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## Uniform lower bound for solutions to a quantum Boltzmann

With Minh-Binh Tran, we establish uniform lower bounds, by a Maxwellian, for positive radial solutions to a Quantum Boltzmann kinetic model for bosons; see arXiv preprint.

When a gas of Bose particles is cooled down to significantly lower temperature, the Bose-Einstein condensation is formed. In this paper, we are interested in the interaction between a Boson particle and such a condensate, modeled by the following kinetic equation (assuming spatial homogeneity):

$\displaystyle \frac{\partial f}{\partial t} = Q[f]$

for excited atom density distribution function ${f(t,p)}$, time ${t\ge 0}$ and momentum ${p \in \mathbb{R}^3}$. Here, ${Q[f]}$ denotes the collision integral operator describing the bosons-condensate interaction.
Roughly speaking, our main result asserts that the positive radial solutions are bounded below by a Gaussian

${e^{-\theta_0|p|^2}}$

for all positive time, provided that initially there is positive mass concentrated near the origin. Physically speaking, this shows that the collision operator ${Q}$ prevents the excited atoms to all fall into the condensate. In other words, 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 positive time.

In the quantum theory of solids, the quantum phonon Boltzmann equation or the Peierls equation is in fact of the same formulation with the equation as the one considered in this paper. To the best of our knowledge, in the context of the study of phonon interactions in anharmonic crystals, the kinetic model is the first kinetic model of weak turbulence. In anharmonic crystals, electronic bands of dielectric crystals are completely filled and separated by an energy gap from the conduction band. As a consequence, electronic energy transport is suppressed and the vibrations of the atoms around their mechanical equilibrium position is the dominant contribution to heat transport. R. Peierls suggested the theoretical option of considering the anharmonicities as a small perturbation to the perfectly harmonic crystal, which leads to a kinetic model of an interacting phonons in terms of a nonlinear Boltzmann equation. The phonon Boltzmann equation is then employed to carry on the actual computation of the thermal conductivity of dielectric crystals. See our paper for further discussions and the references therein.