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Convergence and non-convergence to Bose-Einstein condensation

Shuzhe Cai, Xuguang Lu

TL;DR

This work analyzes how interaction potentials in the space-homogeneous Bose–Einstein Boltzmann equation influence convergence to Bose–Einstein condensation (BEC) at low temperature. By classifying collision kernels into two regimes—one with a lower bound and another with an upper bound on the scattering cross-section—and employing measure-valued isotropic solutions, the authors establish strong convergence to BE equilibrium for the lower-bound regime under suitable initial data, while proving non-convergence in the upper-bound regime when condensation is initially absent. A semi-strong framework links entropy production to a semi-norm distance from BE equilibrium, yielding explicit decay rates that depend on the potential parameter $\eta$. The results highlight the critical effect of the microscopic interaction potential on the macroscopic condensation dynamics, providing both constructive initial data examples and rigorous non-convergence scenarios.

Abstract

The paper is a continuation of our previous work on the strong convergence to equilibrium for the spatially homogeneous Boltzmann equation for Bose-Einstein particles for isotropic solutions at low temperature. Here we study the influence of the particle interaction potentials on the convergence to Bose-Einstein condensation (BEC). Consider two cases of certain potentials that are such that the corresponding scattering cross sections are bounded and 1) have a lower bound ${\rm const.}\min\{1, |{\bf v-v}_*|^{2η}\}$ with ${\rm const}.>0, 0\le η<1$, and 2) have an upper bound ${\rm const.}\min\{1, |{\bf v-v}_*|^{2η}\}$ with $η\ge 1$. For the first case, the long time convergence to BEC i.e. $\lim\limits_{t\to\infty}F_t(\{0\})=F_{\rm be}(\{0\})$ is proved for a class of initial data having very low temperature and thus it holds the strong convergence to equilibrium. For the second case we show that if initially $F_0(\{0\})=0$, then $ F_t(\{0\})=0$ for all $t\ge 0$ and thus there is no convergence to BEC hence no strong convergence to equilibrium.

Convergence and non-convergence to Bose-Einstein condensation

TL;DR

This work analyzes how interaction potentials in the space-homogeneous Bose–Einstein Boltzmann equation influence convergence to Bose–Einstein condensation (BEC) at low temperature. By classifying collision kernels into two regimes—one with a lower bound and another with an upper bound on the scattering cross-section—and employing measure-valued isotropic solutions, the authors establish strong convergence to BE equilibrium for the lower-bound regime under suitable initial data, while proving non-convergence in the upper-bound regime when condensation is initially absent. A semi-strong framework links entropy production to a semi-norm distance from BE equilibrium, yielding explicit decay rates that depend on the potential parameter . The results highlight the critical effect of the microscopic interaction potential on the macroscopic condensation dynamics, providing both constructive initial data examples and rigorous non-convergence scenarios.

Abstract

The paper is a continuation of our previous work on the strong convergence to equilibrium for the spatially homogeneous Boltzmann equation for Bose-Einstein particles for isotropic solutions at low temperature. Here we study the influence of the particle interaction potentials on the convergence to Bose-Einstein condensation (BEC). Consider two cases of certain potentials that are such that the corresponding scattering cross sections are bounded and 1) have a lower bound with , and 2) have an upper bound with . For the first case, the long time convergence to BEC i.e. is proved for a class of initial data having very low temperature and thus it holds the strong convergence to equilibrium. For the second case we show that if initially , then for all and thus there is no convergence to BEC hence no strong convergence to equilibrium.
Paper Structure (4 sections, 8 theorems, 171 equations)

This paper contains 4 sections, 8 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Semi-Strong Convergence to Equilibrium
  3. Some Lemmas for Condensation
  4. Proof of Theorem \ref{['theorem1.2']}

Key Result

Theorem 1.4

Let $F_0\in {\mathcal{B}}_1^{+}({\mathbb R}_{\ge 0})$ with $N=N(F_0)>0, E=E(F_0)>0$ and let $F_{\rm be}$ be the unique Bose-Einstein distribution with the same mass $N$ and energy $E$. (I) Let $B({\bf {\bf v-v}_*},\omega)$ satisfy the Assumption assp with $0\le \eta<1$. Let $0\le\alpha<1-\eta, \beta Let $F_t\in {\cal B}_{1}^{+}({{\mathbb R}}_{\ge 0})$ be a conservative measure-valued isotropic sol

Theorems & Definitions (10)

  • Definition 1.1
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5