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 & 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.  doi: 10.1007/s00220-012-1415-1.  Google Scholar

[2]

L. Arkeryd and A. Nouri, Bose condensates in interaction with excitations - a two-component space-dependent model close to equilibrium, preprint,, , ().   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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.  doi: 10.1007/s00222-004-0389-9.  Google Scholar

[11]

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

[12]

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

[13]

H. Grad, Asymptotic theory of the Boltzmann equation {II},, in Rarefied Gas Dynamics, (1962), 26.   Google Scholar

[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.   Google Scholar

[15]

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

[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.  doi: 10.1007/BF00681309.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

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.  doi: 10.1007/s00220-012-1415-1.  Google Scholar

[2]

L. Arkeryd and A. Nouri, Bose condensates in interaction with excitations - a two-component space-dependent model close to equilibrium, preprint,, , ().   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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.  doi: 10.1007/s00222-004-0389-9.  Google Scholar

[11]

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

[12]

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

[13]

H. Grad, Asymptotic theory of the Boltzmann equation {II},, in Rarefied Gas Dynamics, (1962), 26.   Google Scholar

[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.   Google Scholar

[15]

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

[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.  doi: 10.1007/BF00681309.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

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