April 2019, 12(2): 323-346. doi: 10.3934/krm.2019014

On a Boltzmann equation for Haldane statistics

1. 

Mathematical Sciences, 41296 Göteborg, Sweden

2. 

Aix-Marseille University, CNRS, Centrale Marseille, I2M UMR 7373, 13453 Marseille, France

Received  January 2018 Revised  June 2018 Published  November 2018

The study of quantum quasi-particles at low temperatures including their statistics, is a frontier area in modern physics. In a seminal paper Haldane [10] proposed a definition based on a generalization of the Pauli exclusion principle for fractional quantum statistics. The present paper is a study of quantum quasi-particles obeying Haldane statistics in a fully non-linear kinetic Boltzmann equation model with large initial data on a torus. Strong $L^1$ solutions are obtained for the Cauchy problem. The main results concern existence, uniqueness and stabililty. Depending on the space dimension and the collision kernel, the results obtained are local or global in time.

Citation: Leif Arkeryd, Anne Nouri. On a Boltzmann equation for Haldane statistics. Kinetic & Related Models, 2019, 12 (2) : 323-346. doi: 10.3934/krm.2019014
References:
[1]

L. Arkeryd, A quantum Boltzmann equation for Haldane statistics and hard forces; the spacehomogeneous initial value problem, Comm. Math. Phys., 298 (2010), 573-583. doi: 10.1007/s00220-010-1046-3.

[2]

L. Arkeryd and A. Nouri, Well-posedness of the Cauchy problem for a space-dependent anyon Boltzmann equation, SIAM J. Math. Anal., 47 (2015), 4720-4742. doi: 10.1137/15M1012335.

[3]

L. Arkeryd and A. Nouri, On the Cauchy problem with large data for a space-dependent Boltzmann-Nordheim boson equation, Comm. Math. Sci., 15 (2017), 1247-1264. doi: 10.4310/CMS.2017.v15.n5.a4.

[4]

L. Arkeryd and A. Nouri, On the Cauchy problem with large data for the space-dependent Boltzmann-Nordheim equation Ⅲ, Preprint (2018), arXiv: 1801.02494.

[5]

R. K. BhaduriR. S. Bhalero and M. V. Murthy, Haldane exclusion statistics and the Boltzmann equation, J. Stat. Phys., 82 (1996), 1659-1668.

[6]

J.-M. Bony, Solutions globales bornées pour les modèles discrets de l'équation de Boltzmann, en dimension 1 d'espace, Journées "Équations aux dérivées partielles ", Exp. XVI, École Polytech., Palaiseau, (1987), 1-10.

[7]

M. Briant and A. Einav, On the Cauchy problem for the homogeneous Boltzmann-Nordheim equation for bosons: local existence, uniqueness and creation of moments, J. Stat. Phys., 163 (2016), 1108-1156. doi: 10.1007/s10955-016-1517-9.

[8]

C. Cercignani and R. Illner, Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions, Arch Rat. Mech. Anal., 134 (1996), 1-16. doi: 10.1007/BF00376253.

[9]

M. Escobedo and J. L. Velazquez, Finite time blow-up and condensation for the bosonic Nordheim equation, Inv. Math, 200 (2015), 761-847. doi: 10.1007/s00222-014-0539-7.

[10]

F. D. Haldane, Fractional statistics in arbitrary dimensions: A generalization of the Pauli principle, Phys. Rev. Lett., 67 (1991), 937-940. doi: 10.1103/PhysRevLett.67.937.

[11]

J. M. Leinaas and J. Myrheim, On the theory of identical particles, Nuovo Cim. B, 37 (1977), 1-23.

[12]

X. LU, On isotropic distributional solutions to the Boltzmann equation for Bose-Einstein particles, J. Stat. Phys., 116 (2004), 1597-1649. doi: 10.1023/B:JOSS.0000041750.11320.9c.

[13]

X. Lu, The Boltzmann equation for Bose-Einstein particles: Condensation in finite time, J. Stat. Phys., 150 (2013), 1138-1176. doi: 10.1007/s10955-013-0725-9.

[14]

L. W. Nordheim, On the kinetic methods in the new statistics and its applications in the electron theory of conductivity, Proc. Roy. Soc. London Ser. A, 119 (1928), 689-698.

[15]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rat. Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.

show all references

References:
[1]

L. Arkeryd, A quantum Boltzmann equation for Haldane statistics and hard forces; the spacehomogeneous initial value problem, Comm. Math. Phys., 298 (2010), 573-583. doi: 10.1007/s00220-010-1046-3.

[2]

L. Arkeryd and A. Nouri, Well-posedness of the Cauchy problem for a space-dependent anyon Boltzmann equation, SIAM J. Math. Anal., 47 (2015), 4720-4742. doi: 10.1137/15M1012335.

[3]

L. Arkeryd and A. Nouri, On the Cauchy problem with large data for a space-dependent Boltzmann-Nordheim boson equation, Comm. Math. Sci., 15 (2017), 1247-1264. doi: 10.4310/CMS.2017.v15.n5.a4.

[4]

L. Arkeryd and A. Nouri, On the Cauchy problem with large data for the space-dependent Boltzmann-Nordheim equation Ⅲ, Preprint (2018), arXiv: 1801.02494.

[5]

R. K. BhaduriR. S. Bhalero and M. V. Murthy, Haldane exclusion statistics and the Boltzmann equation, J. Stat. Phys., 82 (1996), 1659-1668.

[6]

J.-M. Bony, Solutions globales bornées pour les modèles discrets de l'équation de Boltzmann, en dimension 1 d'espace, Journées "Équations aux dérivées partielles ", Exp. XVI, École Polytech., Palaiseau, (1987), 1-10.

[7]

M. Briant and A. Einav, On the Cauchy problem for the homogeneous Boltzmann-Nordheim equation for bosons: local existence, uniqueness and creation of moments, J. Stat. Phys., 163 (2016), 1108-1156. doi: 10.1007/s10955-016-1517-9.

[8]

C. Cercignani and R. Illner, Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions, Arch Rat. Mech. Anal., 134 (1996), 1-16. doi: 10.1007/BF00376253.

[9]

M. Escobedo and J. L. Velazquez, Finite time blow-up and condensation for the bosonic Nordheim equation, Inv. Math, 200 (2015), 761-847. doi: 10.1007/s00222-014-0539-7.

[10]

F. D. Haldane, Fractional statistics in arbitrary dimensions: A generalization of the Pauli principle, Phys. Rev. Lett., 67 (1991), 937-940. doi: 10.1103/PhysRevLett.67.937.

[11]

J. M. Leinaas and J. Myrheim, On the theory of identical particles, Nuovo Cim. B, 37 (1977), 1-23.

[12]

X. LU, On isotropic distributional solutions to the Boltzmann equation for Bose-Einstein particles, J. Stat. Phys., 116 (2004), 1597-1649. doi: 10.1023/B:JOSS.0000041750.11320.9c.

[13]

X. Lu, The Boltzmann equation for Bose-Einstein particles: Condensation in finite time, J. Stat. Phys., 150 (2013), 1138-1176. doi: 10.1007/s10955-013-0725-9.

[14]

L. W. Nordheim, On the kinetic methods in the new statistics and its applications in the electron theory of conductivity, Proc. Roy. Soc. London Ser. A, 119 (1928), 689-698.

[15]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rat. Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.

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