.

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]