September  2015, 8(3): 493-531. doi: 10.3934/krm.2015.8.493

Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature

1. 

Departamento de Matemáticas, Universidad del País Vasco, (UPV/EHU), Apartado 644, E-48080 Bilbao, Spain

2. 

Basque Center for Applied Mathematics, Mazarredo 14, E48009 Bilbao, Spain

Received  January 2015 Revised  February 2015 Published  June 2015

We consider an approximation of the linearised equation of the homogeneous Boltzmann equation that describes the distribution of quasiparticles in a dilute gas of bosons at low temperature. The corresponding collision frequency is neither bounded from below nor from above. We prove the existence and uniqueness of solutions satisfying the conservation of energy. We show that these solutions converge to the corresponding stationary state, at an algebraic rate as time tends to infinity.
Citation: Miguel Escobedo, Minh-Binh Tran. Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature. Kinetic and Related Models, 2015, 8 (3) : 493-531. doi: 10.3934/krm.2015.8.493
References:
[1]

L. Arkeryd and A. Nouri, Bose condensates in interaction with excitations: A kinetic model, Communications in Mathematical Physics, 310 (2012), 765-788. doi: 10.1007/s00220-012-1415-1.

[2]

L. Arkeryd and A. Nouri, Bose condensates in interaction with excitations - a two-component space-dependent model close to equilibrium, preprint, arXiv:1307.3012.

[3]

L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Arch. Ration. Mech. Anal., 77 (1981), 11-21. doi: 10.1007/BF00280403.

[4]

D. Benin, Phonon viscosity and wide-angle phonon scattering in superfluid helium, Phys. Rev. B, 11 (1975), 145-149. doi: 10.1103/PhysRevB.11.145.

[5]

F. A. Buot, On the relaxation rate spectrum of phonons, J. Phys. C: Solid State Phys., 5 (1972), 5-14. doi: 10.1088/0022-3719/5/1/004.

[6]

R. E. Caflisch, The Boltzmann equation with a soft potential I. Linear, spatially-homogeneous, Comm. Math. Phys., 74 (1980), 71-95. doi: 10.1007/BF01197579.

[7]

T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math., 60 (1933), 91-146. doi: 10.1007/BF02398270.

[8]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory Of Dilute Gases, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[9]

F. H. Claro and G. H. Wannier, Relaxation spectrum of phonons: A solvable model, J. Math. Phys., 12 (1971), 92-95. doi: 10.1063/1.1665492.

[10]

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.

[11]

U. Eckern, Relaxation processes in a condensed Bose gas, J. Low Temp. Phys., 54 (1984), 333-359. doi: 10.1007/BF00683281.

[12]

M. Escobedo, F. Pezzotti and M. Valle, Analytical approach to relaxation dynamics of condensed Bose gases, Ann. Physics, 326 (2011), 808-827. doi: 10.1016/j.aop.2010.11.001.

[13]

H. Grad, Asymptotic theory of the Boltzmann equation {II}, in Rarefied Gas Dynamics, Vol. I (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), Academic Press, New York, 1963, 26-59.

[14]

M. Imamovic-Tomasovic and A. Griffin, Quasiparticle kinetic equation in a trapped Bose gas at low temperatures, J. Low Temp. Phys., 122 (2001), 617-655.

[15]

T. R. Kirkpatrick and J. R. Dorfman, Transport theory for a weakly interacting condensed Bose gas, Phys. Rev. A, 28 (1983), 2576-2579. doi: 10.1103/PhysRevA.28.2576.

[16]

T. R. Kirkpatrick and J. R. Dorfman, Transport in a dilute but condensed nonideal Bose gas: Kinetic equations, J. Low Temp. Phys., 58 (1985), 301-331. doi: 10.1007/BF00681309.

[17]

H. Spohn, The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics, J. Stat. Phys., 124 (2006), 1041-1104. doi: 10.1007/s10955-005-8088-5.

[18]

R. M. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian, Archive for Rational Mechanics and Analysis, 187 (2008), 287-339. doi: 10.1007/s00205-007-0067-3.

[19]

S. Ukai and K. Asano, On the Cauchy problem of the Boltzmann equation with a soft potential, Publ. RIMS, Kyoto Univ., 18 (1982), 57-99. doi: 10.2977/prims/1195183569.

[20]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook Of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam, 2002, 71-305. doi: 10.1016/S1874-5792(02)80004-0.

[21]

G. H. Wannier, Relaxation rate spectrum of photons, Bull. Am. Phys. Soc., 14 (1969), 303-303.

show all references

References:
[1]

L. Arkeryd and A. Nouri, Bose condensates in interaction with excitations: A kinetic model, Communications in Mathematical Physics, 310 (2012), 765-788. doi: 10.1007/s00220-012-1415-1.

[2]

L. Arkeryd and A. Nouri, Bose condensates in interaction with excitations - a two-component space-dependent model close to equilibrium, preprint, arXiv:1307.3012.

[3]

L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Arch. Ration. Mech. Anal., 77 (1981), 11-21. doi: 10.1007/BF00280403.

[4]

D. Benin, Phonon viscosity and wide-angle phonon scattering in superfluid helium, Phys. Rev. B, 11 (1975), 145-149. doi: 10.1103/PhysRevB.11.145.

[5]

F. A. Buot, On the relaxation rate spectrum of phonons, J. Phys. C: Solid State Phys., 5 (1972), 5-14. doi: 10.1088/0022-3719/5/1/004.

[6]

R. E. Caflisch, The Boltzmann equation with a soft potential I. Linear, spatially-homogeneous, Comm. Math. Phys., 74 (1980), 71-95. doi: 10.1007/BF01197579.

[7]

T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math., 60 (1933), 91-146. doi: 10.1007/BF02398270.

[8]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory Of Dilute Gases, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[9]

F. H. Claro and G. H. Wannier, Relaxation spectrum of phonons: A solvable model, J. Math. Phys., 12 (1971), 92-95. doi: 10.1063/1.1665492.

[10]

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.

[11]

U. Eckern, Relaxation processes in a condensed Bose gas, J. Low Temp. Phys., 54 (1984), 333-359. doi: 10.1007/BF00683281.

[12]

M. Escobedo, F. Pezzotti and M. Valle, Analytical approach to relaxation dynamics of condensed Bose gases, Ann. Physics, 326 (2011), 808-827. doi: 10.1016/j.aop.2010.11.001.

[13]

H. Grad, Asymptotic theory of the Boltzmann equation {II}, in Rarefied Gas Dynamics, Vol. I (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), Academic Press, New York, 1963, 26-59.

[14]

M. Imamovic-Tomasovic and A. Griffin, Quasiparticle kinetic equation in a trapped Bose gas at low temperatures, J. Low Temp. Phys., 122 (2001), 617-655.

[15]

T. R. Kirkpatrick and J. R. Dorfman, Transport theory for a weakly interacting condensed Bose gas, Phys. Rev. A, 28 (1983), 2576-2579. doi: 10.1103/PhysRevA.28.2576.

[16]

T. R. Kirkpatrick and J. R. Dorfman, Transport in a dilute but condensed nonideal Bose gas: Kinetic equations, J. Low Temp. Phys., 58 (1985), 301-331. doi: 10.1007/BF00681309.

[17]

H. Spohn, The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics, J. Stat. Phys., 124 (2006), 1041-1104. doi: 10.1007/s10955-005-8088-5.

[18]

R. M. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian, Archive for Rational Mechanics and Analysis, 187 (2008), 287-339. doi: 10.1007/s00205-007-0067-3.

[19]

S. Ukai and K. Asano, On the Cauchy problem of the Boltzmann equation with a soft potential, Publ. RIMS, Kyoto Univ., 18 (1982), 57-99. doi: 10.2977/prims/1195183569.

[20]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook Of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam, 2002, 71-305. doi: 10.1016/S1874-5792(02)80004-0.

[21]

G. H. Wannier, Relaxation rate spectrum of photons, Bull. Am. Phys. Soc., 14 (1969), 303-303.

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