Landau superfluidity criterion

The Landau criterion for superfluidity is the ratio between the energies and momenta of the elementary excitations of the system ( phonons ), which makes it possible to be in a superfluid state.

Content

1 Formulation of the criterion
2 Conclusion of the criterion
3 See also
4 Literature

Criterion wording

A quantum fluid can be in a superfluid state if, for the energy spectrum of its elementary excitations ε ( p ), the minimum value of the ratio of the energy of the quasiparticle to its momentum ε ( p ) / p is greater than zero.

Criterion inference

Consider a fluid moving along the capillary with a speed v = const . In the presence of viscosity, the kinetic energy will dissipate inside the liquid itself and at the point of contact with the capillary and, as a result, the flow velocity will slow down. Dissipation occurs due to the appearance of elementary excitations.

We turn to the coordinate system in which the liquid is at rest, and the capillary moves with speed - v . Let us consider one elementary excitation with momentum p and energy ε ( p ). Then the energy E _{0 of the} liquid (in the coordinate system in which it was originally resting) becomes equal to the energy of this excitation ε , and its momentum P ₀ - to the momentum p . We now turn back to the coordinate system in which the capillary rests. According to the laws of energy and momentum conversion when moving from one inertial reference frame to another (in the nonrelativistic case), the new values of energy and momentum have the form:

$E=E_{0}+\mathbf {P} _{0}\mathbf {v} +{\frac {Mv^{2}}{2}},$ ${\ displaystyle E = E_ {0} + \ mathbf {P} _ {0} \ mathbf {v} + {\ frac {Mv ^ {2}} {2}},}$

$\mathbf {P} =\mathbf {P} _{0}+M\mathbf {v} ,$ ${\ displaystyle \ mathbf {P} = \ mathbf {P} _ {0} + M \ mathbf {v},}$

where M is the mass of liquid. We substitute here the known values of E ₀ and P ₀ , we obtain:

$E=\varepsilon +\mathbf {p} \mathbf {v} +{\frac {Mv^{2}}{2}}.$ {\ displaystyle E = \ varepsilon + \ mathbf {p} \ mathbf {v} + {\ frac {Mv ^ {2}} {2}}.}

Expression $\varepsilon +\mathbf {p} \mathbf {v}$ ${\ displaystyle \ varepsilon + \ mathbf {p} \ mathbf {v}}$ there is a change in the energy of the liquid due to the appearance of excitation. This change should be negative, since dissipative forces act. Hence we obtain the expression for the flow velocity in the presence of friction

$v>{\frac {\varepsilon }{p}}.$ ${\ displaystyle v> {\ frac {\ varepsilon} {p}}.}$

This inequality must be satisfied at least for some values of the momentum p of elementary excitation. Accordingly, in the absence of friction, i.e., when observing superfluidity, for any values of the momentum of elementary excitations p , the inequality

$v\leq {\frac {\varepsilon }{p}}.$ ${\ displaystyle v \ leq {\ frac {\ varepsilon} {p}}.}$

This condition corresponds to the impossibility of the formation of a quasiparticle and, therefore, the impossibility of dissipation. Thus, for the possibility of observing superfluidity in such a system, it is sufficient that the minimum value of the ratio ε ( p ) / p be greater than zero.

Literature

E. M. Lifshits, L.P. Pitaevsky, Theoretical Physics, vol. 9, Statistical Physics, part 2, § 23.

Landau superfluidity criterion

Content

Criterion wording

Criterion inference

See also

Literature

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