Abstract
A master equation is derived microscopically to describe the fluctuating motion of the particle density in μ. space. This equation accounts for the drift motion of particles and is valid for any inhomogeneous gas. The Boltzmann equation is obtained from the first moment of this equation by neglecting the second cumulant (the pair correlation function). The successive moments form coarse-grained BBGKY-like hierarchy equations, in which small spatial regions with rij < the force range are smeared out. These hierarchy equations are convenient for investigating the nonequilibrium long-range pair correlation function, which arises mainly from sequences of isolated binary collisions and gives rise to the much-discussed long-time tail and the logarithmic term in the density expansion of transport coefficients. It is shown to have a spatial long tail, like the Coulombic potential, in a steady laminar flow. The stochastic nature of the nonlinear Boltzmann-Langevin equation is also investigated; the random source term is found to be expressed as a linear superposition of Poisson random variables and to become Gaussian in special cases.
Similar content being viewed by others
References
A. J. F. Siegert,Phys. Rev. 76:1708 (1949).
G. Ludwig,Physica 28:841 (1962).
G. N. Nicolis,J. Stat. Phys. 6:195 (1972).
N. G. van Kampen,Phys. Lett. A 50:237 (1974).
J. Logan and M. Kac,Phys. Rev. A 13:458 (1976).
N. Hashitsume,Prog. Theor. Phys. 15:369 (1956).
B. B. Kadomtsev,Sov. Phys-JETP 5:771 (1957).
M. Bixon and R. Zwanzig,Phys. Rev. 187:267 (1969).
R. F. Fox and G. E. Uhlenbeck,Phys. Fluids 13:2881 (1970).
F. L. Hinton,Phys. Fluids 13:857 (1970).
Sh. M. Kogan and A. Ya. Shul'man,Sov. Phys.-JETP 29:467 (1969).
S. V. Gantsevich, V. L. Gurevich, and R. Katilius,Sov. Phys.-JETP 32:291 (1971).
H. Mori,Prog. Theor. Phys. 49:1516 (1973).
L. D. Landau and F. M. Lifshitz,Fluid Mechanics (Pergamon Press, London, 1959), Chap. XXII.
J. G. Kirkwood,J. Chem. Phys. 14:180 (1946);15:72 (1947).
N. N. Bogoliubov,Studies in Statistical Mechanics, Vol. 1, J. de Boer and G. E. Uhlenbeck, eds. (North-Holland, Amsterdam, 1972).
R. Kubo,Lecture Notes in Physics 31, Transport Phenomena (Springer-Verlag, Berlin, 1974);Report on Progress in Physics, Vol. 29, Part I (1966), p. 225.
H. Mori and H. Fujisaka,Prog. Theor. Phys. 49:763 (1973).
R. Kubo, K. Matsuo, and K. Kitahara,J. Stat. Phys. 9:51 (1973).
M. Tokuyama and H. Mori,Prog. Theor. Phys. 56:1073 (1976).
K. Kawasaki and I. Oppenheim,Phys. Rev. 139A:1763 (1965).
Y. Pomeau and P. Réesibois,Phys. Rep. 19C:63 (1975).
A. Onuki,J. Phys. Soc. Japan 41:9 (1976).
P. Résibois,J. Stat. Phys. 2:21 (1970).
P. Résibois and J. L. Lebowitz,J. Stat. Phys. 12:483 (1975).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Onuki, A. On fluctuations in μ-space. J Stat Phys 18, 475–499 (1978). https://doi.org/10.1007/BF01014519
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01014519