# Chapter 8: Quantum hydrodynamic approximations to finite temperature trapped Bose gases

This post is based on my joint paper with Shi Jin.

At temperature $T$, bosons of mass $m$ can be regarded as quantum-mechanical wavepackets which have an extent on the order of a thermal de Broglie wavelength $\lambda_{dB} = \left(\frac{2 \pi \hbar^2}{m k_B T}\right)^\frac12$, where $k_B$ is the Boltzmann constant. The de Broglie wavelength $\lambda_{dB}$ describes the position uncertainty associated with the thermal momentum distribution. When the gas temperature is high $T>T_{BEC}$, $\lambda_{dB}$ is very small and the weakly interacting gas can be treated as a system of “billiard balls”. The dynamics of the gas is described by the Boltzmann-Norheim (Uehling-Ulenbeck) equation, whose operator sometimes reads

${C}_{22}[f](t,r,p_1)=\int_{{R}^{3}\times{R}^{3}\times{R}^{3}}\delta({p}_1+{p}_2-{p}_3-{p}_4)\delta(\mathcal{E}_{p_1}+\mathcal{E}_{p_2}-\mathcal{E}_{p_3}-\mathcal{E}_{p_4})[(1+\vartheta f_1)(1+\vartheta f_2)f_3f_4-f_1f_2(1+\vartheta f_3)(1+ \vartheta f_4)]d p_2d p_3d p_4,$

where $\vartheta$ is proportional to $\hbar^3$. In the semiclassical limit, as $\vartheta$ tends to $0$, the quantum Boltzmann collision operator becomes the classical one. This means at high temperature, the behavior of the ”billiard balls” Bose gas is, in some sense, still very similar to classical gases.

At the BEC transition temperature, $\lambda_{dB}$ becomes comparable to the distance between atoms. As a result, the atomic wavepackets “overlap” and the indistinguishability of atoms becomes important. At this temperature, bosons undergo a quantum-mechanical phase transition and the Bose-Einstein condensate is formed. When the temperature of the gas is finite $T_{BEC}>T>0$K, the trapped Bose gas is composed of two distinct components: the high-density condensate, being localized at the center of the trapping potential, and the low-density cloud of thermally excited atoms, spreading over a much wider region. The dynamics of the thermal cloud atoms is described by the quantum kinetic system (see Chapter 6).

The system contains two equations: a quantum Boltzmann equation describing the non-condensate atoms (with two types of collisions, one between excited atoms and one between condensate atoms and excited atoms), and a nonlinear Schrodinger (or Gross-Pitaevski) equation for the condensate. The hydrodynamic limits of the system is an interesting mathematical question, first studied in the PhD thesis of Thibaut Allemand (Paris 6, 2010) where an Euler limit has been derived. This derivation relies on the assumption that, in the considered trapped Bose gas, the noncondensate and condensate share the same local equilibrium. It is known that the condition of complete local equilibrium between the condensate and the thermal cloud requires the energy of a condensate atom in the local rest frame of the thermal cloud to be equal to the local thermal cloud chemical potential. When the condition is satisfied, there is no exchange of particles between the condensate and the thermal cloud. As a consequence, in the derived fluid system, the mass of each component – condensate and non-condensate – does not exchange. Note that the two-fluid low-frequency dynamics of superfluid $^4$He was first developed by Tisza and Landau. Their description accounts for the characteristic features associated with superfluidity in terms of the relative motion of superfluid and normal fluid degrees of freedom, and was shown to be a consequence of a Bose broken symmetry. In the Landau two-fluid theory, the two components superfluid and normal fluid exchange mass. In this paper, we revisit the derivation of the Euler hydrodynamic limit of the system by a different point of view: we assume that even if the thermal cloud atoms are in equilibrium among themselves, the noncondensate and condensate parts may not be in local equilibrium with each other. Moreover, the derivation of the Navier-Stokes approximation of the system is also provided via the Chapman-Enskog expansion. In such circumstance, the Euler and Navier-Stokes approximations include the mass exchange between the condensate and the non-condensate.

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