# [Physics] Intuitive explanation for superfluidity

superfluidity

I know that superfluidity is caused by the fluid having zero viscosity. This only happens at very low temperature, so the fluid (e.g. Helium-4) is a Bose-Einstein condensate.

I also know that in a Bose-Einstein condensate all the particles are in the ground state.

Now, that said:

How can this explain superfluidity? Many websites say that the particles behave as a sigle giant matter wave, but what is this due to and what does it mean physically?

Maybe it would be useful to clear up a little bit the all thing :

First of all, Bose-Einstein Condensation does not necessarily implies superfluidity:

1. Bose-Einstein condensation (BEC) is characterized by a macroscopic occupation of a many-particle state $\Phi$ which correspond to the product of individual particle groud-states with zero-momentum $|\textbf{k=0}\rangle$. Therefore, such system shows strong (quantum) phase coherence effects, said "particles are acting as one".

2. Superfluidity characterize a fluid that do not undergoes any dissipation by viscosity phenomena (i.e. its viscosity $\eta=0$). A superfluid has also a quasi-infinite thermal conductivity.

One can show that superfluidity only occurs in a BEC composed by interacting particles. In other words, an ideal gas of non-interacting boson can not be superfluid.

The reason is that interactions between particles allows to conserve the phase coherence for the all system. Since BEC generally occurs at pretty low temperatures, the thermal De Broglie wavelength associated to each particle is pretty large :

$$\lambda_{th}=\sqrt\frac{2\pi\hbar^2}{m\,k_B\,T}\underset{T\rightarrow 0}{\rightarrow}+\infty$$

so that there is an overlap between each particle wave-function in the condensate (hence the single "big" wave-function for the all condensate). This overlap is what I would call "a phase coherence effect".

And, the all game is that interactions between particles preserves such very special state from external perturbations, i.e. from phase decoherence phenomena. Before explaining this, lets introduce a key notion, which is the healing length $\xi$ of a condensate :

$$\xi=\sqrt\frac{\hbar^2}{m\,g\,n}$$

where $g$ is a constante characterizing interactions, and $n$ the density of the BEC. $\xi$ can be interpreted as the length scale in which density and $\Phi$ wave-function phase fluctuations are removed in the BEC.

Let us now considere a BEC flowing in a $D$ diameter pipe.

• Without interaction $g=0$, leading $\xi\rightarrow +\infty$, more precisely $\xi\rightarrow D$. Thus, such condensate can not recover from any external perturbation in any point $\vec{r}$ of the BEC (for instance, lets say that a particle "hitting" the pipe is a phase perturbation). Therefore, the system has to dissipate its energy, leading to $\eta\neq 0$.

• With interactions $g\neq 0$, $\xi$ is finite, eventually microscopic so that $D>>\xi$. In this case, phase coherence will be preserved in any point $\vec{r}$ of the BEC, except in a small $\xi$ width area near the contact surface between the pipe and the BEC. Then, the BEC is disspating energy only in this tiny area, and preserving the phase cohenrence everywhere else. This leads to $\eta\simeq 0$.

So that was for qualitative explanation. For more theoretical insight, one can show that the Gross-Pitaevskii equation, which describes dynamical properties of a BEC, can be equivalently re-written (with some assumptions) like a Navier-Stokes equation for an irrotational inviscid flow.