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Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion
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 |
The study of quantum quasi-particles at low temperatures including their statistics, is a frontier area in modern physics. In a seminal paper Haldane [
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. Bhaduri, R. 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. Bhaduri, R. 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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