Chaotic dynamics, fluctuations, nonequilibrium ensembles

Chaos

Volumn 8, Issue 2, 1998, Pages 384-392

  Gallavotti, Giovanni ^a  

^a SAPIENZA UNIVERSITY OF ROME   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001369553     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.166320     Document Type: Article

References (41)

1. L. Boltzmann, "Über die Eigenshaften monozyklischer und anderer damit verwandter Systeme," in Wissenshafltliche Abhandlungen, edited by F. P. Hasenöhrl (Chelsea, New York, 1968, reprint), Vol. III.
2. L. Boltzmann, Entgegnung auf die wärmetheoretischen Betrachtungen des Hrn. E. Zermelo, engl. transl.: S. Brush, Kinetic Theory (Pergamon, New York), Vol. 2, p. 218; Zu Hrn. Zermelo's Abhandlung "Ueber die mechanische Erklärung irreversibler Vorgänge", engl. trans. in S. Brush, Kinetic Theory (Pergamon, New York), Vol. 2, p. 238; Lectures on Gas Theory, English edition annotated by S. Brush (University of California Press, Berkeley, 1964).
3. L. Boltzmann, Entgegnung auf die wärmetheoretischen Betrachtungen des Hrn. E. Zermelo, engl. transl.: S. Brush, Kinetic Theory (Pergamon, New York), Vol. 2, p. 218; Zu Hrn. Zermelo's Abhandlung "Ueber die mechanische Erklärung irreversibler Vorgänge", engl. trans. in S. Brush, Kinetic Theory (Pergamon, New York), Vol. 2, p. 238; Lectures on Gas Theory, English edition annotated by S. Brush (University of California Press, Berkeley, 1964).
4. L. Boltzmann, Entgegnung auf die wärmetheoretischen Betrachtungen des Hrn. E. Zermelo, engl. transl.: S. Brush, Kinetic Theory (Pergamon, New York), Vol. 2, p. 218; Zu Hrn. Zermelo's Abhandlung "Ueber die mechanische Erklärung irreversibler Vorgänge", engl. trans. in S. Brush, Kinetic Theory (Pergamon, New York), Vol. 2, p. 238; Lectures on Gas Theory, English edition annotated by S. Brush (University of California Press, Berkeley, 1964).
5. J. L. Lebowitz, "Boltzmann's entropy and time's arrow," Phys. Today, 32-38 (1993); "Microscopic Reversibility and Macroscopic Behavior: Physical Explanations and Mathematical Derivations," in 25 Years of Non-equilibrium Statistical Mechanics," edited by J. Brey, J. Marro, J. Rubi, and M. San Miguel, Lecture Notes in Physics (Springer, 1995).
6. J. L. Lebowitz, "Boltzmann's entropy and time's arrow," Phys. Today, 32-38 (1993); "Microscopic Reversibility and Macroscopic Behavior: Physical Explanations and Mathematical Derivations," in 25 Years of Non-equilibrium Statistical Mechanics," edited by J. Brey, J. Marro, J. Rubi, and M. San Miguel, Lecture Notes in Physics (Springer, 1995).
7. D. Ruelle, "A measure associated with Axiom A attractors," Am. J. Math. 98, 619-654 (1976); "Measures describing a turbulent flow," Ann. (N.Y.) Acad. Sci. 357, 1-9 (1980); R. Bowen and D. Ruelle, "Ergodic theory of Axiom A flows," Invent. Math. 2, 181-202 (1975). See also D. Ruelle, Chaotic Motions and Strange Attractors, Lezioni Lincee, notes by S. Isola, Accademia Nazionale dei Lincei (Cambridge University Press, Cambridge, England, 1989).
8. D. Ruelle, "A measure associated with Axiom A attractors," Am. J. Math. 98, 619-654 (1976); "Measures describing a turbulent flow," Ann. (N.Y.) Acad. Sci. 357, 1-9 (1980); R. Bowen and D. Ruelle, "Ergodic theory of Axiom A flows," Invent. Math. 2, 181-202 (1975). See also D. Ruelle, Chaotic Motions and Strange Attractors, Lezioni Lincee, notes by S. Isola, Accademia Nazionale dei Lincei (Cambridge University Press, Cambridge, England, 1989).
9. D. Ruelle, "A measure associated with Axiom A attractors," Am. J. Math. 98, 619-654 (1976); "Measures describing a turbulent flow," Ann. (N.Y.) Acad. Sci. 357, 1-9 (1980); R. Bowen and D. Ruelle, "Ergodic theory of Axiom A flows," Invent. Math. 2, 181-202 (1975). See also D. Ruelle, Chaotic Motions and Strange Attractors, Lezioni Lincee, notes by S. Isola, Accademia Nazionale dei Lincei (Cambridge University Press, Cambridge, England, 1989).
10. D. Ruelle, "A measure associated with Axiom A attractors," Am. J. Math. 98, 619-654 (1976); "Measures describing a turbulent flow," Ann. (N.Y.) Acad. Sci. 357, 1-9 (1980); R. Bowen and D. Ruelle, "Ergodic theory of Axiom A flows," Invent. Math. 2, 181-202 (1975). See also D. Ruelle, Chaotic Motions and Strange Attractors, Lezioni Lincee, notes by S. Isola, Accademia Nazionale dei Lincei (Cambridge University Press, Cambridge, England, 1989).
11. It is impossible to account here, analytically, for the vast literature on the subject. The following papers are highlights and contain detailed references to related papers: B. L. Holian, W. G. Hoover, and H. A. Posch, "Resolution of Loschmidt's paradox: the origin of irreversible behavior in reversible atomistic dynamics," Phys. Rev. Lett. 59, 10-13 (1987); D. J. Evans and G. P. Morriss, Statistical Mechanics of Non-equilibrium Fluids (Academic, New York, 1990); D. J. Evans, E. G. D. Cohen, and G. P. Morriss, "Viscosity of a simple fluid from its maximal Lyapunov exponents," Phys. Rev. A 42, 5990-5997 (1990); C. Dellago, H. Posch, and W. Hoover, "Lyapunov instability in system of hard disks in equilibrium and non-equilibrium steady states," Phys. Rev. E 53, 1485-1501 (1996).
12. It is impossible to account here, analytically, for the vast literature on the subject. The following papers are highlights and contain detailed references to related papers: B. L. Holian, W. G. Hoover, and H. A. Posch, "Resolution of Loschmidt's paradox: the origin of irreversible behavior in reversible atomistic dynamics," Phys. Rev. Lett. 59, 10-13 (1987); D. J. Evans and G. P. Morriss, Statistical Mechanics of Non-equilibrium Fluids (Academic, New York, 1990); D. J. Evans, E. G. D. Cohen, and G. P. Morriss, "Viscosity of a simple fluid from its maximal Lyapunov exponents," Phys. Rev. A 42, 5990-5997 (1990); C. Dellago, H. Posch, and W. Hoover, "Lyapunov instability in system of hard disks in equilibrium and non-equilibrium steady states," Phys. Rev. E 53, 1485-1501 (1996).
13. It is impossible to account here, analytically, for the vast literature on the subject. The following papers are highlights and contain detailed references to related papers: B. L. Holian, W. G. Hoover, and H. A. Posch, "Resolution of Loschmidt's paradox: the origin of irreversible behavior in reversible atomistic dynamics," Phys. Rev. Lett. 59, 10-13 (1987); D. J. Evans and G. P. Morriss, Statistical Mechanics of Non-equilibrium Fluids (Academic, New York, 1990); D. J. Evans, E. G. D. Cohen, and G. P. Morriss, "Viscosity of a simple fluid from its maximal Lyapunov exponents," Phys. Rev. A 42, 5990-5997 (1990); C. Dellago, H. Posch, and W. Hoover, "Lyapunov instability in system of hard disks in equilibrium and non-equilibrium steady states," Phys. Rev. E 53, 1485-1501 (1996).
14. It is impossible to account here, analytically, for the vast literature on the subject. The following papers are highlights and contain detailed references to related papers: B. L. Holian, W. G. Hoover, and H. A. Posch, "Resolution of Loschmidt's paradox: the origin of irreversible behavior in reversible atomistic dynamics," Phys. Rev. Lett. 59, 10-13 (1987); D. J. Evans and G. P. Morriss, Statistical Mechanics of Non-equilibrium Fluids (Academic, New York, 1990); D. J. Evans, E. G. D. Cohen, and G. P. Morriss, "Viscosity of a simple fluid from its maximal Lyapunov exponents," Phys. Rev. A 42, 5990-5997 (1990); C. Dellago, H. Posch, and W. Hoover, "Lyapunov instability in system of hard disks in equilibrium and non-equilibrium steady states," Phys. Rev. E 53, 1485-1501 (1996).
15. D. J. Evans, E. G. D. Cohen, and G. P. Morriss, "Probability of second law violations in shearing steady flows," Phys. Rev. Lett. 71, 2401-2404 (1993).
16. G. Gallavotti and E. G. D. Cohen, "Dynamical ensembles in non-equilibrium statistical mechanics," Phys. Rev. Lett. 74, 2694-2697 (1995); G. Gallavotti and E. G. D. Cohen, "Dynamical ensembles in stationary states," J. Stat. Phys. 80, 931-970 (1995).
17. G. Gallavotti and E. G. D. Cohen, "Dynamical ensembles in non-equilibrium statistical mechanics," Phys. Rev. Lett. 74, 2694-2697 (1995); G. Gallavotti and E. G. D. Cohen, "Dynamical ensembles in stationary states," J. Stat. Phys. 80, 931-970 (1995).
18. Y. G. Sinai, "Gibbs measures in ergodic theory," Russ. Math. Surveys 27, 21-69 (1972). Also: Introduction to Ergodic Theory (Princeton University Press, Princeton, NJ, 1977).
19. Y. G. Sinai, "Gibbs measures in ergodic theory," Russ. Math. Surveys 27, 21-69 (1972). Also: Introduction to Ergodic Theory (Princeton University Press, Princeton, NJ, 1977).
20. A. Lasota and J. A. Yorke, "On the existence of invariant measures for piecewise monotonie transformations," Trans. Am. Math. Soc. 183, 481 (1973).
21. G. Gentile, "Large deviation rule for Anosov flows," Forum Mathematicum 10, 89-118 (1998).
22. G. Gallavotti, New Methods in Non-equilibrium Gases and Fluids, Maribor meeting lectures, June 1996-5 July 1996, mp_arc@mathutexas.edu #96-533, in print in Open Systems & Information Dynamics; G. Gallavotti, "Topics in chaotic dynamics," Lect. Notes Phys. 448, 271-311 (1995).
23. G. Gallavotti, New Methods in Non-equilibrium Gases and Fluids, Maribor meeting lectures, June 1996-5 July 1996, mp_arc@mathutexas.edu #96-533, in print in Open Systems & Information Dynamics; G. Gallavotti, "Topics in chaotic dynamics," Lect. Notes Phys. 448, 271-311 (1995).
24. ± (1/2)T and arbitrary elsewhere outside the box (using that the range is nearest neighbor). Finally the analogy is not accidental: the theory of the SRB states is done precisely by showing that they can be thought of as Gibbs states for a one-dimensional short range Ising model (Refs. 8, 4, and 11).
25. D. Ruelle, "Positivity of entropy production in nonequilibrium statistical mechanics," J. Stat. Phys. 85, 1-25 (1996).
26. G. Gallavotti, "Reversible Anosov maps and large deviations," Math. Phys. Electron. J. MPEJ (http://mpej.unige.ch), 1, 1-12 (1995).
27. D. Ruelle, "New theoretical ideas in non-equilibrium statistical mechanics," Lecture notes at Rutgers University, 1997.
28. G. P. Morriss and L. Rondoni, "Applications of periodic orbit theory to N-particle systems," J. Stat. Phys. 86, 991 (1997).
29. J. Kurchan, "Fluctuation Theorem for stochastic dynamics," preprint chao-dyn@xyz.lanl.gov, #9709304.
30. F. Bonetto, G. Gallavotti, and P. Garrido, "Chaotic principle: an experimental test," Physica D 105, 226-252 (1997).
31. G. Gallavotti, "Extension of Onsager's reciprocity to large fields and the chaotic hypothesis," Phys. Rev. Lett. 77, 4334-4337 (1996).
32. G. Gallavotti and D. Ruelle, "SRB states and non-equilibrium statistical mechanics close to equilibrium," Commun. Math. Phys. 190, 279-285 (1997).
33. Equation (16) immediately and remarkably follows from the fluctuation theorem Eq. (11): it was pointed out by F. Bonetto.
34. D. Ruelle, "Differentiation of SRB states," Commun. Math. Phys. 187, 227-241 (1997).
35. G. Gallavotti, "Fluctuation patterns and conditional reversibility in non-equilibrium systems," in print on Annales de l' Institut H. Poincaré, chao-dyn@xyz.lanl.gov #9703007.
36. G. Gallavotti, "New methods in non-equilibrium gases and fluids," Proceedings of the conference Let's face chaos through nonlinear dynamics, U. of Maribor, 1996, edited by M. Robnik (unpublished).
37. D. Ruelle, Statistical Mechanics (Benjamin, New York, 1969).
38. G. Gallavotti, "Equivalence of dynamical ensembles and Navier Stokes equations," Phys. Lett. A 223, 91-95 (1996); "Dynamical ensembles equivalence in fluid mechanics," Physica D 105, 163-184 (1997).
39. G. Gallavotti, "Equivalence of dynamical ensembles and Navier Stokes equations," Phys. Lett. A 223, 91-95 (1996); "Dynamical ensembles equivalence in fluid mechanics," Physica D 105, 163-184 (1997).
40. F. Bonetto and G. Gallavotti, "Reversibility, coarse graining and the chaoticity principle," Commun. Math. Phys. 189, 263-276 (1997).
41. S. Ciliberto and C. Laroche, "An experimental verification of the Gallavotti-Cohen fluctuation theorem," preprint Ecole Normale Superieure, CNRS, Lyon, 1998.