to compare the kinetic energy variation and the height of the repulsive potential barrier. All the states l, with velocities vl much larger than c, have a kinetic energy much bigger than the maximum value of the potential energy: no energy barrier can be formed. On the other hand, all the states l with velocities vl much smaller than c cannot lower their rotational state without going over a potential barrier.
In between these two extreme cases, there exists (for a given g) a “critical” value lc corresponding to the onset of the barrier. It is associated with a “critical velocity” vc = lcħ/mr, of the order of the sound velocity c, fixing the maximum value of vl for which this potential barrier exists. If the fluid rotational velocity in the torus is greater than vc, the liquid can slow down its rotation without going over an energy barrier, and dissipation occurs as in an ordinary viscous liquid – the fluid is said to be “normal”. If, however, the fluid velocity is less than the critical velocity, the physical system must necessarily go over a potential barrier (or more) to continuously tend towards l = 0. As this barrier results from the repulsion between all the particles and their neighbors, it has a macroscopic value. In principle, any barrier can be overcome, be it by thermal excitation, or by the quantum tunnel effect. However the time needed for this passage may take a gigantic value. First of all, it is extremely unlikely for a thermal fluctuation to reach a macroscopic energy value. As for the tunnel effect, its transition probability decreases exponentially with the barrier height and becomes extremely low for a macroscopic object. Consequently, the relaxation times of the fluid velocity may become extraordinarily large, and, on the human scale, the rotation can be considered to last indefinitely. This phenomenon is called “superfluidity”.
Figure 3: Plots of the energy of a rotating repulsive boson system, in a coherent superposition of the state l and the state l′, as a function of its average angular momentum 〈Jz〉, expressed in units of ħ. The lower dotted curve corresponds to the case where l′ = l — 1 and the interaction constant g is small (almost ideal gas). The potential energy is then negligible and the total energy presents a single minimum in 〈Jz〉 = 0. Consequently, whatever the initial rotational state of the fluid, it will relax to a motionless state l = 0 without having to go over any energy barrier, and its rotational kinetic energy will dissipate: it behaves as a normal fluid. The other two curves correspond to a much larger value of g - therefore, according to (39) to a much higher value of c. The dashed curve still corresponds to a superposition of the rotational states l and l′ = l — 1, and the solid line to the direct superposition of the state l = 3 (shown with a circle in the figure) and the ground state l′ = 0. The solid line curve presents the smallest barrier, hence determining the metastability of the current.
The higher the coupling constant g, the more l states presenting a minimum in the potential energy appear. They correspond to flow velocities in the torus that are smaller than the critical velocity. To go from the rotational state l = 1 to the motionless state l = 0, the system must go over a macroscopic energy barrier, which only occurs with a probability so small it can be considered equal to zero. The rotational current is therefore permanent, lasting for years, and the system is said to be superfluid. On the other hand, the states with higher values of l, for which the curve presents no minima, correspond to a normal fluid, whose rotation can slow down because of the viscosity (dissipation of the kinetic energy into heat).
3-d. Generalization; topological aspects
Our argument remained qualitative for several reasons. To begin with, we showed the existence for the fluid of a critical velocity vc, of the order of c, without giving its precise value. It would require a more detailed study of the potential curves such as the ones plotted in Figure 3, to obtain the precise values of the parameters for which the potential barrier appears or disappears. We also limited ourselves to simple geometries that could be described by a single variable φ not taking into account other possible deformations of the wave function. Various situations could occur, such as the creation of vortices or more complex processes, which would require a more elaborate mathematical treatment. In other words, we would have to take into account the existence of other relaxation channels for the moving fluid to come to rest, and look for the one leading to the lowest potential barrier, thereby determining the lifetime of the superfluid current.
There is, however, a more general way to address the problem, which shows that our basic conclusions are not limited to the particular case we have studied. It is based on the topological aspects of the wave function phase. When this phase varies by 2lπ as we go around the torus, it expresses a topological property characterized by the winding number l, which is an integer and cannot vary continuously. This is why, as long as the phase is well defined everywhere – i.e. as long as the wave function does not go to zero – we cannot go continuously from l to l ± 1. We already saw this in the particular example of the wave function (63): when the modulus of cl (t) varies in time from 1 to 0, while the modulus of cl′ (t) does the opposite, we necessarily went through a situation where the wave function went to zero through interference, in a plane corresponding to a certain value of φ; but the phase of the wave function is undetermined in this plane, and as we cross it, the phase undergoes a discontinuous jump. Now the canceling of the wave function of a great number of condensed bosons means the density must also be zero at that point, hence larger in other points of space. This spatial density variation introduces an energy increase, due to the finite compressibility of the fluid (as we saw in § 3-b, the energy increase in the high density regions is larger than the energy decrease in low density regions). This means there is an energy barrier opposing the change in the number of turns l of the phase. The height of this barrier must now be compared with the kinetic energy variation. As seen above, there is a drastic change in the flow regime, depending on whether the fluid velocity is smaller or larger than a certain critical velocity vc. In the first case, superfluidity allows a current to flow without dissipation, lasting practically indefinitely. In the second, no energy consideration opposes dissipation, and the rotation slows down progressively, as in an ordinary liquid.
The essential idea to remember is that superfluidity comes from the repulsive interactions, and for two reasons. First of all, they explain the presence of the energy barrier, responsible for the metastability. The second reason, even more essential, is that the repulsion between bosons constantly tends to put all the fluid particles in the same quantum state – see § 4-c of Complement CXV; thanks to this property, we were able to characterize the intermediate rotational states by a very simple wave function (63). This implies that the quantum fluid can only occupy a very limited number of states, compared to a situation where the particles would be distinguishable. Consequently, it has a hard time dissipating its kinetic energy into heat, as a classical fluid would do, and it therefore maintains its rotation over such long times that a slowing down is practically impossible to observe.
1 1 It is a “total derivative” term (the derivative describing, in a fluid, the motion of each particle). As the velocity field has a zero curl according to (51), a simple vector analysis calculation shows this term to be equal to m(v · ▽)v; it can therefore be accounted for by replacing on the left-hand side of (57) the partial derivative ∂/∂t by the total derivative d/dt = ∂/∂t + v · ∇.
2 2