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Numerical schemes for a multi-species quantum BGK model

Gi-Chan Bae, Marlies Pirner, Sandra Warnecke

TL;DR

The paper tackles the numerical solution of a quantum multi-species BGK model with classical and quantum particles by developing an IMEX time discretization that handles collision stiffness while preserving fundamental structure. A key innovation is a convex-optimization–based approach (via Lagrange multipliers) to update quantum local equilibria without explicit inversion of density-energy relations, ensuring conservation laws and entropy production. The authors provide a rigorous macroscopic convergence analysis in the space-homogeneous setting, showing exponential relaxation of mean velocities to a common value and differentiating the behavior of kinetic versus physical temperatures in the quantum case, supported by extensive numerical experiments including homogeneous relaxation, a sulfur-fluorine-electron test, and a Sod problem. The framework is extensible to $N$-species mixtures and includes a rigorous entropy minimization viewpoint for mixture equilibria, highlighting the practical impact for simulating quantum gas mixtures with robust, physically faithful, and efficient schemes.

Abstract

This work is devoted to the numerical implementation of the quantum Bhatnagar- Gross-Krook (BGK) model for gas mixtures consisting of classical and quantum particles (fermions, bosons). We consider the model proposed by Bae, Klingenberg, Pirner, and Yun in 2021 and implement an Implicit-Explicit (IMEX) scheme due to the stiffness of the collision operator. A major obstacle is updating the parameters of quantum local equilibrium, which requires computing by inverting the relation between density and energy at every grid point in space and time. We address this difficulty by using the Lagrange multiplier method to minimize a potential function subject to constraints defined by specific moment equalities. Moreover, we analyze the convergence of mean velocity and temperature between the species both analytically and numerically. When a quantum component is included, we observe that the converging quantity is physical temperature, not the kinetic temperature. This differs from the mixture of classical species.

Numerical schemes for a multi-species quantum BGK model

TL;DR

The paper tackles the numerical solution of a quantum multi-species BGK model with classical and quantum particles by developing an IMEX time discretization that handles collision stiffness while preserving fundamental structure. A key innovation is a convex-optimization–based approach (via Lagrange multipliers) to update quantum local equilibria without explicit inversion of density-energy relations, ensuring conservation laws and entropy production. The authors provide a rigorous macroscopic convergence analysis in the space-homogeneous setting, showing exponential relaxation of mean velocities to a common value and differentiating the behavior of kinetic versus physical temperatures in the quantum case, supported by extensive numerical experiments including homogeneous relaxation, a sulfur-fluorine-electron test, and a Sod problem. The framework is extensible to -species mixtures and includes a rigorous entropy minimization viewpoint for mixture equilibria, highlighting the practical impact for simulating quantum gas mixtures with robust, physically faithful, and efficient schemes.

Abstract

This work is devoted to the numerical implementation of the quantum Bhatnagar- Gross-Krook (BGK) model for gas mixtures consisting of classical and quantum particles (fermions, bosons). We consider the model proposed by Bae, Klingenberg, Pirner, and Yun in 2021 and implement an Implicit-Explicit (IMEX) scheme due to the stiffness of the collision operator. A major obstacle is updating the parameters of quantum local equilibrium, which requires computing by inverting the relation between density and energy at every grid point in space and time. We address this difficulty by using the Lagrange multiplier method to minimize a potential function subject to constraints defined by specific moment equalities. Moreover, we analyze the convergence of mean velocity and temperature between the species both analytically and numerically. When a quantum component is included, we observe that the converging quantity is physical temperature, not the kinetic temperature. This differs from the mixture of classical species.
Paper Structure (27 sections, 15 theorems, 141 equations, 6 figures)

This paper contains 27 sections, 15 theorems, 141 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The multi-species quantum BGK model
  3. Macroscopic equations and convergence rates for the velocities and kinetic temperatures in the space-homogeneous case
  4. Macroscopic equations
  5. Convergence rate for the velocities and kinetic temperatures in the space-homo-ge-neous case
  6. Numerical scheme
  7. Time discretization
  8. First-order splitting
  9. Relaxation.
  10. Transport.
  11. Second-order IMEX Runge-Kutta
  12. General implicit solver
  13. Space discretization
  14. Properties of the semi-discrete scheme
  15. Positivity of distribution functions
  16. ...and 12 more sections

Key Result

theorem 1

Assume $\nu_{12} n_2 = \nu_{21} n_1$. Let $(f_1,f_2)$ be a solution to BGK_quantum, then we obtain the following formal conservation laws where the exchange terms of momentum can be computed as Furthermore, we define the functions $\eta_{\tau}(c)=\int \frac{1}{e^{|p|^2+c}+\tau}$ and $\eta^E_{\tau}(c)=\int \frac{|p|^2}{e^{|p|^2+c}+\tau} dp,$ and obtain for the exchange of energy

Figures (6)

  • Figure 1: Entropy and entropy dissipation for the test case in Section \ref{['test:decay-rates']}, exemplary for fermion-fermion interactions. The entropy decays monotonically.
  • Figure 2: Illustration of the conservation properties for the test case in Section \ref{['test:decay-rates']}, exemplary for fermion-fermion interactions. The mass densities of each species ($\rho_k=m_k n_k$), the total momentum ($M$) and total energy ($E$) have small oscillations of the order of $10^{-14}$.
  • Figure 3: Mean velocities for the test case in Section \ref{['test:decay-rates']}, exemplary for fermion-fermion interactions. The mean velocities converge exponentially fast to a common value, and the numerical decay rate coincides very well with the analytical one.
  • Figure 4: Evolution of the temperatures for the test case in Section \ref{['test:decay-rates']}. First column: kinetic temperatures $T_k$; whenever a quantum particle is involved, the kinetic temperatures do not converge to a common value. Second column: decay rates for kinetic temperatures in logarithmic scale --- numerical and analytical values coincide very well. Additionally, the difference between the physical temperatures $\vartheta_k$ is displayed which decays exponentially fast, whereas the kinetic temperatures $T_k$ behave differently for quantum particles.
  • Figure 5: Evolution of the physical temperatures for the Sulfur-Fluorine-electrons quantum test case in Section \ref{['test:SFe-quantum']}. When both the ions and the electrons are treated classically (lines with dots), the physical temperatures (which coincide with the kinetic temperatures \ref{['eq:T_kinetic']}) converge to the mixture temperature $T_{\rm eq}$ defined in \ref{['eq:T_eq']}. When the electrons are treated like fermions instead, the physical temperatures do converge to a common value as predicted by the theory. However, this value differs from $T_{\rm eq}$.
  • ...and 1 more figures

Theorems & Definitions (31)

  • theorem 1
  • proof
  • remark 1
  • remark 2
  • theorem 2
  • proof
  • remark 3
  • theorem 3
  • proof
  • remark 4
  • ...and 21 more