# Chapter 7: Coupling kinetic and Schrodinger equations – A simplified model by Soffer and Tran.

This post is based on my joint paper with Avy Soffer.

When a bose gas is cooled below the Bose-Einstein critical temperature, the Bose-Einstein condensate is formed, consisting of a macroscopic number of particles, all in the ground state of the system. As we have already known from previous chapters (Chapters 1 &3), a finite temperature trapped Bose gas is composed of two distinct components, the Bose-Einstein Condensate and the noncondensate – thermal cloud. The density function of the thermal cloud satisfies a quantum Boltzmann equation and the wave function of the condensate follows the nonlinear Gross-Pitaevski equation. The coupled dynamics of the kinetic and Gross-Pitaevskii equations brings in a whole new class of phenomena.

In this joint work with Avy Soffer, we are interested in the long time dynamics of the kinetics-Schrodinger coupling system. In order to explain the physical intuition behind our work, let us look at the classical example of surface waves on ocean. Wind blowing along the air-water interface is the reason that creates ocean surface waves. As wind continues to blow, it forms a steadily disturbance on the surface, that leads to the rise of the wave crests. Surface waves are the waves we see at beaches and they occur all over the globe. The coupling of the two states of matters gas – liquid of this phenomenon is, in some sense, very similar to the coupling thermal cloud – Bose-Einstein Condensate. Suppose that the air above the ocean, after quite a long time, stands still and its density function reaches the equilibrium distribution. If this happens, we will not see ocean waves anymore, under the assumption that tidal waves, tsunamis and other waves are negligible. That means the ocean wave function is scattered into a constant function. This gives us an intuition for the long time dynamics of the system thermal cloud – Bose-Einstein Condensate: The thermal cloud would converge to equilibrium, as a normal gas; On the other hand, the wave function of the condensate would also converge to a constant function. In other words, we hope to prove the convergence to equilibrium to the solution of the quantum Boltzmann equation, and the scattering theory for the solution of the nonlinear Schrodinger equation.

With our current technologies, in order to study the scattering theory for the solution of the nonlinear Schrodinger equation, we will need to solve at least two problems:

1. The nonhomogeneous quantum Boltzmann equation has a strong, unique global equation, since we will need to put this solution back into the Schodinger equation to do the coupling.
2. The solution of the nonhomogeneous quantum Boltzmann equation converges to equilibrium with a sufficiently fast rate (exponentially), since the coupling behaves like a confining potential for the Schodinger equation and we will want this potential to nicely behave.

Unfortunately, both of these two problems still remains opened, even in the context of the classical Boltzmann equation. Therefore, as the first step to understand the long time dynamics of the kinetics-Schodinger coupling, let us simplify the system by replacing the nonhomogeneous quantum Boltzmann equation by a linear quantum Boltzmann equation, and study the following coupling system:

$\frac{\partial f}{\partial t}(t,r,p)+{p}\cdot\nabla_{{r}} f(t,r,p) = N_c(t,r)L[f](t,r,p), (t,r,p) \in \mathbb{R}_+\times\mathbb{R}^3\times\mathbb{R}^3,$

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

$i \frac{\partial \Psi(t,r)}{\partial t} = \Big(-{ \Delta_{{r}}} + |\Psi(t,r)|^2 + U(t,r) \Big)\Psi(t,r),(t,r) \in \mathbb{R}_+\times\mathbb{R}^3,$

$\Psi(0,r)=\Psi_0(r), \forall r\in\mathbb{R}^3,$

$\rho[f](t,r)=\int_{\mathbb{R}^3}f(t,r,p')dp',$

$N_c(t,r)=C^*_\vartheta\int_{\mathbb{R}^3}|\Psi|^2(t,r)e^{-|r-r'|/\vartheta}dr',$

$U(t,r)=-1-V(t,r), \ V(t,r) = \rho(t,r),$

where $L$ is some linear quantum Boltzmann operator and $\vartheta$ is some positive constant, $C^*_\vartheta$ is the normalized constant such that $C^*_\vartheta \int_{\mathbb{R}^3}e^{-|p|/\vartheta}dp=1.$

We impose the following boundary condition on $\Psi$

$\lim_{|r|\to\infty}\Psi=1.$

We can see that the coupling lies in the confining potential $U$ of the nonlinear Schrodinger equation and the function $N_c$ in front of the collision operator $L$.

The following result (Soffer-Tran 2016) is proved for this coupling system:

There exist universal constants $\mathcal{C}_1,\mathcal{C}_2>0$, such that

$\|f(t)\|_{\mathcal{L}} + \|\nabla f(t)\|_{\mathcal{L}}\le \mathcal{C}_1e^{-\mathcal{C}_2t},$

for some norm $\|\cdot\|_{\mathcal{L}}$. Moreover, the second component satisfies $\Psi=1+u$ and

$\|u_{1}(t)\|_{L^\infty}\leq O\left({{(t+1)}^{-1}}\right),~~~~\|u_{2}(t)\|_{L^\infty}\leq O\left({{(t+1)}^{-9/10}}\right).$

References for the picture:

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