Abstract
A review is given of research activities since 1976 on the nonlinear Boltzmann equation and related equations of Boltzmann type, in which several rediscoveries have been made and several conjectures have been disproved. Subjects are (i) the BKW solution of the Boltzmann equation for Maxwell molecules, first discovered by Krupp in 1967, and the Krook-Wu conjecture concerning the universal significance of the BKW solution for the large(v, t) behavior of the velocity distribution functionf (v, t); (ii) moment equations and polynomial expansions off (v, t); (iii) model Boltzmann equation for a spatially uniform system of very hard particles, that can be solved in closed form for general initial conditions; (iv) for Maxwell and non-Maxwell-type molecules there exist solutions of the nonlinear Boltzmann equation with algebraic decrease at υ→∞; connections with nonuniqueness and violation of conservation laws; (v) conjectured super-H-theorem and the BKW solution; (vi) exactly soluble one-dimensional Boltzmann equation with spatial dependence.
Similar content being viewed by others
References
R. S. Krupp, A Nonequilibrium Solution of the Fourier Transformed Boltzmann Equation, M.Sc. thesis, MIT (1967).
A. V. Bobylev,Sov. Phys. Dokl. 20:820, 822 (1976).
M. Krook and T. T. Wu,Phys. Rev. Lett. 36:1107 (1976);Phys. Fluids 20:1589 (1979).
M. R. Hoare,Advances in Physical Chemistry (J. Wiley, New York, 1983).
J. L. Lebowitz and E. W. Montroll, eds.Nonequilibrium Phenomena I, The Boltzmann Equation (North-Holland Publ. Co., Amsterdam, 1983).
M. H. Ernst, Ref. 5, Chap. 3;Phys. Rep. 78:1 (1981).
W. Greenberg, J. Polewczak, and P. F. Zweifel, Ref. 5, Chap. 2.
W. Fiszdon and A. Palczewski, Max-Planck Institut für Strömungsforschung, Bericht 102/1983 (Göttingen).
C. Cercignani,Theory and Application of the Boltzmann Equation (Scottish Academic Press, Edinburgh, 1975).
S. Nishimura,J. Appl. Prob. 11:266, 703 (1974).
R. L. Curl,A.I.Ch.E.J. 9:175 (1963).
J. A. Tjon and T. T. Wu,Phys. Rev. A 19:883 (1978).
M. H. Ernst,Phys. Lett. 69A:390 (1979).
E. Futcher, M. R. Hoare, E. M. Hendriks, and M. H. Ernst,Physica 101A:185, 375 (1980).
M. Kac, III Berkeley Symposium on Mathematics, Statistics and Probability (Univ. of California Press, 1955), p. 171.
S. Rouse and G. Simons,J. Phys. A, Math. Gen. A11:423 (1978).
E. M. Hendriks and M. H. Ernst,Phys. Lett. 70A:183 (1979);Physica 120A:545 (1983).
J. A. Tjon,Phys. Lett. 70A:390 (1979).
M. Alexanian,Phys. Lett. 74A:1 (1979).
E. H. Hauge,Phys. Lett. 74A:183 (1979); E. H. Hauge and E. Praestgaard,J. Stat. Phys. 24:21 (1981).
H. Cornille and A. Gervois,Inverse Problems and Nonlinear Evolution, Proceedings Workshop Montpellier, 1979 (P. C. Sabatier, Paris, 1980).
A. V. Bobylev,Sov. Phys. Dokl. 26:1127 (1981).
J. J. Brey, J. Gomez Ordonez, and A. Santos,Phys. Rev. A 26:2817 (1982).
H. Grad,Commun. Pure Appl. Math. 4:331 (1949).
C. Truesdell,J. Rat. Mech. Anal. 5:55 (1956).
M. Barnsley and H. Cornille,J. Math. Phys. 25:1176 (1980);Proc. R. Soc. (London) A374:374 (1981).
R. M. Ziff,Phys. Rev. A 23:916 (1981);24:509 (1981).
V. V. Vedenyapin,Sov. Phys. Dokl 26:26 (1981).
E. M. Hendriks and T. M. Nieuwenhuizen,J. Stat. Phys. 29:591 (1982).
A. V. Bobylev,Sov. Phys. Dokl 25:257 (1980).
H. Cornille and A. Gervois,Phys. Lett. 79A:291 (1980);J. Stat. Phys. 26:181 (1981).
H. Cornille and A. Gervois,J. Phys. Lett. (Paris) 41:L581 (1980);Physica 113A:559 (1982).
J. Piasecki and Y. Pomeau,J. Stat. Phys. 28:235 (1982).
M. H. Ernst, K. Hellesøe, and E. H. Hauge,J. Stat. Phys. 27:677 (1982).
H. P. McKean,Arch. Ration. Mech. Anal. 21:343 (1966).
S. Harris,J. Math. Phys. 8:2407 (1967).
R. M. Ziff, S. D. Merajver, and G. Stell,Phys. Rev. Lett. 47:1493 (1981).
K. Olaussen,Phys. Rev. A 25:3393 (1982).
A. J. M. Garret,J. Phys. A., Math. Gen. 15:L351 (1982).
E. H. Lieb,Phys. Rev. Lett. 48:1057 (1982).
Th. W. Ruijgrok and T. T. Wu,Phys. Lett. 85A:420 (1981);Physica 113A:401 (1982).
Author information
Authors and Affiliations
Additional information
Reference due to C. Cercignani.
Rights and permissions
About this article
Cite this article
Ernst, M.H. Exact solutions of the nonlinear Boltzmann equation. J Stat Phys 34, 1001–1017 (1984). https://doi.org/10.1007/BF01009454
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01009454