August  2018, 11(4): 715-734. doi: 10.3934/krm.2018029

Long time strong convergence to Bose-Einstein distribution for low temperature

Deartment of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received  August 2017 Revised  November 2017 Published  April 2018

Fund Project: This work was supported by National Natural Science Foundation of China Grant No.11571196.

We study the long time behavior of measure-valued isotropic solutions $F_t$ of the Boltzmann equation for Bose-Einstein particles for low temperature. The global in time existence of such solutions $F_t$ that converge at least semi-strongly to equilibrium (the Bose-Einstein distribution) has been proven in previous work and it has been known that the long time strong convergence to equilibrium is equivalent to the long time convergence to the Bose-Einstein condensation. Here we show that if such a solution $F_t$ as a family of Borel measures satisfies a uniform double-size condition (which is also necessary for the strong convergence), then $F_t$ converges strongly to equilibrium as $t$ tends to infinity. We also propose a new condition on the initial datum $F_0$ such that a corresponding solution $F_t$ converges strongly to equilibrium.

Citation: Xuguang Lu. Long time strong convergence to Bose-Einstein distribution for low temperature. Kinetic & Related Models, 2018, 11 (4) : 715-734. doi: 10.3934/krm.2018029
References:
[1]

L. Arkeryd, On low temperature kinetic theory; spin diffusion, Bose Einstein condensates, anyons, J. Stat. Phys., 150 (2013), 1063-1079.  doi: 10.1007/s10955-013-0695-y.  Google Scholar

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L. Arkeryd and A. Nouri, Bose condensates in interaction with excitations: A kinetic model, Comm. Math. Phys., 310 (2012), 765-788.  doi: 10.1007/s00220-012-1415-1.  Google Scholar

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J. Bandyopadhyay and J. J. L. Velázquez, Blow-up rate estimates for the solutions of the bosonic Boltzmann-Nordheim equation, J. Math. Phys., 56 (2015), 063302, 27 pp.  doi: 10.1063/1.4921917.  Google Scholar

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D. BenedettoM. PulvirentiF. Castella and R. Esposito, On the weak-coupling limit for bosons and fermions, Math. Models Methods Appl. Sci., 15 (2005), 1811-1843.  doi: 10.1142/S0218202505000984.  Google Scholar

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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.  Google Scholar

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L. ErdösM. Salmhofer and H.-T. Yau, On the quantum Boltzmann equation, J. Stat. Phys., 116 (2004), 367-380.  doi: 10.1023/B:JOSS.0000037224.56191.ed.  Google Scholar

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M. Escobedo, S. Mischler and M. A. Valle, Homogeneous Boltzmann Equation in Quantum Relativistic Kinetic Theory, Electronic Journal of Differential Equations, Monograph, 4. Southwest Texas State University, San Marcos, TX, 2003. 85 pp.  Google Scholar

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M. EscobedoS. Mischler and J. J. L. Velázquez, Singular solutions for the Uehling-Uhlenbeck equation, Proc. Roy. Soc. Edinburgh, 138 (2008), 67-107.  doi: 10.1017/S0308210506000655.  Google Scholar

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M. Escobedo and J. J. L. Velázquez, On the blow up and condensation of supercritical solutions of the Nordheim equation for bosons, Comm. Math. Phys., 330 (2014), 331-365.  doi: 10.1007/s00220-014-2034-9.  Google Scholar

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

[12]

C. JosserandY. Pomeau and S. Rica, Self-similar singularities in the kinetics of condensation, J. Low Temp. Phys., 145 (2006), 231-265 (2006).   Google Scholar

[13]

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

[14]

X. Lu, The Boltzmann equation for Bose-Einstein particles: Velocity concentration and convergence to equilibrium, J. Statist. Phys., 119 (2005), 1027-1067.  doi: 10.1007/s10955-005-3767-9.  Google Scholar

[15]

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.  Google Scholar

[16]

X. Lu, The Boltzmann equation for Bose-Einstein particles: Regularity and condensation, J. Stat. Phys., 156 (2014), 493-545.  doi: 10.1007/s10955-014-1026-7.  Google Scholar

[17]

X. Lu, Long time convergence of the Bose-Einstein condensation, J. Stat. Phys., 162 (2016), 652-670.  doi: 10.1007/s10955-015-1427-2.  Google Scholar

[18]

J. Lukkarinen and H. Spohn, Not to normal order--notes on the kinetic limit for weakly interacting quantum fluids, J. Stat. Phys., 134 (2009), 1133-1172.  doi: 10.1007/s10955-009-9682-8.  Google Scholar

[19]

P. A. Markowich and L. Pareschi, Fast conservative and entropic numerical methods for the boson Boltzmann equation, Numer. Math., 99 (2005), 509-532.  doi: 10.1007/s00211-004-0570-5.  Google Scholar

[20]

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.  doi: 10.1098/rspa.1928.0126.  Google Scholar

[21]

A. Nouri, Bose-Einstein condensates at very low temperatures: A mathematical result in the isotropic case, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), 649-666.   Google Scholar

[22]

W. Rudin, Real and Complex Analysis, Third edition. McGraw-Hill Book Co., New York, 1987. xiv+416 pp. ISBN: 0-07-054234-1 00A05 (26-01 30-01 46-01).  Google Scholar

[23]

D. V. Semikov and I. I. Tkachev, Kinetics of Bose condensation, Phys. Rev. Lett., 74 (1995), 3093-3097.   Google Scholar

[24]

D. V. Semikov and I. I. Tkachev, Condensation of Bose in the kinetic regime, Phys. Rev. D, 55 (1997), 489-502.   Google Scholar

[25]

H. Spohn, Kinetics of the Bose-Einstein condensation, Physica D, 239 (2010), 627-634.  doi: 10.1016/j.physd.2010.01.018.  Google Scholar

[26]

E. A. Uehling and G. E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases, I, Phys. Rev., 43 (1933), 552-561.  doi: 10.1103/PhysRev.43.552.  Google Scholar

show all references

References:
[1]

L. Arkeryd, On low temperature kinetic theory; spin diffusion, Bose Einstein condensates, anyons, J. Stat. Phys., 150 (2013), 1063-1079.  doi: 10.1007/s10955-013-0695-y.  Google Scholar

[2]

L. Arkeryd and A. Nouri, Bose condensates in interaction with excitations: A kinetic model, Comm. Math. Phys., 310 (2012), 765-788.  doi: 10.1007/s00220-012-1415-1.  Google Scholar

[3]

J. Bandyopadhyay and J. J. L. Velázquez, Blow-up rate estimates for the solutions of the bosonic Boltzmann-Nordheim equation, J. Math. Phys., 56 (2015), 063302, 27 pp.  doi: 10.1063/1.4921917.  Google Scholar

[4]

D. BenedettoM. PulvirentiF. Castella and R. Esposito, On the weak-coupling limit for bosons and fermions, Math. Models Methods Appl. Sci., 15 (2005), 1811-1843.  doi: 10.1142/S0218202505000984.  Google Scholar

[5]

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.  Google Scholar

[6] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Third Edition, Cambridge University Press, Cambridge, 1939.   Google Scholar
[7]

L. ErdösM. Salmhofer and H.-T. Yau, On the quantum Boltzmann equation, J. Stat. Phys., 116 (2004), 367-380.  doi: 10.1023/B:JOSS.0000037224.56191.ed.  Google Scholar

[8]

M. Escobedo, S. Mischler and M. A. Valle, Homogeneous Boltzmann Equation in Quantum Relativistic Kinetic Theory, Electronic Journal of Differential Equations, Monograph, 4. Southwest Texas State University, San Marcos, TX, 2003. 85 pp.  Google Scholar

[9]

M. EscobedoS. Mischler and J. J. L. Velázquez, Singular solutions for the Uehling-Uhlenbeck equation, Proc. Roy. Soc. Edinburgh, 138 (2008), 67-107.  doi: 10.1017/S0308210506000655.  Google Scholar

[10]

M. Escobedo and J. J. L. Velázquez, On the blow up and condensation of supercritical solutions of the Nordheim equation for bosons, Comm. Math. Phys., 330 (2014), 331-365.  doi: 10.1007/s00220-014-2034-9.  Google Scholar

[11]

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

[12]

C. JosserandY. Pomeau and S. Rica, Self-similar singularities in the kinetics of condensation, J. Low Temp. Phys., 145 (2006), 231-265 (2006).   Google Scholar

[13]

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

[14]

X. Lu, The Boltzmann equation for Bose-Einstein particles: Velocity concentration and convergence to equilibrium, J. Statist. Phys., 119 (2005), 1027-1067.  doi: 10.1007/s10955-005-3767-9.  Google Scholar

[15]

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.  Google Scholar

[16]

X. Lu, The Boltzmann equation for Bose-Einstein particles: Regularity and condensation, J. Stat. Phys., 156 (2014), 493-545.  doi: 10.1007/s10955-014-1026-7.  Google Scholar

[17]

X. Lu, Long time convergence of the Bose-Einstein condensation, J. Stat. Phys., 162 (2016), 652-670.  doi: 10.1007/s10955-015-1427-2.  Google Scholar

[18]

J. Lukkarinen and H. Spohn, Not to normal order--notes on the kinetic limit for weakly interacting quantum fluids, J. Stat. Phys., 134 (2009), 1133-1172.  doi: 10.1007/s10955-009-9682-8.  Google Scholar

[19]

P. A. Markowich and L. Pareschi, Fast conservative and entropic numerical methods for the boson Boltzmann equation, Numer. Math., 99 (2005), 509-532.  doi: 10.1007/s00211-004-0570-5.  Google Scholar

[20]

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.  doi: 10.1098/rspa.1928.0126.  Google Scholar

[21]

A. Nouri, Bose-Einstein condensates at very low temperatures: A mathematical result in the isotropic case, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), 649-666.   Google Scholar

[22]

W. Rudin, Real and Complex Analysis, Third edition. McGraw-Hill Book Co., New York, 1987. xiv+416 pp. ISBN: 0-07-054234-1 00A05 (26-01 30-01 46-01).  Google Scholar

[23]

D. V. Semikov and I. I. Tkachev, Kinetics of Bose condensation, Phys. Rev. Lett., 74 (1995), 3093-3097.   Google Scholar

[24]

D. V. Semikov and I. I. Tkachev, Condensation of Bose in the kinetic regime, Phys. Rev. D, 55 (1997), 489-502.   Google Scholar

[25]

H. Spohn, Kinetics of the Bose-Einstein condensation, Physica D, 239 (2010), 627-634.  doi: 10.1016/j.physd.2010.01.018.  Google Scholar

[26]

E. A. Uehling and G. E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases, I, Phys. Rev., 43 (1933), 552-561.  doi: 10.1103/PhysRev.43.552.  Google Scholar

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