Abstract
We apply the periodic orbit expansion to the calculation of transport, thermodynamic, and chaotic properties of the finite-horizon triangular Lorentz gas. We show numerically that the inverse of the normalized Lyapunov number is a good estimate of the probability of an individual periodic orbit. We investigate the convergence of the periodic orbit expansion and compare it with the convergence of the cycle expansions obtained from the Ruelle dynamical σ-function. For this system with severe pruning we find that applying standard convergence acceleration schemes to the periodic orbit expansion is superior to the dynamical σ-function approach. The averages obtained from the periodic orbit expansion are within 8% of the values obtained from direct numerical time and ensemble averaging. None of the periodic orbit expansions used here is computationally competitive with the standard simulation approaches for calculating averages. However, we believe that these expansion methods are of fundamental importance, because they give a direct route to the phase space distribution function.
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References
D. J. Evans, E. G. D. Cohen, and G. P. Morris,Phys. Rev. A 42:5990 (1990).
P. Gaspard and S. A. Rice,J. Chem. Phys. 90:2225, 2242, 2255 (1989).
J. Machta and R. Zwanzig,Phys. Rev. Lett. 50:1959 (1983).
A. J. C. Ladd and W. G. Hoover,J. Stat. Phys. 38:973 (1985); B. Moran and W. G. Hoover,J. Stat. Phys. 48:709 (1987).
P. Gaspard and F. Baras, inMicroscopic Simulations of Complex: Hydrodynamic Phenomena, M. Mareschal and B. L. Holian, eds. (Plenum Press, New York, 1992).
N. I. Chernov, G. L. Eyink, J. L. Lebowitz, and Ya. G. Sinai,Phys. Rev. Lett. 70: 2209 (1993).
W. N. Vance,Phys. Rev. Lett. 69:1356 (1992).
P. Cvitanovic, P. Gaspard and T. Schreiber,Chaos 2:85 (1992).
D. J. Evans and G. P. Morriss,Statistical Mechanics of Nonequilibrium-Liquids (Academic Press, London, 1990).
A. Baranyai, D. J. Evans, and E. G. D. Cohen,J. Stat. Phys. 70:1085 (1993).
P. Cvitanovic,Phys. Rev. Lett. 61:2729 (1988).
R. Artuso, E. Aurell, and P. Cvitanovic,Nonlinearity 3:325, 361 (1990).
W. Parry,Commun. Math. Phys. 106:267 (1986).
C. Grebogi, E. Ott, and J. A. Yorke,Phys. Rev. A 37(5):1711 (1988).
D. Auerbach, et al.,Phys. Rev. Lett. 58:2387 (1987).
J. H. Hannay and A. M. Ozorio de Almeida,J. Phys. A,17:3429 (1984).
P. Gaspard and D. Alonso Ramirez,Phys. Rev. A 45:8383 (1992).
D. Ruelle,Thermodynamic Formalism, (Addison-Wesley, Reading, Massachusetts, 1978).
P. Cvitanovic, Private communication.
P. Cvitanovic and B. Eckhardt,Nonlinearity, in press.
C. M. Bender and S. A. Orszag,Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
W. H. Press et al.,Numerical Recipes in C (Cambridge University Press, Cambridge, 1988; G. Arfken,Mathematical Methods for Physicists (Academic Press, London, 1985).
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Morris, G.P., Rondoni, L. Periodic orbit expansions for the Lorentz gas. J Stat Phys 75, 553–584 (1994). https://doi.org/10.1007/BF02186872
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DOI: https://doi.org/10.1007/BF02186872