Homogeneous Boltzmann equation in quantum relativistic kinetic theory
DOI:
https://doi.org/10.58997/ejde.mon.04Abstract
We consider some mathematical questions about Boltzmann equations for quantum particles, relativistic or non relativistic. Relevant particular cases such as Bose, Bose-Fermi, and photon-electron gases are studied. We also consider some simplifications such as the isotropy of the distribution functions and the asymptotic limits (systems where one of the species is at equilibrium). This gives rise to interesting mathematical questions from a physical point of view. New results are presented about the existence and long time behaviour of the solutions to some of these problems.
For more information see https://ejde.math.txstate.edu/Monographs/04/abstr.html
References
F. Abrahamsson, Strong convergence to equilibrium without entropy conditions for the spatially homogeneous Boltzmann equation, Comm. PDE, 24, (1999), 1501-1535.
H. Andreasson, Regularity of the gain term and strong L1 convergence to equilibrium for the relativistic Boltzmann equation, SIAM J. Math. Anal. 27, (1996), 1386–1405.
L. Arkeryd, On the Boltzmann equation, I and II, Arch. Rat. Mech. Anal. 45, (1972), 1-34.
L. Arkeryd, Asymptotic behaviour of the Boltzmann equation with infinite range forces, Comm. Math. Phys. 86, (1982), 475-484.
L. Boltzmann, Weitere Studien ¨uber das W¨armegleichgewicht unter Gasmolekulen, Sitzungsberichte der Akademie der Wissenschaften Wien, 66, (1872), 275-370.
S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Z. Phys. 26 (1924), 178–181.
R.E. Caflisch, C.D. Levermore, Equilibrium for radiation in a homogeneous plasma, Phys. Fluids 29, (1986), 748-752.
C. Cercignani, The Boltzmann equation and its applications; Applied Math. Sciences 67. Springer Verlag.
C. Cercignani and G. M. Kremer, The relativistic Boltzmann equation: theory and applications; Birkh¨auser Verlag 2002.
G. Chapline, G. Cooper and S. Slutz, Condensation of photons in a hot plasma; Phys. Rev. A 9, (1974), 1273–1277.
S. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases; Cambridge University Press, 1970. Third edition.
N. A. Chernikov, Equilibrium distributions of the relativistic gas, Acta Phys. Pol. 26, (1964), 1069–1092.
G. Cooper, Compton Fokker-Planck equation for hot plasmas, Phys. Rev. D 3, (1974), 2412-2416.
J. Dolbeault, Kinetic models and quantum effects, a modified Boltzmann equation for Fermi-Dirac particles, Arch. Rat. Mech. Anal. 127, 101-131.
H. Dreicer, Kinetic theory of an electron-photon Gas, Phys. Fluids 7, (1964), 735-753.
M. Dudynsky and M. Ekiel Jezewska, Global existence proof for relativistic Boltzmann equations, J. Stat. Physics 66, (1992), 991-1001.
J. Ehlers, Akad. Wiss. Lit. Mainz,, Abh. Math.-Naturw. Kl. 11, (1961), 1.
A. Einstein, Quantentheorie des einatomingen idealen gases, Stiz. resussische
Akademie der Wissenschaften Phys-math. Klasse, Sitzungsberichte 23, (1925), 3-14.
A. Einstein, Zur quantentheorie des idealen gases, Stiz. Presussische
Akademie der Wissenschaften, Phys-math. Klasse, Sitzungsberichte 23, (1925), 18-25.
T. Elmroth, Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range, Arch. Rat. Mech. Anal. 82, (1983), 1-12.
M. Escobedo, M. A. Herrero, J. J. L. Velazquez, A nonlinear Fokker-Planck equation modelling the approach to thermal equilibrium in a homogeneous plasma, Trans. Amer. Math. Soc. 350, (1998), 3837-3901.
M. Escobedo, S. Mischler, Equation de Boltzmann quantique homogene: existence et comportement asymptotique, C. R. Acad. Sci. Paris 329 SerieI (1999) 593-598.
M. Escobedo, S. Mischler, On a quantum Boltzmann equation for a gas of photons, J. Math. Pures Appl. 80, 5 (2001), pp. 471-515.
M. Escobedo, S. Mischler and M. A. Valle, Maximization entropy problem for quantum relativistic particles. Submited.
T. Goudon, On Boltzmann equation and Fokker Planck asymptotics: influence of grazing collisions, J. Stat. Physics, 89, (1997), 751–776.
R. T. Glassey, The Cauchy Problem in Kinetic Theory; SIAM, Philadelphia 1996.
R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian Publ. Res. Inst. Math. Sciences, Kyoto University, 29, (1993), 301–347.
R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments, Trans. Th. Stat. Phys. 24, (1995), 657-678.
S. R. Groot, W. A. van Leeuwen, Ch. G. van Weert, Relativistic kinetic
theory, North Holland Publishing Company, 1980.
E. Hanamura and H. Hang, Phys. Rep. C33, 210 (1977).
F. J¨uttner, Das Maxwellsche Gesetz der geschwindigkeitsverteilung in der relativtheorie, Ann. Phys. (Leipzig) 34, (1911), 856–882.
C. Josserand and Y. Pomeau, Nonlinear aspects of the theory of Bose-Einstein condensates.
A.S. Kompaneets, The establishment of thermal equilibrium between quanta and electrons, Soviet Physics JETP, (1957).
L. D. Landau, E. M. Lifshitz Course of theoretical physics, Pergamon Press, 1977. Third Edition.
B. G. Levich Theoretical physics: an advanced text, North Holland 1970.
E. Levich and V. Yakhot, Time evolution of a Bose system passing through the critical point, Phys. Rev. B 15, (1977), 243–251 .
E. Levich and V. Yakhot, Time development of coherent and superfluid
properties in a course of a −transition, J. Phys. A, Math. Gen., 11, (1978), 2237–2254.
A. Lichnerowicz and R.Marrot, Propri´et´es statistiques des ensembles de particules en relativit´e restreinte, C.R.A.S. 210, (1940), 759–761.
Lions, P.L. (1994) Compactness in Boltzmann’s equation via Fourier integral operators and applications, parts I, II, III, J. Math. Kyoto Univ. 34, 391-427, 427-461, 539-584
X.G. Lu, A modified Boltzmann equation for Bose-Einstein particles: isotropic solutions and long time behaviour, J. Stat. Phys., 98, (2000), 1335–1394.
S. Mischler, B. Wennberg, On the spatially homogeneous Boltzmann equation, Ann. Inst. Henri Poincar´e 16 4, (1999), 467-501.
G. L. Naber The geometry of Minkowski spacetime, Springer Verlag, 1992.
L.W. Nordheim, On the kinetic method in the new statistics and its applications
in the electron theory of conductivity, Proc. Roy. Soc. London, A 119 (1928) 689-698.
P.J.E. Peebles, Principles of physical cosmology, Princeton University Press, 1993.
A. Ja. Povzner, On the Boltzmann equation in the kinetic theory of gases.(Russian), Mat. Sb. (N.S.) 58, 1962, 65–86.
M. E. Peskin, D.V. Schroeder, An introduction to quantum field theory, Addison-Wesley, 1995.
M. Reed, B. Simon, Methods of modern mathematical physics III: Scattering Theory; Academic Press, 1979.
D.V. Semikov, I.I. Tkachev, Kinetics of Bose condensation, Phys. Rev. Lett. 74 (1995) 3093-3097.
D.V. Semikov, I.I. Tkachev, Condensation of Bosons in the kinetic regime, Phys. Rev. D 55, 2, (1997) 489-502.
D. Snoke, J.P. Wolfe and A. Mysyrowicz, Phys. Rev. Lett. 59, 827 (1987).
J. M. Stewart, Non equilibrium relativistic theory. Lecture Notes in Physics No.10, Springer Verlag 1971.
E. A. Uehling, G. E. Uhlenbeck, Transport phenomena in Einstein- Bose and Fermi-Dirac gases. I Physical Review 43 (1933) 552-561.
G. E. Tayler and J. W. Weinberg, Internal state of a gravitating gas, Phys. Rev. 122, (1961), 1342–1365
G. Toscani, C. Villani, Sharp entropy dissipation bounds and explicit rate
of trend to equilibrium for spatially homogeneous Boltzmann equation,
Comm. Math. Phys. 203, (1999), 667- 706.
C. Villani, A review of mathematical topics on collisional kinetic theory. To appear in Handbook of fluid dynamics, S. Friedlander and D. Serre (Eds.) 2001.
B. Wennberg, Stability and exponential convergence in Lp for the spatially
homogeneous Boltzmann equation, Nonl. Anal. 20, (1993), 935–964.
Downloads
Published
License
Copyright (c) 2023 Miguel Escobedo, Stephane Mischler, Manuel A. Valle
This work is licensed under a Creative Commons Attribution 4.0 International License.
