Abstract
A number of new relations between the Kaplan–Yorke dimension, phase space contraction, transport coefficients and the maximal Lyapunov exponents are given for dissipative thermostatted systems, subject to a small but non-zero external field in a nonequilibrium stationary state. A condition for the extensivity of phase space dimension reduction is given. A new expression for the linear transport coefficients in terms of the Kaplan–Yorke dimension is derived. Alternatively, the Kaplan–Yorke dimension for a dissipative macroscopic system can be expressed in terms of the linear transport coefficients of the system. The agreement with computer simulations for an atomic fluid at small shear rates is very good.
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REFERENCES
P. Frederickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, J. Differ. Equations 49:183 (1983).
L.-S. Young, Ergod. Th. and Dynam. Sys. 2:109 (1982).
F. Ledrappier and L.-S. Young, Comm. Math. Phys 117:529 (1988).
F. Ledrappier and L.-S. Young, Annals of Mathematics 122:540 (1985).
D. Ruelle, private communication.
Wm. G. Hoover, Computational Statistical Mechanics (Elsevier, Amsterdam, 1991).
J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57:617 (1985).
H. A. Posch and W. G. Hoover, Phys. Rev. A 38:479 (1988). We note that the entropy production rate σ in this paper, as well as in ref. 20, derived from the rate of change of the Gibbs entropy which leads to the wrong sign for σ. However, this does not affect their results.
D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Liquids (Academic Press, London, 1990).
E. G. D. Cohen and L. Rondoni, Chaos 8:357 (1998).
S. R. de Groot and P. Mazur, Nonequilibrium Thermodynamics (Dover Publications, New York, 1984).
W. N. Vance, Phys. Rev. Lett. 69:1356 (1992).
D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. A 42:5990 (1990).
C. P. Dettmann and G. P. Morriss, Phys. Rev. E 53:R5541 (1996); 55:3963 (1997); F. Bonetto, E. G. D. Cohen, and C. Pugh, J. Stat. Phys. 92:587 (1998).
D. J. Searles, D. J. Evans, and D. J. Isbister, Chaos 8:309 (1998).
Ch. Dellago, L. Glatz, and H. A. Posch, Phys. Rev. E 52:4817 (1995).
D. Ruelle, J. Stat. Phys. 95:393 (1999).
Ch. Dellago and H. A. Posch, Physica A 68:240 (1997).
See Ya. G. Sinai, J. Bifur. Chaos Appl. Sci. Engrg 6:1137 (1996).
W. G. Hoover and H. A. Posch, Phys. Rev. E 49:1913 (1994).
G. Gallavotti and E. G. D. Cohen, J. Stat. Phys. 80:931 (1995).
Ch. Dellago, W. G. Hoover, and H. A. Posch, Phys. Rev. E 57:4969 (1998).
J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem. Phys. 54:5237 (1971).
J. Petravic and D. J. Evans, Mol. Phys. 95:219 (1998).
See ref. 7, p. 641, Fig. 18.
K. P. Travis, D. J. Searles, and D. J. Evans, Mol. Phys. 95:195 (1998).
G. Ayton and D. J. Evans, J. Stat. Phys. 97:811 (1999).
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Evans, D.J., Cohen, E.G.D., Searles, D.J. et al. Note on the Kaplan–Yorke Dimension and Linear Transport Coefficients. Journal of Statistical Physics 101, 17–34 (2000). https://doi.org/10.1023/A:1026449702528
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DOI: https://doi.org/10.1023/A:1026449702528