Chapter 2: Uniform in time lower bound for solutions to a quantum Boltzmann equation of bosons

This post is based on my paper.

Milady de Winter is a spy for Cardinal Richelieu and the most beautiful female fictional character in the classic ”The Three Musketeers”

Chapter 2 Pic 1

Figure 1: Milady de Winter by Milla Jovovich

by Alexandre Dumas (father). Being twenty-two, uncommonly beautiful, with brilliant blue eyes and a bewitching voice, Milady can seduce any men she wants.

Our question in this chapter is if the greedy Milady-condensate can absorb all of the men-excited atoms around her. If the answer is yes, there would be a moment, when all of excited atoms fell into the condensate i.e. there exists a point      (t,p)\in \mathbb{R}_+\times\mathbb{R}^3 such that the density function f(t,p) of the excited atoms becomes     0. In order to see this, let us recall the equation of f(t,p) from Chapter 1:

\frac{df}{dt}=C_{12}[f], f(0,\cdot)=f_0, (1)

where the interaction operator is defined as

C_{12}[f]:=\int _{ \mathbb{R}^3}\int _{ \mathbb{R}^3}dp_1dp_2 [R(p, p_1, p_2)-R(p_1, p, p_2)-R(p_2, p_1, p) ],

R(p, p_1, p_2):= |\mathcal{M}(p, p_1, p_2)|^2 [\delta (\mathcal{E}_p-\mathcal{E}_{p_1}-\mathcal{E}_{p_2}) \delta (p-p_1-p_2)][ f(p_1)f(p_2)(1+f(p))-(1+f(p_1)(1+f(p_2))f(p)].

Here, we use the Bogoliubov dispersion law form of the particle energy   \mathcal{E}_p=\sqrt{\kappa_1 |p|^2 + \kappa_2 |p|^4} (2), for    \kappa_1,    \kappa_2 are positive constants.

The Dirac delta function in (1) ensures the conservation of momentum and energy after collision: p = p_1 + p_2, \mathcal{E}_p = \mathcal{E}_{p_1} + \mathcal{E}_{p_2}.

The above conservation laws dictate the structure of the energy level sets in \mathbb{R}^3, for each p, defined by  S_p: = \{ p_2\in \mathbb{R}^3~:~\mathcal{E}_{p-p_2} + \mathcal{E}_{p_2}= \mathcal{E}_{p} \}, S'_p : = \{p_2\in \mathbb{R}^3~:~ \mathcal{E}_{p+p_2} = \mathcal{E}_p+\mathcal{E}_{p_2} \}, and  S''_p : = \{p_*\in \mathbb{R}^3~:~ \mathcal{E}_{p_*}= \mathcal{E}_{p}+\mathcal{E}_{p_*-p}\}.

The Bogoliubov dispersion law form of energy functions (2) significantly complicates the analysis in treating the collision integral operator   C_{12}[f], which is now reduced to the surface integral on the energy surfaces   S_p,   S'_p and   S''_p. For instance, it is not clear whether the second moment of   f on these surfaces is bounded, even the second moment of   f in    \mathbb{R}^3 is bounded.

In this work with Toan Nguyen (Penn State), we proved that under some conditions on the initial data, for any time   T>0, there exist positive constants   \theta_0, \theta_1 such that positive radial solutions  f(t,p) to the quantum Boltzmann equations (1), satisfy     f(t,p)\geq \theta_1\exp(-\theta_0|p|^2),    \forall t\ge T.

By the H-Theorem, the equation has a family of equilibria:    n_0(p )=\frac {1} {e^{\frac{\mathcal{E}_p}{k_B T}}-1}, \beta >0, where   k_B is the Boltzmann’s constant and  T the temperature of the quasiparticles whose distribution is  n_0. It is clear that   n_0(p) is above a Gaussian, as a consequence, our result is quite reasonable, mathematically speaking.

Physically speaking, our result asserts that given a condensate and its thermal cloud, we can prove that there will be some portion of excited atoms which remain outside of the condensate and the density of such atoms will be greater than a Gaussian, uniformly in time   t\geq T for any time   T>0.

Chapter 2 Pic 2

Figure 2: The Bose-Einstein Condensate (BEC) and the excited atoms.

In other words, the greedy Milady-condensate cannot seduce all of the men-excited atoms around her!

Up to now, we have been discussing about the condensate growth term. A natural question would be: how does the full system look like? This is the topic of Chapter 3:

What else besides the condensate growth term?




Reference for the picture:


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