June  2018, 11(3): 521-545. doi: 10.3934/krm.2018023

Landau damping in the multiscale Vlasov theory

1. 

École Polytechnique de Montréal, C.P.6079 suc. Centre-ville, Montréal, H3C 3A7, Québec, Canada

2. 

Mathematical Institute, Faculty of Mathematics, Charles University, Sokolovská 83, 18675 Prague, Czech Republic

* Corresponding author: Miroslav Grmela

Received  March 2017 Revised  August 2017 Published  March 2018

Fund Project: This research was supported by the Natural Sciences and Engineering Research Council of Canada and by the Czech Science Foundation, project no. 17-15498Y.

Vlasov kinetic theory is extended by adopting an extra one particle distribution function as an additional state variable characterizing the micro-turbulence internal structure. The extended Vlasov equation keeps the reversibility, the Hamiltonian structure, and the entropy conservation of the original Vlasov equation. In the setting of the extended Vlasov theory we then argue that the Fokker-Planck type damping in the velocity dependence of the extra distribution function induces the Landau damping. The same type of extension is made also in the setting of fluid mechanics.

Citation: Miroslav Grmela, Michal Pavelka. Landau damping in the multiscale Vlasov theory. Kinetic & Related Models, 2018, 11 (3) : 521-545. doi: 10.3934/krm.2018023
References:
[1]

S. Ansumali, I. V. Karlin and H. C. Öttinger, Thermodynamic Theory of Incomressible Hydrodynamics, Phys. Rev. Lett., 94 (2005), 080602. Google Scholar

[2]

V. I. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infini et ses applications dans l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier, 16 (1966), 319-361.  doi: 10.5802/aif.233.  Google Scholar

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R. B. Bird, R. C. Armstrong and C. F. Curtiss, Dynamics of Polymeric Liquids: Volume 2 : Kinetic Theory, v. 2 ISBN 9780471015963, Board of advisors, enigineering, Wiley, 1997. Google Scholar

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A. Clebsch, Über die Integration der hydrodynamische Gleichungen, J. Reine Angew. Math., 56 (1859), 1-10.   Google Scholar

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L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9.  Google Scholar

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O. EsenM. Pavelka and M. Grmela, Hamiltonian coupling of electromagnetic field and matter, Int. J. Adv. Eng. Sci. Appl. Math., 9 (2017), 3-20.  doi: 10.1007/s12572-017-0179-4.  Google Scholar

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E. Feireisl and A. Novotný, The low Mach number limit for the full Navier-Stokes-Fourier system, Arch. Rational Mech. Anal., 186 (2007), 77-107.  doi: 10.1007/s00205-007-0066-4.  Google Scholar

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H. Grad, On Boltzmann's H-theorem, J. Soc. Indust. Math., 13 (1965), 259-277.  doi: 10.1137/0113016.  Google Scholar

[9]

M. Grmela, Particle and bracket formulations of kinetic equations, Contemp. Math., 28 (1984), 125-132.   Google Scholar

[10]

M. Grmela, Extensions of classical hydrodynamics, J. Stat. Phys., 132 (2008), 581-602.  doi: 10.1007/s10955-008-9558-3.  Google Scholar

[11]

M. Grmela, Fluctuations in extended mass-action-law dynamics, Physica D, 241 (2012), 976-986.  doi: 10.1016/j.physd.2012.02.008.  Google Scholar

[12]

M. Grmela, Contact geometry of mesoscopic thermodynamics and dynamics, Entropy, 16 (2014), 1652-1686.  doi: 10.3390/e16031652.  Google Scholar

[13]

B. Haspot and E. Zatorska, From the highly compressible Navier-Stokes equations to the porous medium equation -rate of convergence, Discrete and Continuous Dynamical Systems, 36 (2016), 3107-3123.   Google Scholar

[14]

D. Holm, Geometric Mechanics: Part Ⅰ. Dynamics and Symmetry, Imperial College Press, London, UK, 2011.  Google Scholar

[15]

D. Jou, G. Lebon and J. Casas-V{á}zquez, Extended Irreversible Thermodynamics, Springer-Verlag, Berlin, 1993.  Google Scholar

[16]

J. G. Kirkwood, The statistical mechanical theory of transport processes, Ⅰ, Ⅱ, J. Chem. Phys., 14 (1946), 180-201, 15 (1947), 72-76. Google Scholar

[17]

L. D. Landau, On the vibration of the electronic plasma, Zhurnal eksperimantalnoi teoreticheskoi fiziki, 16 (1946), 574-586.   Google Scholar

[18]

E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, Vol. 10 (1st ed.), Pergamon Press, ISBN 978-0-7506-2635-4,1981. Google Scholar

[19]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., 71 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6.  Google Scholar

[20]

A. Majda, Comressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, Berlin, 1984.  Google Scholar

[21]

J. Malmberg and C. Wharton, Collisionless damping of electrostatic plasma waves, Phys. Rev. Lett., 13 (1964), 184-186.  doi: 10.1103/PhysRevLett.13.184.  Google Scholar

[22]

J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices and Clebsch variables for incompressible fluids, Physica D, 7 (1983), 305-323.  doi: 10.1016/0167-2789(83)90134-3.  Google Scholar

[23]

J. E. MarsdenT. S. Ratiu and A. Weinstein, Reduction and Hamiltonian structures on duals of semidirect product lie algebras, Cont. Math. AMS, 28 (1984), 55-100.   Google Scholar

[24]

C. Mouhot and C. Villani, On Landau damping, Acta Mathematica, 207 (2011), 29-201.  doi: 10.1007/s11511-011-0068-9.  Google Scholar

[25]

I. Müller and T. Ruggeri, Rational Extended Thermodynamics, ISBN 9780387983738, Springer tracts in natural philosophy, 1998, Springer.  Google Scholar

[26]

M. PavelkaV. KlikaO. Esen and M. Grmela, A hierarchy of Poisson brackets in non-equilibrium thermodynamics, Physica D: Nonlinear Phenomena, 335 (2016), 54-69.  doi: 10.1016/j.physd.2016.06.011.  Google Scholar

[27]

S. B. Pope, Turbulent Flows, Cambridge University Press, 2000.  Google Scholar

[28]

R. Robert, Statistical mechanics and hydrodynamical turbulence, In Proceedings of the International Congress of Mathematicians, 1, 2, Zürich (1994), Basel, (1995), Birkhäuser, 1523-1531.  Google Scholar

[29]

T. Ruggeri and M. Sugiyama, Rational Extended Thermodynamics Beyond the Monatomic Gas, Springer, Cham, 2015.  Google Scholar

[30]

B. Turkington, Statistical equilibrium measures and coherent states in two-dimensional turbulence, Comm. Pure Appl. Math., 52 (1999), 781-809.  doi: 10.1002/(SICI)1097-0312(199907)52:7<781::AID-CPA1>3.0.CO;2-C.  Google Scholar

[31]

P. VánM. Pavelka and M. Grmela, Extra mass flux in fluid mechanics, J. Non-Equilibrium Thermodynamics, 42 (2017), 133-151.   Google Scholar

show all references

References:
[1]

S. Ansumali, I. V. Karlin and H. C. Öttinger, Thermodynamic Theory of Incomressible Hydrodynamics, Phys. Rev. Lett., 94 (2005), 080602. Google Scholar

[2]

V. I. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infini et ses applications dans l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier, 16 (1966), 319-361.  doi: 10.5802/aif.233.  Google Scholar

[3]

R. B. Bird, R. C. Armstrong and C. F. Curtiss, Dynamics of Polymeric Liquids: Volume 2 : Kinetic Theory, v. 2 ISBN 9780471015963, Board of advisors, enigineering, Wiley, 1997. Google Scholar

[4]

A. Clebsch, Über die Integration der hydrodynamische Gleichungen, J. Reine Angew. Math., 56 (1859), 1-10.   Google Scholar

[5]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9.  Google Scholar

[6]

O. EsenM. Pavelka and M. Grmela, Hamiltonian coupling of electromagnetic field and matter, Int. J. Adv. Eng. Sci. Appl. Math., 9 (2017), 3-20.  doi: 10.1007/s12572-017-0179-4.  Google Scholar

[7]

E. Feireisl and A. Novotný, The low Mach number limit for the full Navier-Stokes-Fourier system, Arch. Rational Mech. Anal., 186 (2007), 77-107.  doi: 10.1007/s00205-007-0066-4.  Google Scholar

[8]

H. Grad, On Boltzmann's H-theorem, J. Soc. Indust. Math., 13 (1965), 259-277.  doi: 10.1137/0113016.  Google Scholar

[9]

M. Grmela, Particle and bracket formulations of kinetic equations, Contemp. Math., 28 (1984), 125-132.   Google Scholar

[10]

M. Grmela, Extensions of classical hydrodynamics, J. Stat. Phys., 132 (2008), 581-602.  doi: 10.1007/s10955-008-9558-3.  Google Scholar

[11]

M. Grmela, Fluctuations in extended mass-action-law dynamics, Physica D, 241 (2012), 976-986.  doi: 10.1016/j.physd.2012.02.008.  Google Scholar

[12]

M. Grmela, Contact geometry of mesoscopic thermodynamics and dynamics, Entropy, 16 (2014), 1652-1686.  doi: 10.3390/e16031652.  Google Scholar

[13]

B. Haspot and E. Zatorska, From the highly compressible Navier-Stokes equations to the porous medium equation -rate of convergence, Discrete and Continuous Dynamical Systems, 36 (2016), 3107-3123.   Google Scholar

[14]

D. Holm, Geometric Mechanics: Part Ⅰ. Dynamics and Symmetry, Imperial College Press, London, UK, 2011.  Google Scholar

[15]

D. Jou, G. Lebon and J. Casas-V{á}zquez, Extended Irreversible Thermodynamics, Springer-Verlag, Berlin, 1993.  Google Scholar

[16]

J. G. Kirkwood, The statistical mechanical theory of transport processes, Ⅰ, Ⅱ, J. Chem. Phys., 14 (1946), 180-201, 15 (1947), 72-76. Google Scholar

[17]

L. D. Landau, On the vibration of the electronic plasma, Zhurnal eksperimantalnoi teoreticheskoi fiziki, 16 (1946), 574-586.   Google Scholar

[18]

E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, Vol. 10 (1st ed.), Pergamon Press, ISBN 978-0-7506-2635-4,1981. Google Scholar

[19]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., 71 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6.  Google Scholar

[20]

A. Majda, Comressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, Berlin, 1984.  Google Scholar

[21]

J. Malmberg and C. Wharton, Collisionless damping of electrostatic plasma waves, Phys. Rev. Lett., 13 (1964), 184-186.  doi: 10.1103/PhysRevLett.13.184.  Google Scholar

[22]

J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices and Clebsch variables for incompressible fluids, Physica D, 7 (1983), 305-323.  doi: 10.1016/0167-2789(83)90134-3.  Google Scholar

[23]

J. E. MarsdenT. S. Ratiu and A. Weinstein, Reduction and Hamiltonian structures on duals of semidirect product lie algebras, Cont. Math. AMS, 28 (1984), 55-100.   Google Scholar

[24]

C. Mouhot and C. Villani, On Landau damping, Acta Mathematica, 207 (2011), 29-201.  doi: 10.1007/s11511-011-0068-9.  Google Scholar

[25]

I. Müller and T. Ruggeri, Rational Extended Thermodynamics, ISBN 9780387983738, Springer tracts in natural philosophy, 1998, Springer.  Google Scholar

[26]

M. PavelkaV. KlikaO. Esen and M. Grmela, A hierarchy of Poisson brackets in non-equilibrium thermodynamics, Physica D: Nonlinear Phenomena, 335 (2016), 54-69.  doi: 10.1016/j.physd.2016.06.011.  Google Scholar

[27]

S. B. Pope, Turbulent Flows, Cambridge University Press, 2000.  Google Scholar

[28]

R. Robert, Statistical mechanics and hydrodynamical turbulence, In Proceedings of the International Congress of Mathematicians, 1, 2, Zürich (1994), Basel, (1995), Birkhäuser, 1523-1531.  Google Scholar

[29]

T. Ruggeri and M. Sugiyama, Rational Extended Thermodynamics Beyond the Monatomic Gas, Springer, Cham, 2015.  Google Scholar

[30]

B. Turkington, Statistical equilibrium measures and coherent states in two-dimensional turbulence, Comm. Pure Appl. Math., 52 (1999), 781-809.  doi: 10.1002/(SICI)1097-0312(199907)52:7<781::AID-CPA1>3.0.CO;2-C.  Google Scholar

[31]

P. VánM. Pavelka and M. Grmela, Extra mass flux in fluid mechanics, J. Non-Equilibrium Thermodynamics, 42 (2017), 133-151.   Google Scholar

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