A fast spectral method for the Uehling-Uhlenbeck equation for quantum gas mixtures: homogeneous relaxation and transport coefficients

Research output: Contribution to journalArticle

Abstract

A fast spectral method (FSM) is developed to solve the Uehling-Uhlenbeck equation for quantum gas mixtures with generalized differential cross-sections. The computational cost of the proposed FSM is O(Mdv-1Ndv+1log N), where dv is the dimension of the problem, Mdv-1 is the number of discrete solid angles, and N is the number of frequency nodes in each direction. Spatially-homogeneous relaxation problems are used to demonstrate that the FSM conserves mass and momentum/energy to the machine and spectral accuracy, respectively. Based on the variational principle, transport coefficients such as the shear viscosity, thermal conductivity, and diffusion are calculated by the FSM, which agree well with the analytical solutions. Then, the FSM is applied to find the accurate transport coefficients through an iterative scheme for the linearized quantum Boltzmann equation. The shear viscosity and thermal conductivity of three-dimensional quantum Fermi and Bose gases interacting through hard-sphere potential are calculated. For Fermi gas, the relative difference between the accurate and variational transport coefficients increases with fugacity; for Bose gas, the relative difference in thermal conductivity has similar behavior as the gas moves from the classical to degenerate limits, but the relative difference in shear viscosity decreases when the fugacity increases. Finally, the viscosity and diffusion coefficients have been calculated for a two-dimensional equal-mole mixture of Fermi gases. When the molecular masses of the two components are the same, our numerical results agree with the variational solutions. However, when the molecular mass ratio is not one, large discrepancies between the accurate and variational results are observed; our results are reliable because (i) the method does not rely on any assumption on the form of velocity distribution function and (ii) the ratio between shear viscosity and entropy density satisfies the minimum bound predicted by the string theory.
LanguageEnglish
Pages1-27
Number of pages27
JournalJournal of Computational Physics
Publication statusAccepted/In press - 29 Aug 2019

Fingerprint

Gas Mixture
Transport Coefficients
spectral methods
Spectral Methods
Shear Viscosity
Gas mixtures
Shear viscosity
gas mixtures
transport properties
viscosity
Thermal Conductivity
shear
coefficients
Thermal conductivity
Gases
Bose Gas
gases
thermal conductivity
Molecular mass
String theory

Keywords

  • quantum Boltzmann equation
  • fast spectral method
  • gas mixture
  • shear viscosity
  • thermal conductivity
  • spin diffusion

Cite this

@article{ae15a364b0d049dd9ccefae999f1ab99,
title = "A fast spectral method for the Uehling-Uhlenbeck equation for quantum gas mixtures: homogeneous relaxation and transport coefficients",
abstract = "A fast spectral method (FSM) is developed to solve the Uehling-Uhlenbeck equation for quantum gas mixtures with generalized differential cross-sections. The computational cost of the proposed FSM is O(Mdv-1Ndv+1log N), where dv is the dimension of the problem, Mdv-1 is the number of discrete solid angles, and N is the number of frequency nodes in each direction. Spatially-homogeneous relaxation problems are used to demonstrate that the FSM conserves mass and momentum/energy to the machine and spectral accuracy, respectively. Based on the variational principle, transport coefficients such as the shear viscosity, thermal conductivity, and diffusion are calculated by the FSM, which agree well with the analytical solutions. Then, the FSM is applied to find the accurate transport coefficients through an iterative scheme for the linearized quantum Boltzmann equation. The shear viscosity and thermal conductivity of three-dimensional quantum Fermi and Bose gases interacting through hard-sphere potential are calculated. For Fermi gas, the relative difference between the accurate and variational transport coefficients increases with fugacity; for Bose gas, the relative difference in thermal conductivity has similar behavior as the gas moves from the classical to degenerate limits, but the relative difference in shear viscosity decreases when the fugacity increases. Finally, the viscosity and diffusion coefficients have been calculated for a two-dimensional equal-mole mixture of Fermi gases. When the molecular masses of the two components are the same, our numerical results agree with the variational solutions. However, when the molecular mass ratio is not one, large discrepancies between the accurate and variational results are observed; our results are reliable because (i) the method does not rely on any assumption on the form of velocity distribution function and (ii) the ratio between shear viscosity and entropy density satisfies the minimum bound predicted by the string theory.",
keywords = "quantum Boltzmann equation, fast spectral method, gas mixture, shear viscosity, thermal conductivity, spin diffusion",
author = "Lei Wu",
year = "2019",
month = "8",
day = "29",
language = "English",
pages = "1--27",
journal = "Journal of Computational Physics",
issn = "0021-9991",

}

TY - JOUR

T1 - A fast spectral method for the Uehling-Uhlenbeck equation for quantum gas mixtures

T2 - Journal of Computational Physics

AU - Wu, Lei

PY - 2019/8/29

Y1 - 2019/8/29

N2 - A fast spectral method (FSM) is developed to solve the Uehling-Uhlenbeck equation for quantum gas mixtures with generalized differential cross-sections. The computational cost of the proposed FSM is O(Mdv-1Ndv+1log N), where dv is the dimension of the problem, Mdv-1 is the number of discrete solid angles, and N is the number of frequency nodes in each direction. Spatially-homogeneous relaxation problems are used to demonstrate that the FSM conserves mass and momentum/energy to the machine and spectral accuracy, respectively. Based on the variational principle, transport coefficients such as the shear viscosity, thermal conductivity, and diffusion are calculated by the FSM, which agree well with the analytical solutions. Then, the FSM is applied to find the accurate transport coefficients through an iterative scheme for the linearized quantum Boltzmann equation. The shear viscosity and thermal conductivity of three-dimensional quantum Fermi and Bose gases interacting through hard-sphere potential are calculated. For Fermi gas, the relative difference between the accurate and variational transport coefficients increases with fugacity; for Bose gas, the relative difference in thermal conductivity has similar behavior as the gas moves from the classical to degenerate limits, but the relative difference in shear viscosity decreases when the fugacity increases. Finally, the viscosity and diffusion coefficients have been calculated for a two-dimensional equal-mole mixture of Fermi gases. When the molecular masses of the two components are the same, our numerical results agree with the variational solutions. However, when the molecular mass ratio is not one, large discrepancies between the accurate and variational results are observed; our results are reliable because (i) the method does not rely on any assumption on the form of velocity distribution function and (ii) the ratio between shear viscosity and entropy density satisfies the minimum bound predicted by the string theory.

AB - A fast spectral method (FSM) is developed to solve the Uehling-Uhlenbeck equation for quantum gas mixtures with generalized differential cross-sections. The computational cost of the proposed FSM is O(Mdv-1Ndv+1log N), where dv is the dimension of the problem, Mdv-1 is the number of discrete solid angles, and N is the number of frequency nodes in each direction. Spatially-homogeneous relaxation problems are used to demonstrate that the FSM conserves mass and momentum/energy to the machine and spectral accuracy, respectively. Based on the variational principle, transport coefficients such as the shear viscosity, thermal conductivity, and diffusion are calculated by the FSM, which agree well with the analytical solutions. Then, the FSM is applied to find the accurate transport coefficients through an iterative scheme for the linearized quantum Boltzmann equation. The shear viscosity and thermal conductivity of three-dimensional quantum Fermi and Bose gases interacting through hard-sphere potential are calculated. For Fermi gas, the relative difference between the accurate and variational transport coefficients increases with fugacity; for Bose gas, the relative difference in thermal conductivity has similar behavior as the gas moves from the classical to degenerate limits, but the relative difference in shear viscosity decreases when the fugacity increases. Finally, the viscosity and diffusion coefficients have been calculated for a two-dimensional equal-mole mixture of Fermi gases. When the molecular masses of the two components are the same, our numerical results agree with the variational solutions. However, when the molecular mass ratio is not one, large discrepancies between the accurate and variational results are observed; our results are reliable because (i) the method does not rely on any assumption on the form of velocity distribution function and (ii) the ratio between shear viscosity and entropy density satisfies the minimum bound predicted by the string theory.

KW - quantum Boltzmann equation

KW - fast spectral method

KW - gas mixture

KW - shear viscosity

KW - thermal conductivity

KW - spin diffusion

UR - https://www.sciencedirect.com/journal/journal-of-computational-physics

M3 - Article

SP - 1

EP - 27

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -