Кванттық турбуленттік: Кванттық сұйықтардағы хаостық ағын

Quantum turbulence is the name given to the turbulent flow – the chaotic motion of a fluid at high flow rates – of quantum fluids, such as superfluids. The idea that a form of turbulence might be possible in a superfluid via the quantized vortex lines was first suggested by Richard Feynman. The dynamics of quantum fluids are governed by quantum mechanics, rather than classical physics which govern classical (ordinary) fluids. Some examples of quantum fluids include superfluid helium (4He and Cooper pairs of 3He), Bose–Einstein condensates (BECs), polariton condensates, and nuclear pasta theorized to exist inside neutron stars. Quantum fluids exist at temperatures below the critical temperature at which Bose Einstein condensation takes place.

The turbulence of quantum fluids has been studied primarily in two quantum fluids: liquid Helium and atomic condensates. Experimental observations have been made in the two stable isotopes of Helium, the common 4He and the rare 3He. The latter isotope has two phases, named the A phase and the B phase. The A phase is strongly anisotropic, and although it has very interesting hydrodynamic properties, turbulence experiments have been performed almost exclusively in the B phase. Helium liquidizes at a temperature of approximately 4K. At this temperature, the fluid behaves like a classical fluid with extraordinarily small viscosity, referred to as helium I. After further cooling, Helium I undergoes Bose Einstein condensation into a superfluid, referred to as helium II. The critical temperature for Bose Einstein condensation of helium is 2.17K (at the saturated vapour pressure), while only approximately a few mK for 3He B. Although in atomic condensates there is not as much experimental evidence for turbulence as in Helium, experiments have been performed with rubidium, sodium, caesium, lithium and other elements. The critical temperature for these systems is of the order of micro Kelvin. There are two fundamental properties of quantum fluids that distinguish them from classical fluids: superfluidity and quantized circulation.

Superfluidity arises as a consequence of the dispersion relation of elementary excitations, and fluids that exhibit this behaviour flow without viscosity. This is a vital property for quantum turbulence as viscosity in classical fluids causes dissipation of kinetic energy into heat, damping out motion of the fluid. Landau predicted that if a superfluid flows faster than a certain critical velocity (or alternatively an object moves faster than in a static fluid) thermal excitations (rotons) are emitted as it becomes energetically favourable to generate quasiparticles, resulting in the fluid no longer exhibiting superfluid properties. For helium II, this critical velocity is .

Vortex lines are topological line defects of the phase. Their nucleation makes the quantum fluid's region to become a multiply connected region. As given by Fig 2, density depletion can be observed near the axis, with on the vortex line. The size of the vortex core varies between different quantum fluids. The size of the vortex core is around for helium II, for 3He B and for typical atomic condensates The simplest vortex system in a quantum fluid consists of a single straight vortex line; the velocity field of such configuration is purely azimuthal given by This is the same formula as for a classical vortex line solution of the Euler equation, however, classically, this model is physically unrealistic as the velocity diverges as This leads to the idea of the Rankine vortex as shown in fig 2, which combines solid body rotation for small and vortex motion for large values of , and is a more realistic model of ordinary classical vortices. Many similarities can be drawn with vortices in classical fluids, for example the fact that vortex lines obey the classical Kelvin circulation theorem: the circulation is conserved and the vortex lines must terminate at boundaries or exist in the shape of closed loops. In the zero temperature limit, a point on a vortex line will travel accordingly to the velocity field that is generated at that point by the other parts of the vortex line, provided that the vortex line is not straight (an isolated straight vortex does not move). The velocity can also be generated by any other vortex lines in the fluid, a phenomenon also present in classical fluids. A simple example of this is a vortex ring (a torus shaped vortex) which moves at a self induced velocity inversely proportional to the radius of the ring , where The whole ring moves at a velocity

Vortex lattices are laminar (ordered) configurations of vortex lines that can be created by rotating the system. For a cylindrical vessel of radius , a condition can be derived for the formation of a vortex lattice by minimising the expression , where is the free energy, is the angular momentum of the fluid and is the rotation, with magnitude and axial direction. The critical velocity for the appearance of a vortex lattice is then
Exceeding this velocity allows for a vortex to form in the fluid. States with more vortices can be formed by increasing the rotation further, past the next critical velocities The vortices arrange themselves into ordered configurations that are called vortex lattices.

For temperatures low enough for quantum mechanical effects to govern the fluid, quantum turbulence is a seemingly chaotic tangle of vortex lines with a highly knotted topology, which move each other and reconnect when they collide. In a pure superfluid, there is no normal component to carry the entropy of the system and therefore the fluid flows without viscosity, resulting in the lack of a dissipation scale Analogously to classical fluids, a quantum length scale (and the corresponding value in k space ) can be introduced by replacing the kinematic viscosity in the Kolmogorov length scale with the quantum of circulation where both above and below the critical temperature a Kolmogorov energy spectrum was observed that is indistinguishable from those observed in the turbulence of classical fluids. For higher temperatures, the existence of the normal fluid component leads to the presence of viscous forces and eventual heat dissipation which warms the system. As a consequence of this friction the vortices become smoother, and the Kelvin waves that arise due to vortex reconnections are smoother than in low temperature quantum turbulence. Kolmogorov turbulence arises in quantum fluids for energy input at large length scales, where the energy spectrum follows in the inertial range For length scales smaller than , instead the energy spectrum follows a regime. For temperatures in the zero limit, the undamped Kelvin waves result in more kinks appearing in the shapes of the vortices. For large length scales the quantum turbulence manifests as a Kolmogorov energy cascade (numerical simulations using the Gross Pitaevskii equation and the vortex filament model confirmed this effect ), with the energy spectrum following Lacking thermal dissipation, it is intuitive to assume that quantum turbulence in the low temperature limit does not decay as it would for higher temperatures, however experimental evidence showed that this was not the case: quantum turbulence decays even at very low temperatures. The Kelvin waves interact and create shorter Kelvin waves, until they are short enough that emission of sound (phonons), which results in the conversion of kinetic energy into heat, thus dissipation of energy. This process which shifts energy to smaller and smaller length scales at wavenumbers larger than is called the Kelvin wave cascade and proceeds on individual vortices. Low temperature quantum turbulence should thus consist of a double cascade: a Kolmogorov regime (a cascade of eddies) in the inertial range , followed by a bottle neck plateau, followed by the Kelvin wave cascade (a cascade of waves) that obeys the same law but with different physical origin. This is at current consensus, but it must be stressed that it arises from theory and numerical simulations only: there is currently no direct experimental evidence for the Kelvin wave cascade due to the difficulty of observing and measuring at such small length scales.

Quantum turbulence can be detected in 3He B in two ways: nuclear magnetic resonance (NMR) and by Andreev scattering of thermal quasiparticles. For atomic condensates, it is typical that the condensate must be expanded (by switching off the trapping potential) so that is sufficiently large for an image to be taken. This procedure has a disadvantage as it leads to the condensate being destroyed. The outcome leads to a 2 dimensional image which allows for the study of 2 dimensional quantum turbulence, but imposes a constraint studying 3 dimensional quantum turbulence using this method. Individual quantum vortices have been observed in 3 dimensions, moving and reconnecting using a technique which extracts small fractions of the condensate at a time, allowing for the observation of a time sequence of the same vortex configuration.