Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation

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• 作者：Niclas Bernhoff (1)
  
  1. Department of Mathematics and Computer Science ; Karlstad University ; 651 88 ; Karlstad ; Sweden
  
• 关键词：Bose鈥揈instein condensate ; Low temperature kinetics ; Discrete kinetic equation ; Milne problem ; Kramer problem
• 刊名：Journal of Statistical Physics
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：159
• 期：2
• 页码：358-379
• 全文大小：256 KB
• 参考文献：1. Arkeryd, L (2013) On low temperature kinetic theory; spin diffusion, Bose鈥揈instein condensates, anyons. J. Stat. Phys. 150: pp. 1063-1079 CrossRef
  2. Arkeryd, L, Nouri, A (2013) A Milne problem from a Bose condensate with excitations. Kinet. Relat. Models 6: pp. 671-686 CrossRef
  3. Bardos, C, Caflisch, RE, Nicolaenko, B (1986) The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas. Comm. Pure Appl. Math. 39: pp. 323-352 CrossRef
  4. Bardos, C, Golse, F, Sone, Y (2006) Half-space problems for the Boltzmann equation: a survey. J. Stat. Phys. 124: pp. 275-300 CrossRef
  5. Bernhoff, N (2008) On half-space problems for the linearized discrete Boltzmann equation. Riv. Mat. Univ. Parma 9: pp. 73-124
  6. Bernhoff, N (2010) On half-space problems for the weakly non-linear discrete Boltzmann equation. Kinet. Relat. Models 3: pp. 195-222 CrossRef
  7. Bernhoff, N, Bobylev, A (2007) Weak shock waves for the general discrete velocity model of the Boltzmann equation. Commun. Math. Sci. 5: pp. 815-832 CrossRef
  8. Bhaduri, RK, Bhalerao, RS, Murthy, MVN (1996) Haldane exclusion statistics and the Boltzmann equation. J. Stat. Phys. 82: pp. 1659-1668 CrossRef
  9. Bobylev, A.V., Bernhoff, N.: Discrete velocity models and dynamical systems. In: N. Bellomo, R. Gatignol (eds.) Lecture Notes on the Discretization of the Boltzmann Equation, pp. 203鈥?22. World Scientific, Singapore (2003)
  10. Bobylev, AV, Cercignani, C (1999) Discrete velocity models without non-physical invariants. J. Stat. Phys. 97: pp. 677-686 CrossRef
  11. Bobylev, A.V., Palczewski, A., Schneider, J.: On approximation of the Boltzmann equation by discrete velocity models. C. R. Acad. Sci. Paris S茅r. I Math. 320, 639鈥?44 (1995)
  12. Bobylev, AV, Vinerean, MC (2008) Construction of discrete kinetic models with given invariants. J. Stat. Phys. 132: pp. 153-170 CrossRef
  13. Bobylev, AV, Vinerean, MC, Windf盲ll, 脜 (2010) Discrete velocity models of the Boltzmann equation and conservation laws. Kinet. Relat. Models 3: pp. 35-58 CrossRef
  14. Cabannes, H.: The discrete Boltzmann equation (1980, 2003). Lecture notes given at the University of California at Berkeley, : revised with R. Gatignol and L-S, Luo (1980). 2003
  15. Cercignani, C.: The Boltzmann Equation and its Applications. Springer, Berlin (1988).
  16. Cercignani, C.: Rarefied Gas Dynamics. Cambridge University Press, Cambridge (2000)
  17. Coron, F, Golse, F, Sulem, C (1988) A classification of well-posed kinetic layer problems. Comm. Pure Appl. Math. 41: pp. 409-435 CrossRef
  18. Fainsilber, L, Kurlberg, P, Wennberg, B (2006) Lattice points on circles and discrete velocity models for the Boltzmann equation. Siam J. Math. Anal. 37: pp. 1903-1922 CrossRef
  19. Golse, F (2008) Analysis of the boundary layer equation in the kinetic theory of gases. Bull. Inst. Math. Acad. Sin. 3: pp. 211-242
  20. Golse, F, Poupaud, F (1989) Stationary solutions of the linearized Boltzmann equation in a halfspace. Math. Methods Appl. Sci. 11: pp. 483-502 CrossRef
  21. Kawashima, S, Nishibata, S (1999) Existence of a stationary wave for the discrete Boltzmann equation in the half space. Comm. Math. Phys. 207: pp. 385-409 CrossRef
  22. Kawashima, S, Nishibata, S (2000) Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries. Comm. Math. Phys. 211: pp. 183-206 CrossRef
  23. Kirkpatrick, TR, Dorfman, JR (1985) Transport in a dilute but condensed nonideal Bose gas: kinetic equations. J. Low Temp. Phys. 58: pp. 301-331 CrossRef
  24. Nordheim, LW (1928) On the kinetic methods in the new statistics and its applications in the electron theory of conductivity. Proc. Roy. Soc. London Ser. A 119: pp. 689-698 CrossRef
  25. Palczewski, A, Schneider, J, Bobylev, AV (1997) A consistency result for a discrete-velocity model of the Boltzmann equation. SIAM J. Numer. Anal. 34: pp. 1865-1883 CrossRef
  26. Sone, Y.: Kinetic Theory and Fluid Dynamics. Birkh盲user, Basel (2002)
  27. Sone, Y.: Molecular Gas Dynamics. Birkh盲user, Basel (2007)
  28. Uehling, EA, Uhlenbeck, GE (1933) Transport phenomena in Einstein-Bose and Fermi-Dirac gases. Phys. Rev. 43: pp. 552-561 CrossRef
  29. Ukai, S.: On the half-space problem for the discrete velocity model of the Boltzmann equation. In: S. Kawashima, T. Yanagisawa (eds.) Advances in Nonlinear Partial Differential Equations and Stochastics, pp. 160鈥?74. World Scientific, Singapore (1998)
  30. Ukai, S, Yang, T, Yu, SH (2003) Nonlinear boundary layers of the Boltzmann equation: I. Existence. Comm. Math. Phys. 236: pp. 373-393 CrossRef
  31. Wu, Y-S (1994) Statistical distribution for generalized ideal gas of fractional-statistics particles. Phys. Rev. Lett. 73: pp. 922-925 CrossRef
  32. Yang, X (2011) The solutions for the boundary layer problem of Boltzmann equation in a halfspace. J. Stat. Phys. 143: pp. 168-196 CrossRef
  33. Zaremba, E, Nikuni, T, Griffin, A (1999) Dynamics of trapped Bose gases at finite temperatures. J. Low Temp. Phys. 116: pp. 277-345 CrossRef
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Statistical Physics
  Mathematical and Computational Physics
  Physical Chemistry
  Quantum Physics
  
• 出版者：Springer Netherlands
• ISSN：1572-9613

文摘

We study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for a general discrete model of a quantum kinetic equation for excitations in a Bose gas. In the discrete case the plane stationary quantum kinetic equation reduces to a system of ordinary differential equations. These systems are studied close to equilibrium and are proved to have the same structure as corresponding systems for the discrete Boltzmann equation. Then a classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete kinetic equation can be made. The number of additional conditions that need to be imposed for well-posedness is given by some characteristic numbers. These characteristic numbers are calculated for discrete models axially symmetric with respect to the x-axis. When the characteristic numbers change is found in the discrete as well as the continuous case. As an illustration explicit solutions are found for a small-sized model.