Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-25T13:21:35.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Aspects of the kinetic equations for a special one-dimensional system

Published online by Cambridge University Press:  17 February 2009

J. W. Evans
J. W. Evans
Affiliation:
Department of Mathematical Physics, The University of Adelaide, Adelaide, 5001, South Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Some initial value problems are considered which arise in the treatment of a one-dimensional gas of point particles interacting with a “hard-core” potential.

Two basic types of initial conditions are considered. For the first, one particle is specified to be at the origin with a given velocity. The positions in phase space of the remaining background of particles are represented by continuous distribution functions. The second problem is a periodic analogue of the first.

Exact equations for the delta-function part of the single particle distribution functions are derived for the non-periodic case and approximate equations for the periodic case. These take the form of differential operator equations. The spectral and asymptotic properties of the operators associated with the two cases are examined and compared. The behaviour of the solutions is also considered.

Type
Research Article
Information
The ANZIAM Journal , Volume 21 , Issue 2 , August 1979 , pp. 145 - 175
Copyright
Copyright © Australian Mathematical Society 1979

References

[1]Anstis, G. R., Green, H. S. and Hoffman, D. K., “Kinetic theory of a one-dimensional model”, J. Math. Phys. 14 (1973), 1473.CrossRefGoogle Scholar
[2]Bogoliubov, N. N., in Studies in statistical mechanics, ed. by de Boer, J. and Uhlenbeck, G. E. (Interscience, New York, 1962).Google Scholar
[3]Carrier, G. F., Krook, M. and Pearson, C. E., Functions of a complex variable (McGraw-Hill, New York, 1966).Google Scholar
[4]Doplicher, S., Kadison, R. V., Kastler, D. and Robinson, D. W., “Asymptotically abelian systems”, Comm. Math. Phys. 6 (1967), 101.CrossRefGoogle Scholar
[5]Dunford, N. and Schwartz, J. T., Linear operators. Part 1 (Interscience, New York).Google Scholar
[6]Dunford, N. and Schwartz, J. T., Linear operators. Part 2 (Interscience, New York.)Google Scholar
[7]Gervois, A. and Pomeau, Y., “On the absence of diffusion in a semi-infinite one-dimensional system”, J. Stat. Phys. 14 (1976), 483.CrossRefGoogle Scholar
[8]Halmos, P. R., A Hubert space problem book (Springer-Verlag, Graduate Texts in Mathematics 1974).CrossRefGoogle Scholar
[9]Hille, E., Functional analysis and semigroups (published by Amer. Math. Soc. 1948).Google Scholar
Wiener, N., “Tauberian theorems”, Annals Math. 2nd Series 33 (1932) 1.,CrossRefGoogle Scholar
[10]Jepsen, D. W., “Dynamics of a simple many body system of hard rods”, J. Math. Phys. 6 (1965), 405.CrossRefGoogle Scholar
[11]Lebowitz, J. L. and Percus, J. K., “Kinetic equations and density expansions: exactly solvable one-dimensional system”, Phys. Rev. 155 (1967), 122.CrossRefGoogle Scholar
[12]Lighthill, M. J., Introduction to Fourier analysis and generalized functions (Cambridge University Press, 1958).CrossRefGoogle Scholar
[13]Taylor, A. E., Introduction to functional analysis (Wiley, New York, 1958).Google Scholar
[14]Thompson, C. J., Mathematical statistical mechanics (Macmillan, London, 1972).Google Scholar
[15]Ziock, K., Basic quantum mechanics (Wiley, New York, 1969).Google Scholar