分子气体稀薄效应的动理学建模

曾嘉楠; 李琪; 吴雷

doi:10.7638/kqdlxxb-2021.0378

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分子气体稀薄效应的动理学建模

综述 | 更新时间：2026-05-28

- 分子气体稀薄效应的动理学建模
- Kinetic modeling of rarefied molecular gas dynamics
- 空气动力学学报 2022年40卷第2期页码：1-30
- 作者机构：
  
  南方科技大学力学与航空航天工程系,深圳,518055
- 作者简介：
  
  曾嘉楠（1996-），男，江西抚州人，博士研究生，研究方向：稀薄气体动力学. E-mail：12031255@mail.sustech.edu.cn
  吴雷*，博士，长聘副教授，研究方向：稀薄气体动力学、计算流体动力学. E-mail：wul@sustech.edu.cn
- 基金信息：
  
  国家自然科学基金（12172162）
- DOI：10.7638/kqdlxxb-2021.0378
  中图分类号： O356;V211.25
- 纸质出版：2022
- 稿件说明：
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曾嘉楠, 李琪, 吴雷. 分子气体稀薄效应的动理学建模[J]. 空气动力学学报, 2022, 40(2): 1−30.

ZENG Jianan, LI Qi, WU Lei. Kinetic modeling of rarefied molecular gas dynamics[J]. 2022, 40(2): 1-30.
曾嘉楠, 李琪, 吴雷. 分子气体稀薄效应的动理学建模[J]. 空气动力学学报, 2022, 40(2): 1−30. DOI： 10.7638/kqdlxxb-2021.0378

ZENG Jianan, LI Qi, WU Lei. Kinetic modeling of rarefied molecular gas dynamics[J]. 2022, 40(2): 1-30. DOI： 10.7638/kqdlxxb-2021.0378.

摘要

稀薄气体效应，是指气体在特征尺度与其分子平均自由程相当的系统中流动时出现的非平衡效应。相比于单原子气体，分子气体（每个气体分子包含两个及以上原子）流动因同时具有转动、振动等多种自由度的非平衡过程，其稀薄效应更为复杂。分子气体稀薄效应在航空航天、微机电系统和页岩气开采等民生、科技领域广泛存在，而描述该效应的动理学模型与数值模拟方法尚不成熟。本文从单原子气体与玻尔兹曼方程出发，介绍气体动理学建模的相关研究现状，针对分子气体特性详细讨论气体弛豫过程与输运系数的关系。针对典型的稀薄气体流动验证常用的模型方程的精度，并指出直接模拟蒙特卡罗方法在分子气体稀薄流动应用中存在的问题，即在体积黏性确定的情况下无法指定热流弛豫速率及恢复热导率。随后针对此问题，使用吴模型量化研究该弛豫速率在分子气体稀薄流动模拟中导致的宏观量的不确定性，并讨论从分子动力学模拟和瑞利-布里渊散射实验中减小甚至消除不确定性的方法。本文对涉及化学反应的稀薄气体流动建模有指导意义。

Abstract

The rarefaction effects arise when the mean free path of gas molecules is comparable to the characteristic flow length. Compared to monatomic gas

the dynamics of rarefied molecular gas

which consists of two or more atoms in one gas molecule

is more complicated

due to the additional non-equilibrium effects in the rotational and vibrational motions. As the major constituent in the atmosphere of Earth and Mars

rarefied molecular gas dynamics has strong applications in aerospace science

micro-electro-mechanical system

shale gas extraction

and so on. However

the corresponding models and simulation methods are not sophisticated. Starting from the monatomic gas and the Boltzmann equation

we review the development of the kinetic modeling of rarefied gas and the essential connection between the transport coefficients and relaxation processes. Then

we focus on the relaxation process and transport coefficients of molecular gas

and introduce typical kinetic models with the systematic assessment of their accuracy. Meanwhile

we discuss the defect of direct simulation Monte Carlo method in the modeling of rarefied molecular gas

i.e.

it cannot specify the thermal relaxation rates (hence cannot recover the thermal conductivities) when the bulk viscosity is determined. Finally

we use the Wu model to quantify the uncertainties

and discuss how to reduce or even eliminate the uncertainties based on the data from the molecular dynamics simulation and Rayleigh-Brillouin scattering experiment. This present article is instructive for the modeling of rarefied gas flows involving chemical reactions.

关键词

Keywords

references

VOTTA R, SCHETTINO A, BONFIGLIOLI A. Hypersonic high altitude aerothermodynamics of a space re-entry vehicle[J]. Aerospace Science and Technology, 2013, 25(1): 253-265. DOI:10.1016/j.ast.2012.02.001

IVANOV M S, GIMELSHEIN S F. Computational hypersonic rarefied flows[J]. Annual Review of Fluid Mechanics, 1998, 30: 469-505. DOI:10.1146/annurev.fluid.30.1.469

SCANLON T J, WHITE C, BORG M K, et al. Open-source direct simulation Monte Carlo chemistry modeling for hypersonic flows[J]. AIAA Journal, 2015, 53(6): 1670-1680. DOI:10.2514/1.j053370

LI Z H, LIANG J, LI Z H, et al. Simulation methods of aerodynamics covering various flow regimes and applications to aerodynamic characteristics of re-entry spacecraft module[J]. Acta Aerodynamica Sinica, 2018, 36(5): 826-847. (in Chinese)

LIANG J, LI Z H, LI Q, et al. Numerical simulation of aerodynamic and trim characteristics across different flow regimes for reentry module[J]. Acta Aerodynamica Sinica, 2018, 36(5): 848-855. (in Chinese)

WANG X W, SU T Y, ZHANG W Q, et al. Knudsen pumps: a review[J]. Microsystems Nanoengineering, 2020, 6: 26. DOI:10.1038/s41378-020-0135-5

AOKI K, TAKATA S, TATSUMI E, et al. Rarefied gas flows through a curved channel: application of a diffusion-type equation[J]. Physics of Fluids, 2010, 22(11): 112001. DOI:10.1063/1.3496315

KUGIMOTO K, HIROTA Y, KIZAKI Y, et al. Performance prediction method for a multi-stage Knudsen pump[J]. Physics of Fluids, 2017, 29(12): 122002. DOI:10.1063/1.5001213

WANG P, SU W, WU L. Thermal transpiration in molecular gas[J]. Physics of Fluids, 2020, 32(8): 082005. DOI:10.1063/5.0018505

WU L, HO M T, GERMANOU L, et al. On the apparent permeability of porous media in rarefied gas flows[J]. Journal of Fluid Mechanics, 2017, 822: 398-417. DOI:10.1017/jfm.2017.300

YE Y D. Advances and prospects in hypersonic aerodynamics[J]. Chinese Science Bulletin, 2015, 60(12): 1095-1103. (in Chinese)

庄逢甘, 黄志澄. 未来高技术战争对空气动力学创新发展的需求[C]//2003空气动力学前沿研究学术研讨会论文集, 中国空气动力学会, 2003: 73–79.

DAUDÉ B, ELANDALOUSSI H, JANSSEN C. On the gas dependence of thermal transpiration and a critical appraisal of correction methods for capacitive diaphragm gauges[J]. Vacuum, 2014, 104: 77-87. DOI:10.1016/j.vacuum.2014.01.002

WU L, LIU H H, REESE J M, et al. Non-equilibrium dynamics of dense gas under tight confinement[J]. Journal of Fluid Mechanics, 2016, 794: 252-266. DOI:10.1017/jfm.2016.173

ALSMEYER H. Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam[J]. Journal of Fluid Mechanics, 1976, 74(3): 497-513. DOI:10.1017/s0022112076001912

SCHOTTER R. Rarefied gas acoustics in the noble gases[J]. The Physics of Fluids, 1974, 17(6): 1163-1168. DOI:10.1063/1.1694859

WU L, GU X J. On the accuracy of macroscopic equations for linearized rarefied gas flows[J]. Advances in Aerodynamics, 2020, 2: 2. DOI:10.1186/s42774-019-0025-4

EWART T, PERRIER P, GRAUR I A, et al. Mass flow rate measurements in a microchannel, from hydrodynamic to near free molecular regimes[J]. Journal of Fluid Mechanics, 2007, 584: 337-356. DOI:10.1017/s0022112007006374

JIANG D W, MAO M L, LI J, et al. An implicit parallel UGKS solver for flows covering various regimes[J]. Advances in Aerodynamics, 2019, 1: 8. DOI:10.1186/s42774-019-0008-5

LI Z H, JIANG X Y, WU J L, et al. Gas-kinetic unified algorithm for Boltzmann model equation in rotational nonequilibrium and its application to the whole range flow regimes[J]. Chinese Journal of Theoretical and Applied Mechanics, 2014, 46(3): 336-351. (in Chinese)

CHAPMAN S, COWLING T G, CERCIGNANI C. The mathematical theory of non-uniform gases[M]. Cambridge University Press, 1970.

GRAD H. On the kinetic theory of rarefied gases[J]. Communications on Pure and Applied Mathematics, 1949, 2(4): 331-407. DOI:10.1002/cpa.3160020403

GU X J, EMERSON D R. A high-order moment approach for capturing non-equilibrium phenomena in the transition regime[J]. Journal of Fluid Mechanics, 2009, 636: 177-216. DOI:10.1017/s002211200900768x

BIRD G A. Molecular gas dynamics and the direct simulation of gas flows[M]. ‎ New York : Oxford Science Publications, Oxford University Press Inc, 1994.

WAGNER W. A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation[J]. Journal of Statistical Physics, 1992, 66(3-4): 1011-1044. DOI:10.1007/BF01055714

LI Z H, FANG M, JIANG X Y, et al. Convergence proof of the DSMC method and the Gas-Kinetic Unified Algorithm for the Boltzmann equation[J]. Science China Physics, Mechanics and Astronomy, 2013, 56(2): 404-417. DOI:10.1007/s11433-013-4999-3

HADJICONSTANTINOU N G, GARCIA A L, BAZANT M Z, et al. Statistical error in particle simulations of hydrodynamic phenomena[J]. Journal of Computational Physics, 2003, 187(1): 274-297. DOI:10.1016/S0021-9991(03)00099-8

TCHEREMISSINE F G. Solution to the Boltzmann kinetic equation for high-speed flows[J]. Computational Mathematics and Mathematical Physics, 2006, 46(2): 315-329. DOI:10.1134/s0965542506020138

WU L, REESE J M, ZHANG Y H. Solving the Boltzmann equation deterministically by the fast spectral method: application to gas microflows[J]. Journal of Fluid Mechanics, 2014, 746: 53-84. DOI:10.1017/jfm.2014.79

WU L, WHITE C, SCANLON T J, et al. Deterministic numerical solutions of the Boltzmann equation using the fast spectral method[J]. Journal of Computational Physics, 2013, 250: 27-52. DOI:10.1016/j.jcp.2013.05.003

WU L, ZHANG Y H, LI Z H. Computable model on the collision integral of Boltzmann equation and application to rarefied aerodynamics[J]. SCIENTIA SINICA Physica, Mechanica Astronomica, 2017, 47(7): 28-45. (in Chinese)

BHATNAGAR P L, GROSS E P, KROOK M. A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems[J]. Physical Review, 1954, 94(3): 511-525. DOI:10.1103/physrev.94.511

HOLWAY L H. New statistical models for kinetic theory: methods of construction[J]. The Physics of Fluids, 1966, 9(9): 1658-1673. DOI:10.1063/1.1761920

ANDRIES P, LE TALLEC P, PERLAT J P, et al. The Gaussian-BGK model of Boltzmann equation with small Prandtl number[J]. European Journal of Mechanics - B/Fluids, 2000, 19(6): 813-830. DOI:10.1016/S0997-7546(00)01103-1

SHAKHOV E M. Approximate kinetic equations in rarefied gas theory[J]. Fluid Dynamics, 1968, 3(1): 112-115. DOI:10.1007/BF01016254

LI Z H, ZHANG H X. Study on gas kinetic unified algorithm for flows from rarefied transition to continuum[J]. Journal of Computational Physics, 2004, 193(2): 708-738. DOI:10.1016/j.jcp.2003.08.022

LI Z H, PENG A P, MA Q, et al. Gas-kinetic unified algorithm for computable modeling of Boltzmann equation and application to aerothermodynamics for falling disintegration of uncontrolled Tiangong-No. 1 spacecraft[J]. Advances in Aerodynamics, 2019, 1: 4. DOI:10.1186/s42774-019-0009-4

WU J L, LI Z H, ZHANG Z B, et al. On derivation and verification of a kinetic model for quantum vibrational energy of polyatomic gases in the gas-kinetic unified algorithm[J]. Journal of Computational Physics, 2021, 435: 109938. DOI:10.1016/j.jcp.2020.109938

LI Z H, ZHANG H X. Gas-kinetic numerical studies of three-dimensional complex flows on spacecraft re-entry[J]. Journal of Computational Physics, 2009, 228(4): 1116-1138. DOI:10.1016/j.jcp.2008.10.013

XU K, HUANG J C. A unified gas-kinetic scheme for continuum and rarefied flows[J]. Journal of Computational Physics, 2010, 229(20): 7747-7764. DOI:10.1016/j.jcp.2010.06.032

YANG L M, SHU C, YANG W M, et al. An improved discrete velocity method (DVM) for efficient simulation of flows in all flow regimes[J]. Physics of Fluids, 2018, 30(6): 062005. DOI:10.1063/1.5039479

YUAN R F, ZHONG C W. A conservative implicit scheme for steady state solutions of diatomic gas flow in all flow regimes[J]. Computer Physics Communications, 2020, 247: 106972. DOI:10.1016/j.cpc.2019.106972

GUO Z L, XU K, WANG R J. Discrete unified gas kinetic scheme for all Knudsen number flows: low-speed isothermal case[J]. Physical Review E, Statistical, Nonlinear, and Soft Matter Physics, 2013, 88(3): 033305. DOI:10.1103/PhysRevE.88.033305

SU W, ZHU L H, WANG P, et al. Can we find steady-state solutions to multiscale rarefied gas flows within dozens of iterations? [J]. Journal of Computational Physics, 2020, 407: 109245. Doi:10.1016/j.jcp.2020.109245

SU W, ZHU L H, WU L. Fast convergence and asymptotic preserving of the general synthetic iterative scheme[J]. SIAM Journal on Scientific Computing, 2020, 42(6): B1517. DOI:10.1137/20m132691x

ZHU L H, PI X C, SU W, et al. General synthetic iterative scheme for nonlinear gas kinetic simulation of multi-scale rarefied gas flows[J]. Journal of Computational Physics, 2021, 430: 110091. DOI:10.1016/j.jcp.2020.110091

SU W, ZHANG Y H, WU L. Multiscale simulation of molecular gas flows by the general synthetic iterative scheme[J]. Computer Methods in Applied Mechanics and Engineering, 2021, 373: 113548. DOI:10.1016/j.cma.2020.113548

ANDRIES P, BOURGAT J F, LE TALLEC P, et al. Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases[J]. Computer Methods in Applied Mechanics and Engineering, 2002, 191(31): 3369-3390. DOI:10.1016/S0045-7825(02)00253-0

MIEUSSENS L, STRUCHTRUP H. Numerical comparison of Bhatnagar-Gross-Krook models with proper Prandtl number[J]. Physics of Fluids, 2004, 16(8): 2797-2813. DOI:10.1063/1.1758217

GORJI M H, TORRILHON M, JENNY P. Fokker–Planck model for computational studies of monatomic rarefied gas flows[J]. Journal of Fluid Mechanics, 2011, 680: 574-601. DOI:10.1017/jfm.2011.188

欧阳水吾, 谢中强. 高温非平衡空气绕流[M]. 北京: 国防工业出版社, 2001.

WANG-CHANG C S, UHLENBECK G E. Transport phenomena in polyatomic gases[R ] . University of Michigan, UMR4405, 1951. http://deepblue.lib.umich.edu/bitstream/2027.42/8195/5/bad7743.0001.001.pdf

MASON E A, MONCHICK L. Heat conductivity of polyatomic and polar gases[J]. The Journal of Chemical Physics, 1962, 36(6): 1622-1639. DOI:10.1063/1.1732790

WU L, LI Q, LIU H H, et al. Extraction of the translational Eucken factor from light scattering by molecular gas[J]. Journal of Fluid Mechanics, 2020, 901: A23. DOI:10.1017/jfm.2020.568

BORGNAKKE C, LARSEN P S. Statistical collision model for Monte Carlo simulation of polyatomic gas mixture[J]. Journal of Computational Physics, 1975, 18(4): 405-420. DOI:10.1016/0021-9991(75)90094-7

BOYD I D. Rotational-translational energy transfer in rarefied nonequilibrium flows[J]. Physics of Fluids A: Fluid Dynamics, 1990, 2(3): 447-452. DOI:10.1063/1.857740

HAAS B L, HASH D B, BIRD G A, et al. Rates of thermal relaxation in direct simulation Monte Carlo methods[J]. Physics of Fluids, 1994, 6(6): 2191-2201. DOI:10.1063/1.868221

GIMELSHEIN N E, GIMELSHEIN S F, LEVIN D A. Vibrational relaxation rates in the direct simulation Monte Carlo method[J]. Physics of Fluids, 2002, 14(12): 4452-4455. DOI:10.1063/1.1517297

PORODNOV B T, KULEV A N, TUCHVETOV F T. Thermal transpiration in a circular capillary with a small temperature difference[J]. Journal of Fluid Mechanics, 1978, 88(4): 609-622. DOI:10.1017/s002211207800230x

LI Q, ZENG J N, SU W, et al. Uncertainty quantification in rarefied dynamics of molecular gas: rate effect of thermal relaxation[J]. Journal of Fluid Mechanics, 2021, 917: A58. DOI:10.1017/jfm.2021.338

RYKOV V A, SKOBELKIN V N. Macroscopic description of the motions of a gas with rotational degrees of freedom[J]. Fluid Dynamics, 1978, 13(1): 144-147. DOI:10.1007/BF01094479

SHARIPOV F. Influence of quantum intermolecular interaction on internal flows of rarefied gases[J]. Vacuum, 2018, 156: 146-153. DOI:10.1016/j.vacuum.2018.07.022

SONE Y. Kinetic Theory and Fluid Dynamics[M ] . Boston, MA: Birkhuser Boston, 2002. DOI: 10.1007/978-1-4612-0061-1

HILBERT D. Begründung der kinetischen gastheorie[J]. Mathematische Annalen, 1912, 72(4): 562-577. DOI:10.1007/BF01456676

Chapman S. VI. On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas[J]. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 1916, 216(538-548): 279-348. DOI:10.1098/rsta.1916.0006

ENSKOG D. Kinetische theorie der vorgänge in mässig verdünnten gasen (German edition)[M]. University of Michigan Library, 1917. (in German)

STRUCHTRUP H. Macroscopic transport equations for rarefied gas flows: approximation methods in Kinetic theory[M ] // Interaction of Mechanics and Mathematics, Heidelberg, Germany: Springer, 2021. doi: 10.1007/3-540-32386-4

CHEN S Z, XU K, CAI Q D. A comparison and unification of ellipsoidal statistical and shakhov BGK models[J]. Advances in Applied Mathematics and Mechanics, 2015, 7(2): 245-266. DOI:10.4208/aamm.2014.m559

WU L. Deterministic numerical simulation of the Boltzmann and kinetic model equations for classical and quantum dilute gases[D]. Glasgow: University of Strathclyde, 2013.

BEYLICH A. An interlaced system for Nitrogen gas[R]//Proceedings of CECAM Workshop, ENS de Lyon, France, 2000.

TCHEREMISSINE F G, AGARWAL R K, ABE T. Computation of hypersonic shock waves in diatomic gases using the generalized Boltzmann equation[C ] //AIP Conference Proceedings 1084, Kyoto, Japan, 2008: 427-433. doi: 10.1063/1.3076515

MORSE T F. Kinetic model for gases with internal degrees of freedom[J]. The Physics of Fluids, 1964, 7(2): 159-169. DOI:10.1063/1.1711128

LI X D, HU Z M, JIANG Z L. The continuum theory of bulk viscosity in compressible fluids[J]. SCIENTIA SINICA Physica, Mechanica Astronomica, 2016, 46(3): 034701. (in Chinese)

BRUNO D, FREZZOTTI A, GHIROLDI G P. Oxygen transport properties estimation by classical trajectory-direct simulation Monte Carlo[J]. Physics of Fluids, 2015, 27(5): 057101. DOI:10.1063/1.4921157

MEADOR W E, MINER G A, TOWNSEND L W. Bulk viscosity as a relaxation parameter: fact or fiction? [J]. Physics of Fluids, 1996, 8(1): 258-261. DOI:10.1063/1.868833

MEIJER A S, DE WIJN A S, PETERS M F E, et al. Coherent Rayleigh-Brillouin scattering measurements of bulk viscosity of polar and nonpolar gases, and kinetic theory[J]. The Journal of Chemical Physics, 2010, 133(16): 164315. DOI:10.1063/1.3491513

JAEGER F, MATAR O K, MÜLLER E A. Bulk viscosity of molecular fluids[J]. The Journal of Chemical Physics, 2018, 148(17): 174504. DOI:10.1063/1.5022752

HANSON F B, MORSE T F. Kinetic models for a gas with internal structure[J]. The Physics of Fluids, 1967, 10(2): 345-353. DOI:10.1063/1.1762114

GROSS E P, JACKSON E A. Kinetic models and the linearized Boltzmann equation[J]. The Physics of Fluids, 1959, 2(4): 432-441. DOI:10.1063/1.1724415

BOLEY C D, DESAI R C, TENTI G. Kinetic models and Brillouin scattering in a molecular gas[J]. Canadian Journal of Physics, 1972, 50(18): 2158-2173. DOI:10.1139/p72-286

TENTI G, BOLEY C D, DESAI R C. On the kinetic model description of Rayleigh–Brillouin scattering from molecular gases[J]. Canadian Journal of Physics, 1974, 52(4): 285-290. DOI:10.1139/p74-041

PAN X G, SHNEIDER M N, MILES R B. Coherent Rayleigh-Brillouin scattering in molecular gases[J]. Physical Review A, 2004, 69(3): 033814. DOI:10.1103/physreva.69.033814

WU L, WHITE C, SCANLON T J, et al. A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases[J]. Journal of Fluid Mechanics, 2015, 763: 24-50. DOI:10.1017/jfm.2014.632

WANG Z, YAN H, LI Q B, et al. Unified gas-kinetic scheme for diatomic molecular flow with translational, rotational, and vibrational modes[J]. Journal of Computational Physics, 2017, 350: 237-259. DOI:10.1016/j.jcp.2017.08.045

PENG A P, LI Z H, WU J L, et al. Validation and analysis of gas-kinetic unified algorithm for solving Boltzmann model equation with vibrational energy excitation[J]. Acta Physica Sinica, 2017, 66(20): 204703. (in Chinese)[万方]

TITAREV V A, FROLOVA A A. Application of model kinetic equations to calculations of super- and hypersonic molecular gas flows[J]. Fluid Dynamics, 2018, 53(4): 536-551. DOI:10.1134/s0015462818040110

MATHIAUD J, MIEUSSENS L. BGK and Fokker-Planck models of the Boltzmann equation for gases with discrete levels of vibrational energy[J]. Journal of Statistical Physics, 2020, 178(5): 1076-1095. DOI:10.1007/s10955-020-02490-7

RYKOV V A. A model kinetic equation for a gas with rotational degrees of freedom[J]. Fluid Dynamics, 1975, 10(6): 959-966. DOI:10.1007/BF01023275

GORJI M H, JENNY P. A Fokker-Planck based kinetic model for diatomic rarefied gas flows[J]. Physics of Fluids, 2013, 25(6): 062002. DOI:10.1063/1.4811399

MASON E A. Molecular relaxation times from thermal transpiration measurements[J]. The Journal of Chemical Physics, 1963, 39(3): 522-526. DOI:10.1063/1.1734288

GUPTA A D, STORVICK T S. Analysis of the heat conductivity data for polar and nonpolar gases using thermal transpiration measurements[J]. The Journal of Chemical Physics, 1970, 52(2): 742-749. DOI:10.1063/1.1673048

GREEN M S. Markoff random processes and the statistical mechanics of time-dependent phenomena. II. irreversible processes in fluids[J]. The Journal of Chemical Physics, 1954, 22(3): 398-413. DOI:10.1063/1.1740082

KUBO R. Statistical-mechanical theory of irreversible processes. I. general theory and simple applications to magnetic and conduction problems[J]. Journal of the Physical Society of Japan, 1957, 12(6): 570-586. DOI:10.1143/jpsj.12.570

PAN X G, SHNEIDER M N, MILES R B. Coherent Rayleigh-Brillouin scattering[J]. Physical Review Letters, 2002, 89(18): 183001. DOI:10.1103/physrevlett.89.183001

VIEITEZ M O, VAN DUIJN E J, UBACHS W, et al. Coherent and spontaneous Rayleigh-Brillouin scattering in atomic and molecular gases and gas mixtures[J]. Physical Review A, 2010, 82(4): 043836. DOI:10.1103/physreva.82.043836

GU Z Y, UBACHS W. Temperature-dependent bulk viscosity of nitrogen gas determined from spontaneous Rayleigh-Brillouin scattering[J]. Optics Letters, 2013, 38(7): 1110-1112. DOI:10.1364/OL.38.001110

SUGAWARA A, YIP S, SIROVICH L. Spectrum of density fluctuations in gases[J]. The Physics of Fluids, 1968, 11(5): 925-932. DOI:10.1063/1.1692060

REICHL L E. A Modern Course in Statistical Physics[M ] . Weinheim, Germany: Wiley-VCH Verlag GmbH Co. KGaA, 2016. doi: 10.1002/9783527690497

DAGDIGIAN P J. Vibrational and rotational relaxation in gases (Lambert J D)[J ] . Journal of Chemical Education, 1979, 56(1): A42. https://pubs.acs.org/doi/pdf/10.1021/ed056pA42.1 doi: 10.1021/ed056pA42.1

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