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Evolution of finite temperature Bose-Einstein Condensates: Some rigorous studies on condensate growth

Gigliola Staffilani, Minh-Binh Tran

TL;DR

The paper rigorously proves the onset of condensate growth for a finite-temperature Bose gas by analyzing a radial kinetic equation with three collision channels: $C_{12}$, $C_{22}$, and $C_{31}$. Employing a domain-decomposition multiscale framework, it overcomes sign-indefinite convexity issues to establish immediate Dirac mass formation at the origin for suitable initial data, and also demonstrates finite-time condensation under weaker initial concentration. The authors prove global existence of a mild radial solution, derive robust a priori estimates, and develop a sequence of multiscale estimates that quantify mass transfer from the thermal cloud into the condensate across scales. The results provide a rigorous mathematical foundation for condensate growth, with the energy cascade behavior set for a companion paper and the full dynamic coupling to the Gross-Pitaevskii equation discussed in the physical context. Overall, the work connects microscopic wave interactions to macroscopic condensate growth through a rigorous kinetic-analytic framework.

Abstract

In trapped Bose-Einstein condensates (BECs), \emph{condensate growth} refers to the process in which an increasing number of quasi-particles are immediately transferred from the non-condensate state (the thermal cloud) into the condensate state following the initial formation of the BEC. Despite its physical significance, this phenomenon has not yet been studied rigorously from a mathematical standpoint. In this work, we investigate a kinetic equation whose collision operator includes three types of wave interactions: one corresponding to a 3-wave process, and two classified as 4-wave processes. This wave kinetic equation models the evolution of the density function of the thermal cloud. We establish the immediate formation of condensation in solutions to this equation, thus providing a rigorous demonstration of the condensate growth phenomenon.

Evolution of finite temperature Bose-Einstein Condensates: Some rigorous studies on condensate growth

TL;DR

The paper rigorously proves the onset of condensate growth for a finite-temperature Bose gas by analyzing a radial kinetic equation with three collision channels: , , and . Employing a domain-decomposition multiscale framework, it overcomes sign-indefinite convexity issues to establish immediate Dirac mass formation at the origin for suitable initial data, and also demonstrates finite-time condensation under weaker initial concentration. The authors prove global existence of a mild radial solution, derive robust a priori estimates, and develop a sequence of multiscale estimates that quantify mass transfer from the thermal cloud into the condensate across scales. The results provide a rigorous mathematical foundation for condensate growth, with the energy cascade behavior set for a companion paper and the full dynamic coupling to the Gross-Pitaevskii equation discussed in the physical context. Overall, the work connects microscopic wave interactions to macroscopic condensate growth through a rigorous kinetic-analytic framework.

Abstract

In trapped Bose-Einstein condensates (BECs), \emph{condensate growth} refers to the process in which an increasing number of quasi-particles are immediately transferred from the non-condensate state (the thermal cloud) into the condensate state following the initial formation of the BEC. Despite its physical significance, this phenomenon has not yet been studied rigorously from a mathematical standpoint. In this work, we investigate a kinetic equation whose collision operator includes three types of wave interactions: one corresponding to a 3-wave process, and two classified as 4-wave processes. This wave kinetic equation models the evolution of the density function of the thermal cloud. We establish the immediate formation of condensation in solutions to this equation, thus providing a rigorous demonstration of the condensate growth phenomenon.
Paper Structure (16 sections, 14 theorems, 363 equations, 2 figures)

This paper contains 16 sections, 14 theorems, 363 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The evolution of Bose-Einstein condensations after being formed and condensate growth
  3. Physical context of the model
  4. 3-wave and 4-wave kinetic operators
  5. Main results
  6. Weak formulations
  7. A priori estimates using the initial mass and energy
  8. Global existence result
  9. Multiscale setting
  10. The first multiscale estimates
  11. The second multiscale estimates
  12. The main second estimate
  13. The second multiscale estimates
  14. The final multiscale estimates
  15. Condensation growth
  16. ...and 1 more sections

Key Result

Theorem 3

We assume Assumption A and Assumption B. Let $f_0(k) = f_0(|k|) \ge 0$ be an initial condition satisfying for some constants $\mathscr{M},\mathscr{E}>0$. Then there exists at least one global mild radial solution $f(t,k)$ of 4wave in the sense of 4wavemild such that for all $t \ge 0$. We define for $t \ge 0$, and assume that Moreover, suppose further that there exist constants $C_{\mathrm{ini}

Figures (2)

  • Figure 1: Condensate growth curves for different initial numbers of condensed particles. At time $t = 0$, the condensate initially contains $10^2$, $10^3$, $10^4$, $10^5$, or $10^6$ particles. This is Figure 5 in bijlsma2000condensate.
  • Figure 2: The Bose--Einstein Condensate (BEC) and the excited atoms.

Theorems & Definitions (32)

  • Remark 1
  • Remark 2
  • Definition 1
  • Theorem 3
  • Remark 4
  • Remark 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 22 more