Abstract
A linearized model of the Boltzmann equation for a relativistic gas is shown to be reducible, in the ultrarelativistic limit and for (1 + 1)-dimensional problems, to a system of three uncoupled transport equations, one of which is well known. A general method for solving these equations is recalled, with a few new details, and applied to the solution of two boundary value problems. The first of these describes the propagation of an impulsive change in a half space and is shown to give an explicit example of the recently proved result that no signal can propagate with speed larger than the speed of light, according to the relativistic Boltzmann equation. The second problem deals with steady oscillations in a half space and illustrates the meaning of certain recent results concerning the dispersion relation for linear waves in relativistic gas.
Similar content being viewed by others
References
C. CercignaniPhys. Rev. Lett. 50:1122 (1983).
C. Cercignani Atti V Congresso di Relativité Générale e Fisica della Gravitazione Catania September 1982.
C. Cercignani and A. Majorana,Meccanica 19:175 (1985).
C. Cercignani and A. Majorana,Phys. Fluids 28:1673 (1985).
P. L. Bhatnagar, E. P. Gross and M. Krook,Phys. Rev. 94:511 (1954).
C. Cercignani,Mathematical Methods in Kinetic Theory (Plenum Press, New York, and McMillan, London, 1969).
C. Cercignani,Theory and Application of the Boltzmann Equation (Scottish Academic Press, Edinburgh and Elsevier, New York, 1975).
J. L. Anderson and H. R. Witting,Physica 74:466 (1974).
L. Sirovich and J. K. Thurber,Rarefied Gas Dynamics, Vol. 1, J. A. Laurman, ed. (Academic Press, New York, 1963), p. 159.
L. Sirovich and J. K. Thurber,J. Acoust. Soc. Am. 37:32 (1965).
C. Cercignani, Elementary solutions of linearized kinetic models and boundary value problems in kinetic theory, Brown University Report (1965).
K. Aoki and C. Cercignani,ZAMP 35:345 (1984).
S. Chandrasekhar,Radiative Transfer (Oxford University Press, London, 1950).
K. M. Case and P. F. Zweifel,Linear Transport Theory (Addison-Wesley, Reading, Massachusetts, 1967).
M. Krook,J. Fluid Mech. 6:523 (1959).
C. Cercignani,Ann. Phys. (N.Y.) 40:469 (1966).
R. L. Bowden and C. D. Williams,J. Math. Phys. 5:1527 (1964).
K. M. Case,Ann. Phys. (N.Y.) 9:1 (1960).
C. Cercignani and F. Sernagiotto,Ann. Phys. (N.Y.) 30:15 (1974).
C. Cercignani,Ann. Phys. (N.Y.) 20:219 (1962).
N. I. Muskhelishvili,Singular Integral Equations (Noordhoff, N.V. Groningen, 1953).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cercignani, C. Solution of a linearized kinetic model for an ultrarelativistic gas. J Stat Phys 42, 601–620 (1986). https://doi.org/10.1007/BF01127731
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01127731