# Chapter 3: What else besides the condensate growth term?

This post is based on my paper.

Let us recall from Chapter 1 that the BEC occupies the lowest quantum state, at that point macroscopic quantum phenomena become apparent. Above the BEC are excited atoms, which occupy higher quantum states. The are two types of interactions:

• excited atoms – excited atoms,
• condensate atoms – excited atoms.

“The behaviour of condensates at finite temperatures is a frontier of manybody physics, the experimental exploration of which has so far concentrated mainly on the initial formation of condensates.” (James R. Anglin & Wolfgang Ketterle, Nature, 2002)

Up to now, under some physical constraints, we have assumed that the interactions between excited atoms are much smaller than the interactions between condensate atoms and excited atoms, which are described by the $C_{12}$ operator.

In this chapter, we will consider the full system in which both types of interactions are taken into account. Let us denote the operator that describes the interactions between excited atoms by $C_{22}$. We then have

• excited atoms – excited atoms: $C_{22}$ (Uehling-Ulenbeck term),
• condensate atoms – excited atoms: $C_{12}$ (condensate growth term).

As usual, the dynamics of the condensate itself follows the nonlinear Schrodinger equation (NLS).

Finally, the full system should contain the NLS and a kinetic equation of $C_{12}$ and $C_{22}$. This can be seen in the following Figure:

Figure 1: The Bose-Einstein Condensate (BEC) and the excited atoms.

And the full system for the excited atoms density distribution $f(t,r,p)$ and the condensate wave function $\Phi(t,r)$ reads (we will come back to the derivation of this equation in my joint work with Linda Reichl in Chapter 6):

$\frac{\partial f}{\partial t}+p\cdot\nabla_{r} f=Q[f]:=C_{12}[f]+C_{22}[f], (t,r,p)\in\mathbb{R}_+\times\mathbb{R}^3\times\mathbb{R}^3 (1)$

$f(0,r,p)=f_0(r,p), (r,p)\in\mathbb{R}^3\times\mathbb{R}^3,$

$C_{12}[f](t,r,p_1):=N_c\lambda_1\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}K^{12}({p}_1,{p}_2,{p}_3)\delta({p}_1-{p}_2-{p}_3)\delta(\mathcal{E}_{{p}_1}-\mathcal{E}_{{p}_2}-\mathcal{E}_{{p}_3})[(1+f(t,r,{p}_1))f(t,r,{p}_2)f(t,r,{p}_3)-f(t,r,{p}_1)(1+f(t,r,{p}_2))(1+f(t,r,{p}_3))]d{p}_2d{p}_3-$

$-2N_c\lambda_1\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}K^{12}({p}_1,{p}_2,{p}_3)\delta({p}_2-{p}_1-{p}_3)\delta(\mathcal{E}_{{p}_2}-\mathcal{E}_{{p}_1}-\mathcal{E}_{{p}_3})[(1+f(t,r,{p}_2))f(t,r,{p}_1)f(t,r,{p}_3)-f(t,r,{p}_2)(1+f(t,r,{p}_1))(1+f(t,r,{p}_3))]d{p}_2d{p}_3,$

$C_{22}[f](t,r,p_1):=\lambda_2\iiint_{\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}}K^{22}({p}_1,{p}_2,{p}_3,p_4)\delta({p}_1+{p}_2-{p}_3-{p}_4)\delta(\mathcal{E}_{{p}_1}+\mathcal{E}_{{p}_2}-\mathcal{E}_{{p}_3}-\mathcal{E}_{{p}_4})[(1+f(t,r,{p}_1))(1+f(t,r,{p}_2))f(t,r,{p}_3)f(t,r,{p}_4)-f(t,r,{p}_1)f(t,r,{p}_2)(1+f(t,r,{p}_3))(1+f(t,r,{p}_4))]d{p}_2d{p}_3d{p}_4,$

where $\lambda_1,\lambda_2$ are some constants. $N_c=|\Phi|^2$ is the condensate density, $\Phi$ satisfies

$i \hbar \frac{\partial \Phi(r,t)}{\partial t}=(-\frac{\hbar \Delta_r}{2m}+g|\Phi(r,t)|^2)\Phi(r,t),\Phi(0,r)=\Phi_0(r), \forall r\in\mathbb{R}^3,$

and $\mathcal{E}_{{p}}$ is the Bogoliubov dispersion law

$\mathcal{E}_p=\mathcal{E}(p)=\sqrt{\kappa_1 |p|^2 + \kappa_2 |p|^4} (2),$

for $\kappa_1$, $\kappa_2$ are some positive constants, $m$ is the mass of the particles, $g$ is the interaction coupling constant.

Let us try to understand each term from the equation. From Chapter 1, we know that the condensate growth term $C_{12}$ describes the $1\leftrightarrow 2$ interaction between excited atoms and the condensate. Notice that $C_{12}$ depends on $N_c$ and vanishes when $N_c=0$. The Uehling-Ulenbeck (Boltzmann-Norheim) term $C_{22}$ describes the interaction between excited atoms. Following the same argument for the quantum Boltzmann equation for fermions, we suppose that two microscopic boxes with momenta $p_1$ and $p_2$ collide. After the collision, the particles will move to the new boxes $p_3$ and $p_4$. For the fermion case, the collision contains the term

$f(t,p_3)f(t,p_4)(1-f(t,p_1))(1-f(t,p_2)) -f(t,p_1)f(t,p_2)(1-f(t,p_3))(1-f(t,p_4)),$

where the factor $1-f(t,p_i)$ indicates the fullness of the box $p_i$. From Chapter 1, we also know that bosons do not follow the Pauli Exclusion Principle: the bosons in $p_i$ are very friendly and they are trying to attract their friends to their home. As a result, we use the factor $1+f(t,p_i)$ and modify the collision as

$f(t,p_3)f(t,p_4)(1+f(t,p_1))(1+f(t,p_2)) -f(t,p_1)f(t,p_2)(1+f(t,p_3))(1+f(t,p_4)).$

Taking into account the conservation of momentum and energy, we obtain

$\delta(p_1+p_2-p_3-p_4)\delta(\mathcal{E}_{p_1} + \mathcal{E}_{p_2}- \mathcal{E}_{p_3} - \mathcal{E}_{p_4})[f(t,p_3)f(t,p_4)(1+f(t,p_1))(1+f(t,p_2))-f(t,p_1)f(t,p_2)(1+f(t,p_3))(1+f(t,p_4))],$

which gives the form of $C_{22}.$

The $C_{22}$ operator describes the $2\to2$ interaction or the $4$-wave interaction between excited atoms.

Figure 2: The $C_{22}$ collision operator

Above the BEC critical temperature, the density of the condensate $n_c$ is $0$, then $C_{12}=0$. The homogeneous version of Equation (1) is reduced to the Uehling-Ulenbeck equation

$\frac{\partial f}{\partial t}=C_{22}[f], \forall p\in\mathbb{R}^3, (3)$

which has a blow-up positive radial solution in the $L^\infty$ norm if the mass of the initial data is too concentrated around the origin due to a result by Miguel Escobedo and Juan JL Velazquez (Invent. Math. 2015). Let us mention that when the temperature is above the BEC critical temperature, the energy is of the form $\frac{p^2}{2m}$. The collision of two microscopic boxes of particles with momenta $p_1$ and $p_2$ changes the momenta into $p_3$ and $p_4$; and the conservation laws read: $|p_1|^2+|p_2|^2=|p_3|^2+|p_4|^2, p_1+p_2=p_3+p_4.$

Since $p_1$, $p_2$, $p_3$, $p_4$ belong to the sphere centered at $\frac{p_1+p_2}{2}$ with radius $\frac{|p_1-p_2|}{2}$, the collision operator $C_{22}$ in (2) can be expressed as a integration on a sphere, following the strategy for the classical Boltzmann operator. In (1), $\mathcal{E}_p$ is approximated by the Bogoliubov dispersion law (2) , which means that the collision operators are integrals on much more complicated manifolds. New estimates on these energy manifolds are required. Moreover, (3) conserves the mass of the solution, while the full equation (1) does not.

In this work, we first decouple the QB and NLS equations, then show a global existence and uniqueness result for strong, positive, radial solutions to the spatially homogeneous system.

In the next chapter, we will come back to the condensate growth term to study the convergence to equilibrium.