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The Spatially Homogeneous Boltzmann Equation for Bose-Einstein Particles: Rate of Strong Convergence to Equilibrium

Shuzhe Cai, Xuguang Lu

TL;DR

This work analyzes the spatially homogeneous Boltzmann equation for Bose-Einstein particles with a quantum collision kernel that encompasses the hard-sphere model. The authors develop entropy-based tools, including entropy-entropy dissipation inequalities and Villani’s inequality, and introduce a time-dependent convex combination and convex-positivity of the cubic collision integral to handle condensation. They prove that for radially symmetric, non-singular initial data at low temperature, the BE condensation mass $F_t(\{0\})$ converges to the equilibrium $F_{ m be}(\{0\})$ and establish an explicit algebraic rate of convergence to equilibrium for general temperatures, along with a corresponding rate for the full distribution in strong norms. These results advance the understanding of long-time equilibration and condensation formation in Bose-Einstein systems under spatially homogeneous dynamics, providing quantitative rates and robust convergence guarantees that extend prior work.

Abstract

The paper is a continuation of our previous work on the spatially homogeneous Boltzmann equation for Bose-Einstein particles with quantum collision kernel that includes the hard sphere model. Solutions $F_t$ under consideration that conserve the mass, momentum, and energy and converge at least weakly to equilibrium $F_{\rm be}$ as $t\to\infty$ have been proven to exist at least for radially symmetric and non-singular initial data, and for the case of low temperature, $F_t$ have to be positive Borel measures. The new progress is as follows: we prove that the long time convergence of $F_t(\{0\})$ to the Bose-Einstein condensation $F_{\rm be}(\{0\})$ for low temperature holds for all radially symmetric and non-singular initial data $F_0$. This immediately implies the long time strong convergence to equilibrium. We also obtain an algebraic rate of the strong convergence for arbitrary temperature. Our proofs are based on the entropy control, Villani's inequality for the entropy dissipation, a suitable time-dependent convex combination between the solution and a fixed positive function (in order to overcome the lack of positive lower bound), the convex-positivity of the cubic collision integral, and an iteration technique for obtaining a positive lower bound of condensation.

The Spatially Homogeneous Boltzmann Equation for Bose-Einstein Particles: Rate of Strong Convergence to Equilibrium

TL;DR

This work analyzes the spatially homogeneous Boltzmann equation for Bose-Einstein particles with a quantum collision kernel that encompasses the hard-sphere model. The authors develop entropy-based tools, including entropy-entropy dissipation inequalities and Villani’s inequality, and introduce a time-dependent convex combination and convex-positivity of the cubic collision integral to handle condensation. They prove that for radially symmetric, non-singular initial data at low temperature, the BE condensation mass converges to the equilibrium and establish an explicit algebraic rate of convergence to equilibrium for general temperatures, along with a corresponding rate for the full distribution in strong norms. These results advance the understanding of long-time equilibration and condensation formation in Bose-Einstein systems under spatially homogeneous dynamics, providing quantitative rates and robust convergence guarantees that extend prior work.

Abstract

The paper is a continuation of our previous work on the spatially homogeneous Boltzmann equation for Bose-Einstein particles with quantum collision kernel that includes the hard sphere model. Solutions under consideration that conserve the mass, momentum, and energy and converge at least weakly to equilibrium as have been proven to exist at least for radially symmetric and non-singular initial data, and for the case of low temperature, have to be positive Borel measures. The new progress is as follows: we prove that the long time convergence of to the Bose-Einstein condensation for low temperature holds for all radially symmetric and non-singular initial data . This immediately implies the long time strong convergence to equilibrium. We also obtain an algebraic rate of the strong convergence for arbitrary temperature. Our proofs are based on the entropy control, Villani's inequality for the entropy dissipation, a suitable time-dependent convex combination between the solution and a fixed positive function (in order to overcome the lack of positive lower bound), the convex-positivity of the cubic collision integral, and an iteration technique for obtaining a positive lower bound of condensation.

Paper Structure

This paper contains 6 sections, 24 theorems, 413 equations.

Table of Contents

  1. Introduction
  2. Inequality of Entropy and Entropy Dissipation
  3. Positive Lower Bound of Entropy
  4. Rate of Entropy Convergence
  5. Rate of Convergence to BEC
  6. Appendix

Key Result

Theorem 1.4

Suppose $B({\bf {\bf v-v}_*},\omega)$ satisfy Assumption assp with $0\le\eta<1/4$. Let $F_0\in {\mathcal{B}}_1^{+}({\mathbb R}_{\ge 0})$ satisfy $N(F_0)>0, E(F_0)>0$, let $F_{\rm be}$ be the unique Bose-Einstein distribution with the same mass $N=N(F_0)$ and energy $E=E(F_0)$. Let $\frac{1}{20}<\lam In particular if $\overline{T}/\overline{T}_c<1$ then Here the constant $C>0$ depends only on $N,

Theorems & Definitions (32)

  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • Proposition 2.2
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • ...and 22 more