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An energy cascade finite volume scheme for a mixed 3- and 4-wave kinetic equation arising from the theory of finite-temperature trapped Bose gases

Arijit Das, Minh-Binh Tran

TL;DR

This work develops a finite-volume scheme to simulate a mixed 3- and 4-wave kinetic equation arising from finite-temperature Bose gases, capturing the energy cascade that drives energy to leave bounded domains. By reformulating the isotropic kinetic equation with kernels $\mathcal W_{12}=(\omega\omega_1\omega_2)^{\sigma}$ and $\mathcal W_{22}=(\omega\omega_1\omega_2\omega_3)^{\gamma}$ and dispersion $\omega(k)=|k|^{\rho}$, the authors derive a discretizable form $\partial_t f=\mathcal Q[f]$ with explicit $C_1$ and $C_2$ terms amenable to finite-volume discretization. They prove non-negativity, consistency, and Lipschitz continuity of the discrete operator, establishing convergence with first-order accuracy. Numerical tests across $\rho=2,3$ and multiple $(C_1,C_2,\sigma,\gamma)$ configurations demonstrate robust energy decay and alignment with the predicted energy cascade on a truncated domain, validating the scheme’s effectiveness for finite-temperature Bose gases.

Abstract

Building on recent developments in numerical schemes designed to capture energy cascades for 3-wave kinetic equations~\cite{das2024numerical, walton2022deep, walton2023numerical, walton2024numerical}, we construct in this work a finite-volume algorithm for a significantly more complex wave kinetic equation whose collision operator incorporates both 3-wave and 4-wave interactions. This model arises in the context of finite-temperature Bose-Einstein condensation. We establish theoretical properties of the proposed scheme, and our numerical experiments demonstrate that it successfully captures the energy cascade behavior predicted by the equation.

An energy cascade finite volume scheme for a mixed 3- and 4-wave kinetic equation arising from the theory of finite-temperature trapped Bose gases

TL;DR

This work develops a finite-volume scheme to simulate a mixed 3- and 4-wave kinetic equation arising from finite-temperature Bose gases, capturing the energy cascade that drives energy to leave bounded domains. By reformulating the isotropic kinetic equation with kernels and and dispersion , the authors derive a discretizable form with explicit and terms amenable to finite-volume discretization. They prove non-negativity, consistency, and Lipschitz continuity of the discrete operator, establishing convergence with first-order accuracy. Numerical tests across and multiple configurations demonstrate robust energy decay and alignment with the predicted energy cascade on a truncated domain, validating the scheme’s effectiveness for finite-temperature Bose gases.

Abstract

Building on recent developments in numerical schemes designed to capture energy cascades for 3-wave kinetic equations~\cite{das2024numerical, walton2022deep, walton2023numerical, walton2024numerical}, we construct in this work a finite-volume algorithm for a significantly more complex wave kinetic equation whose collision operator incorporates both 3-wave and 4-wave interactions. This model arises in the context of finite-temperature Bose-Einstein condensation. We establish theoretical properties of the proposed scheme, and our numerical experiments demonstrate that it successfully captures the energy cascade behavior predicted by the equation.

Paper Structure

This paper contains 11 sections, 3 theorems, 83 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Physical context
  3. The $\mathscr{C}_{12}$ collision operator.
  4. The $\mathscr{C}_{22}$ collision operator.
  5. Reformulating \ref{['wavekinetic']}
  6. Formulation of the numerical scheme
  7. Convergence analysis of the numerical scheme
  8. Numerical Tests
  9. Test Case I
  10. Test Case II
  11. Test Case III

Key Result

Proposition 4.1

Assume that the wave kinetic kernels $\mathcal{K}_i$ satisfy the condition 3_5. Then there exists a positive constant $\mathcal{L}(T)$, independent of the mesh, such that

Figures (11)

  • Figure 1: Evolution of wave density $f(\omega)$ at initial and final time.
  • Figure 2: Time evolution of the total energy (A) and third order moment (B) for different values of $C_1$ and $C_2$ when $\sigma=0.50$, $\gamma = 0.50$, and $\left|k\right|\left(\omega\right) = \sqrt{\omega}$.
  • Figure 3: Time evolution of the total energy (A) and third order moment (B) for different values of $C_1$ and $C_2$ when $\sigma=0.50$, $\gamma = 0.50$, and $\left|k\right|\left(\omega\right) = {\omega}^{1/3}$.
  • Figure 4: Time evolution of the total energy (A) and third order moment (B) for different values of $\sigma$ and $\gamma$ when $\left|k\right|\left(\omega\right) = \sqrt{\omega}$.
  • Figure 5: Time evolution of the total energy (A) and third order moment (B) for different values of $C_1$ and $C_2$ when $\sigma=0.50$, $\gamma = 0.50$, and $\left|k\right|\left(\omega\right) = \sqrt{\omega}$.
  • ...and 6 more figures

Theorems & Definitions (8)

  • Remark 1.1
  • Definition 4.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Theorem 4.1
  • proof