# Quantum Fluid

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## Quantum Fluid

a fluid whose properties are determined by quantum effects. An example is liquid helium at a temperature close to absolute zero. Quantum effects begin to be manifested in a fluid at very low temperatures, when the de Broglie wavelength for particles of the fluid, calculated on the basis of their energy of thermal motion, becomes comparable to the distance between them. For liquid helium this condition is satisfied at a temperature of 3°-2°K.

According to the concepts of classical mechanics, the kinetic energy of particles of any body should decrease with temperature. At a sufficiently low temperature the particles in a system of interacting particles will display small oscillations about the positions that correspond to the minimum potential energy of the entire body. At absolute zero the oscillations should cease and the particles should begin to occupy strictly defined positions—that is, any body should become a crystal. Therefore, the very existence of fluids near absolute zero is associated with quantum effects.

The principle that the more precisely the position of a particle is fixed, the greater the spread of values for its velocity, is valid in quantum mechanics. Consequently, even at absolute zero particles cannot occupy strictly defined positions, and their kinetic energy does not become zero. So-called zero-point vibrations remain. The weaker the force of interaction among the particles and the smaller their mass, the greater will be the amplitude of the vibrations. If the amplitude of zero-point vibrations is comparable to the mean distance between particles of the body, then such a body can remain fluid down to absolute zero.

Of all substances only two isotopes of helium (^{4}He and ^{3}He) have sufficiently low mass and weak interaction among atoms at atmospheric pressure to remain fluid close to absolute zero and thus make possible the study of the specifics of a quantum fluid. Electrons in metals also have the properties of a quantum fluid.

Quantum fluids are divided into Bose fluids and Fermi fluids, depending on the difference in the properties of their particles and according to the Bose-Einstein and Fermi-Dirac statistics used to describe them. Only one Bose fluid is known—liquid ^{4}He, whose atoms have a spin (intrinsic angular momentum) of zero. Atoms of the rarer isotope ^{3}He and electrons in a metal have half-integral spin (1/2) and form Fermi fluids.

Liquid ^{4}He was the first quantum fluid to be studied comprehensively. The theoretical concepts developed to explain the basic effects in liquid helium were the basis for the general theory of quantum fluids. At 2.17°K and saturation vapor pressure, ^{4}He undergoes a second-order phase transition to the new state, He II, with specific quantum properties. The very presence of the transition point is connected with the appearance of a Bose condensate—that is, with the transition of a finite fraction of the atoms in a state having a momentum strictly equal to zero. This new state is characterized by superfluidity, which is the flow of He II without any friction through narrow capillaries and slits. Superfluidity was discovered by P. L. Kapitsa (1938) and explained by L. D. Landau (1941).

According to quantum mechanics, any system of interacting particles may exist only in certain quantum states that are characteristic of the entire system as a whole. Here the energy of the entire system may vary only in certain portions—quanta. Like an atom in which energy changes by the emission or absorption of a light quantum, in a quantum fluid a change in energy takes place by emission or absorption of elementary excitations characterized by a certain momentum *p;* energy ∊(*p* which depends on the momentum; and spin. These elementary excitations apply to the entire fluid as a whole rather than to the individual particles and are called quasiparticles because of their properties (such as the presence of momentum and spin). Sonic excitations in He II—phonons with energy ∊ = ℏ*cp*, where ℏ is Planck’s constant divided by 2π and *c* is the speed of sound—are examples of quasiparticles. As long as the number of quasiparticles is small, corresponding to low temperatures, their interaction is slight, and it may be assumed that they form an ideal gas of quasiparticles. Consideration of the properties of quantum fluids on the basis of these concepts proves to be simpler, in one sense, than the study of the properties of ordinary fluids at high temperatures, when the number of excitations is great and their properties are not analogous to the properties of an ideal gas.

If a quantum fluid flows with a certain velocity *v* through a narrow pipe or slit, its deceleration caused by friction consists in the formation of quasiparticles with a momentum opposite the velocity of the current. As a result of deceleration, the energy of the quantum fluid should decline, but not smoothly, in particular portions. For the formation of quasiparticles with the required energy, the velocity of the flow must be no less than *v*_{c} = min [∊(*p*)/*p*]; this velocity is called the critical velocity. Quantum fluids in which *v*_{c} ≠ 0 will be superfluid since at velocities less than *v*_{c} new quasiparticles are not formed and consequently the fluid is not retarded. The energy spectrum ∊ (*P*) of quasiparticles in He II, which was predicted by Landau’s theory and confirmed by experiment, satisfies this requirement.

The impossibility of formation of new quasiparticles in He II at a flow with *v* < *v*_{c} leads to unique two-fluid hydrodynamics. The aggregate of the quasiparticles present in He II is scattered and retarded by the walls of the vessel and constitutes the normal, viscous part of the fluid, and the remainder is superfluid. The appearance of vortices with quantized circulation of the velocity of the superfluid component under certain conditions (such as rotation of the vessel) is characteristic of a superfluid. In He II the propagation of several types of sound is possible: the first of these corresponds to ordinary adiabatic fluctuations of density, and the second corresponds to fluctuations of the density of quasiparticles and, consequently, of temperature. Liquid ^{3}He becomes superfluid at temperatures below 2.68 × 10^{−3}°K and a pressure of 33.87 atmospheres.

The presence of a gas consisting of quasiparticles is equally characteristic of both Bose and Fermi fluids. In a Fermi fluid some of the quasiparticles have half-integral spin and conform to Fermi-Dirac statistics. These are single-particle excitations. In addition, quasiparticles with integral spin that conform to Bose-Einstein statistics also exist in a Fermi fluid. The most interesting of these is the “zero sound,” which was predicted theoretically and was discovered in liquid ^{3}He. Fermi fluids are divided into normal and superfluid, depending on the properties of the quasi-particle spectrum.

Electrons in nonsuperconducting metals in which the energy of single-particle excitations may be as small as desired when the value of the pulse is finite, leading to *v*_{c} = 0, are classified as normal Fermi fluids. The theory of normal Fermi fluids was developed by L. D. Landau (1956–58).

Electrons in superconducting metals and liquid ^{3}He are superfluid Fermi fluids. The theory of superfluid electron Fermi fluids was developed by J. Bardeen, L. Cooper, and J. Schrieffer (1957) and by N. N. Bogoliubov (1957). According to this theory, attraction predominates among the electrons in superconductors, leading to the formation of bound pairs with momentums that are opposite but equal in absolute magnitude and with a total moment equal to zero. Finite energy must be expended for the occurrence of any single-particle excitation—the breakup of a bound pair. This leads, in contrast to normal Fermi fluids, to *v*_{c} ≠ 0, that is, to superfluidity of the electron fluid (superconductivity of the metal). A deep-seated analogy exists between superconductivity and superfluidity. As in ^{4}He, a second-order phase transition associated with the appearance of the Bose condensate of electron pairs occurs in superconducting metals. Under certain conditions vortices with a quantized magnetic flux, which are the analogue of vortices in He II, appear in second-order superconductors in a magnetic field.

In addition to the quantum fluids listed above, mixtures of ^{3}He and ^{4}He, which form a continuous transition from a Fermi fluid to a Bose fluid as the ratio of the components is gradually changed, are classified as quantum fluids. According to theoretical concepts, at exceedingly high pressures and sufficiently low temperatures all substances enter the quantum-fluid state. This may occur, for example, in some stars.

### REFERENCES

Landau, L. D., and E. M. Lifshits.*Statisticheskaia fizika*, 2nd ed. Moscow, 1964.

Abrikosov, A. A., and I. M. Khalatnikov. “Teoriia fermizhidkosti.”

*Uspekhi fizicheskikh nauk*, 1958, vol. 66, fasc. 2, p. 177.

*Fizika nizkikh temperatur*. Moscow, 1959. (Translated from English.)

Pines, D., and P. Nozières.

*Teoriia kvantovykh zhidkostei*Moscow, 1967. (Translated from English.)

S. V. IORDANSKII [11–1701–1; updated]