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Volumn 69, Issue 1 2, 2004, Pages 162051-162058

Coarse-grained entropy and information dimension of dynamical systems: The driven Lorentz gas

Author keywords

[No Author keywords available]

Indexed keywords

ASYMPTOTIC STABILITY; CHAOS THEORY; DECOMPOSITION; EIGENVALUES AND EIGENFUNCTIONS; GIBBS FREE ENERGY; INFORMATION ANALYSIS; INTEGRATION; LYAPUNOV METHODS; MAPS; NUMERICAL METHODS;

EID: 18344410549     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article
Times cited : (4)

References (42)
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    • We refer here to the entropy of the full system
    • We refer here to the entropy of the full system.
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    • Vollmer, J.1    Tél, T.2    Breymann, W.3
  • 18
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    • J. Vollmer, T. Tél, and W. Breymann, Phys. Rev. Lett. 79, 2759 (1997); Phys. Rev. E 58, 1672 (1998); W. Breymann, T. Tél, and J. Vollmer, Chaos 8, 396 (1998).
    • (1998) Phys. Rev. E , vol.58 , pp. 1672
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    • J. Vollmer, T. Tél, and W. Breymann, Phys. Rev. Lett. 79, 2759 (1997); Phys. Rev. E 58, 1672 (1998); W. Breymann, T. Tél, and J. Vollmer, Chaos 8, 396 (1998).
    • (1998) Chaos , vol.8 , pp. 396
    • Breymann, W.1    Tél, T.2    Vollmer, J.3
  • 21
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    • Springer-Verlag, Berlin
    • W.G. Hoover, Molecular Dynamics, (Springer-Verlag, Berlin 1986); C.P. Dettmann, in Hard Ball Systems and the Lorentz Gas, edited by D. Szász (Springer, Berlin, 2000).
    • (1986) Molecular Dynamics
    • Hoover, W.G.1
  • 22
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    • edited by D. Szász (Springer, Berlin)
    • W.G. Hoover, Molecular Dynamics, (Springer-Verlag, Berlin 1986); C.P. Dettmann, in Hard Ball Systems and the Lorentz Gas, edited by D. Szász (Springer, Berlin, 2000).
    • (2000) Hard Ball Systems and the Lorentz Gas
    • Dettmann, C.P.1
  • 23
    • 33645067576 scopus 로고    scopus 로고
    • From the point of view of the dynamical system this constitutes a Poincaré map
    • From the point of view of the dynamical system this constitutes a Poincaré map.
  • 25
    • 33645055610 scopus 로고    scopus 로고
    • note
    • The time step of the method is sufficiently small to ensure that the coarse-grained entropy reaches its saturation much before the numerical errors in the trajectories become comparable to the resolution of the partitioning (i.e., the size of the cells).
  • 27
    • 33645089143 scopus 로고    scopus 로고
    • note
    • In the general definition of the entropy we use coarse graining by means of squares in units of the size of the total phase space. For the Lorentz gas this means that the lengths of the respective sides of a cell of size ε are a fraction ε of those of the full domain ([-1,1] × [0,2π]). The cells are thus rectangles of size 28ε × 2πε. From the point of view of the coarse-grained entropy this does not matter, because only the measure of boxes appears in Eq. (9).
  • 28
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    • Ph.D. thesis, ELTE Budapest (unpublished)
    • L. Mátyás, Ph.D. thesis, ELTE Budapest 2001 (unpublished).
    • (2001)
    • Mátyás, L.1
  • 31
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    • note
    • Common discussions of transport processes from the point of view of kinetic theory of gases characterize local equilibrium by the sufficient, even though not necessary, condition of a Maxwellian velocity distribution of the particles. In contrast, we adopt the definition of de Groot and Mazur [1] [Chapter III, Section 2], which is more general than the one in kinetic theory: "Although the total system is not in equilibrium, there exists within small mass elements a state of "local" equilibrium, for which the local entropy is the same function [of the macroscopic thermodynamic variables] as in real equilibrium. [...] This hypothesis of local equilibrium can, from a macroscopic point of view, only be justified by virtue of the conclusions derived from it."
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    • (1993) Commun. Math. Phys. , vol.154 , pp. 569


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