Abstract
Recent work by several authors has shown that non-equilibrium processes in simple, classical, chaotic systems can be described in terms of fractal structures that develop in the system’s phase space. These structures form exponentially rapidly in phase space as an initial non-equilibrium distribution evolves in time. Since the motion of a region in phase space, for a Hamiltonian system, is measure preserving, the phase space distribution is advected as a passive scalar in the motion of the phase points. Due to the chaotic nature of the motion, the stretching and folding motion in phase space produces very complicated fractal distributions which may vary greatly over regions of small measure. This mechanism is responsible for the formation of the fractals under discussion. Here we illustrate this phenomenon for a few simple models with deterministic diffusion. The origin of the fractals is explained and connected to the microscopic properties of the hydrodynamic modes of the system. These hydrodynamic modes are, in turn, closely related, on averaging, to the van Hove intermediate scattering function. Further we describe the connections of the properties of the fractals with important quantities for transport - transport coefficients and irreversible entropy production. One interesting result is a connection between the coefficient of diffusion for the moving particle in a chaotic Lorentz gas and the Hausdorff dimension of the hydrodynamic modes of diffusion at small wave numbers.
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References
S. Tasaki and P. Gaspard, J. Stat. Phys. 81, 935 (1995).
P. Gaspard, Phys. Rev. E 53, 4379 (1996).
P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge, 1998).
T. Gilbert, J. R. Dorfman and P. Gaspard, Nonlinearity, 14, 339, (2001).
P. Gaspard, I. Claus, T. Gilbert and J. R. Dorfman, Phys. Rev. Lett., 86, 1506, (2001).
T. Gilbert, J. R. Dorfman and P. Gaspard, Phys. Rev. Lett. 85, 1606 (2000).
J. R. Dorfman, P. Gaspard and T. Gilbert, “Entropy production of diffusion in spatially periodic deterministic systems”, arXiv:nlin.CD/0203046; (to appear in Phys. Rev. E).
J. R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics, (Cambridge University Press, Cambridge UK, 1999).
P. Gaspard, J. Stat. Phys., 88, 1215, (1997).
L. Van Hove, Phys. Rev. 95, 249 (1954).
Ya. G. Sinai, Russian Math. Surveys 25, 137 (1970).
P. Gaspard and F. Baras, Phys. Rev. E51, 5332 (1995).
A. Knauf, Commun. Math. Phys. 110, 89 (1987); Ann. Phys. (N. Y.) 191, 205 (1989).
I. Claus, Ph. D. Thesis: Microscopic Chaos, Fractals and Reaction-Diffusion Processes, Universite Libre de Bruxelles, (2002).
S. Viscary, Mémoire: Viscosité et chaos dans un modèle à deux disques durs, Universite Libre de Bruxelles, (2000).
S. Tasaki, T. Gilbert and J. R. Dorfman, CHAOS, 8, 424, (1998).
P. M Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics, (Cambridge University Press, Cambridge, 1995).
S. R. de Groot and P. Mazur, Nonequilibrium Thermodynamics, (Dover Publishing Co., New York, 1984).
References can be found in the paper of L. Rondoni in this volume.
D. McQuarrie, Statistical Mechanics, (Harper and Row, New York, 1976).
R. Kubo, M. Toda and N. Hashitume, Statistical Physics II, 2nd. Edition, (Springer-Verlag, Berlin, 1991).
E. H. Hauge in Transport Phenomena, G. Kirczenow and J. Marro, eds., (Springer-Verlag, Berlin, 1974)
C. P. Dettmann and E. G. D. Cohen, J. Stat. Phys., 101, 775, (2000); 103, 589, (2001).
Ya. G. Sinai, Russian Math. Surveys 27, 21 (1972).
R. Bowen and D. Ruelle, Invent. Math. 29, 181 (1975).
D. Ruelle, Thermodynamic Formalism (Addison-Wesley, Reading MA, 1978).
R. Bowen, Publ. Math. IHES 50, 11 (1976).
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Dorfman, J.R. (2002). Fractal Structures in the Phase Space of Simple Chaotic Systems with Transport. In: Garbaczewski, P., Olkiewicz, R. (eds) Dynamics of Dissipation. Lecture Notes in Physics, vol 597. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46122-1_8
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DOI: https://doi.org/10.1007/3-540-46122-1_8
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