Multibaker map for thermodynamic cross effects in dynamical systems

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 62, Issue 1 A, 2000, Pages 349-365

  Mátyás, László ^a   Tél, Tamás ^a   Vollmer, Jürgen ^b,c  

^a EÖTVÖS LORÁND UNIVERSITY   (Hungary)

^b UNIVERSITY OF DUISBURG ESSEN   (Germany)

^c MAX PLANCK INSTITUTE FOR POLYMER RESEARCH   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

THERMODYNAMICS;

DYNAMICAL SYSTEMS;

STATISTICAL MECHANICS;

EID: 0038740335     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.62.349     Document Type: Article

References (71)

1. J.R. Dorfman, An Introduction to Chaos in Non-Equilibrium Statistical Mechanics (Cambridge University Press, Cambridge, 1999).
2. Chaos and Irreversibility, focus issue of Chaos 8, 309 (1998).
3. P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge, 1998).
4. D.J. Evans and G.P. Morriss, Statistical Mechanics of Non-equilibrium Liquids (Academic Press, London, 1990); W.G. Hoover, Computational Statistical Mechanics (Elsevier, Amsterdam, 1991).
5. D.J. Evans and G.P. Morriss, Statistical Mechanics of Non-equilibrium Liquids (Academic Press, London, 1990); W.G. Hoover, Computational Statistical Mechanics (Elsevier, Amsterdam, 1991).
6. W.N. Vance, Phys. Rev. Lett. 69, 1356 (1992).
7. N.I. Chernov, G.L. Eyink, J.L. Lebowitz, and Ya.G. Sinai, Phys. Rev. Lett. 70, 2209 (1993); Commun. Math. Phys. 154, 569 (1993).
8. N.I. Chernov, G.L. Eyink, J.L. Lebowitz, and Ya.G. Sinai, Phys. Rev. Lett. 70, 2209 (1993); Commun. Math. Phys. 154, 569 (1993).
9. D.J. Evans, E.G.D. Cohen, and G.P. Morriss, Phys. Rev. Lett. 71, 2401 (1993).
10. G. Gallavotti and E.G.D. Cohen, Phys. Rev. Lett. 74, 2694 (1995); J. Stat. Phys. 80, 931 (1995); G. Gallavotti, Phys. Rev. Lett. 77, 4334 (1996); J. Stat. Phys. 86, 907 (1997).
11. G. Gallavotti and E.G.D. Cohen, Phys. Rev. Lett. 74, 2694 (1995); J. Stat. Phys. 80, 931 (1995); G. Gallavotti, Phys. Rev. Lett. 77, 4334 (1996); J. Stat. Phys. 86, 907 (1997).
12. G. Gallavotti and E.G.D. Cohen, Phys. Rev. Lett. 74, 2694 (1995); J. Stat. Phys. 80, 931 (1995); G. Gallavotti, Phys. Rev. Lett. 77, 4334 (1996); J. Stat. Phys. 86, 907 (1997).
13. G. Gallavotti and E.G.D. Cohen, Phys. Rev. Lett. 74, 2694 (1995); J. Stat. Phys. 80, 931 (1995); G. Gallavotti, Phys. Rev. Lett. 77, 4334 (1996); J. Stat. Phys. 86, 907 (1997).
14. E.G.D. Cohen, Physica A 213, 293 (1995); 240, 43 (1997).
15. E.G.D. Cohen, Physica A 213, 293 (1995); 240, 43 (1997).
16. G.P. Morriss and L. Rondoni, J. Stat. Phys. 75, 553 (1994); L. Rondoni and G.P. Morriss, Physica A 233, 767 (1996); Phys. Rep. 290, 173 (1997).
17. G.P. Morriss and L. Rondoni, J. Stat. Phys. 75, 553 (1994); L. Rondoni and G.P. Morriss, Physica A 233, 767 (1996); Phys. Rep. 290, 173 (1997).
18. G.P. Morriss and L. Rondoni, J. Stat. Phys. 75, 553 (1994); L. Rondoni and G.P. Morriss, Physica A 233, 767 (1996); Phys. Rep. 290, 173 (1997).
19. N.I. Chernov and J.L. Lebowitz, Phys. Rev. Lett. 75, 2831 (1995); J. Stat. Phys. 86, 953 (1997); Ch. Dellago and H.A. Posch, ibid. 88, 825 (1997).
20. N.I. Chernov and J.L. Lebowitz, Phys. Rev. Lett. 75, 2831 (1995); J. Stat. Phys. 86, 953 (1997); Ch. Dellago and H.A. Posch, ibid. 88, 825 (1997).
21. N.I. Chernov and J.L. Lebowitz, Phys. Rev. Lett. 75, 2831 (1995); J. Stat. Phys. 86, 953 (1997); Ch. Dellago and H.A. Posch, ibid. 88, 825 (1997).
22. D. Ruelle, J. Stat. Phys. 85, 1 (1996); 86, 935 (1997).
23. D. Ruelle, J. Stat. Phys. 85, 1 (1996); 86, 935 (1997).
24. W. Breymann, T. Tél, and J. Vollmer, Phys. Rev. Lett. 77, 2945 (1996).
25. G. Nicolis and D. Daems, J. Chem. Phys. 1000, 19 187 (1996); D. Daems and G. Nicolis, Phys. Rev. E 59, 4000 (1999).
26. G. Nicolis and D. Daems, J. Chem. Phys. 1000, 19 187 (1996); D. Daems and G. Nicolis, Phys. Rev. E 59, 4000 (1999).
27. P. Gaspard, Physica A 240, 54 (1997); J. Stat. Phys. 88, 1215 (1997).
28. P. Gaspard, Physica A 240, 54 (1997); J. Stat. Phys. 88, 1215 (1997).
29. J. Vollmer, T. Tél, and W. Breymann, Phys. Rev. Lett. 79, 2759 (1997).
30. J. Vollmer, T. Tél, and W. Breymann, Phys. Rev. E 58, 1672 (1998).
31. W. Breymann, T. Tél, and J. Vollmer, Chaos 8, 396 (1998).
32. T. Gilbert, C.D. Ferguson, and J.R. Dorfman, Phys. Rev. E 59, 364 (1999).
33. T. Gilbert and J.R. Dorfman, J. Stat. Phys. 96, 225 (1999).
34. P. Gaspard, J. Stat. Phys. 68, 673 (1992).
35. S. Tasaki and P. Gaspard, J. Stat. Phys. 81, 935 (1995).
36. P. Gaspard and G. Nicolis, Phys. Rev. Lett. 65, 1693 (1990); P. Gaspard and F. Baras, Phys. Rev. E 51, 5332 (1995); J.R. Dorfman and P. Gaspard, ibid. 51, 28 (1995); 52, 3525 (1995); H. van Beijeren and J.R. Dorfman, Phys. Rev. Lett. 74, 4412 (1995).
37. P. Gaspard and G. Nicolis, Phys. Rev. Lett. 65, 1693 (1990); P. Gaspard and F. Baras, Phys. Rev. E 51, 5332 (1995); J.R. Dorfman and P. Gaspard, ibid. 51, 28 (1995); 52, 3525 (1995); H. van Beijeren and J.R. Dorfman, Phys. Rev. Lett. 74, 4412 (1995).
38. P. Gaspard and G. Nicolis, Phys. Rev. Lett. 65, 1693 (1990); P. Gaspard and F. Baras, Phys. Rev. E 51, 5332 (1995); J.R. Dorfman and P. Gaspard, ibid. 51, 28 (1995); 52, 3525 (1995); H. van Beijeren and J.R. Dorfman, Phys. Rev. Lett. 74, 4412 (1995).
39. P. Gaspard and G. Nicolis, Phys. Rev. Lett. 65, 1693 (1990); P. Gaspard and F. Baras, Phys. Rev. E 51, 5332 (1995); J.R. Dorfman and P. Gaspard, ibid. 51, 28 (1995); 52, 3525 (1995); H. van Beijeren and J.R. Dorfman, Phys. Rev. Lett. 74, 4412 (1995).
40. P. Gaspard and G. Nicolis, Phys. Rev. Lett. 65, 1693 (1990); P. Gaspard and F. Baras, Phys. Rev. E 51, 5332 (1995); J.R. Dorfman and P. Gaspard, ibid. 51, 28 (1995); 52, 3525 (1995); H. van Beijeren and J.R. Dorfman, Phys. Rev. Lett. 74, 4412 (1995).
41. T. Tél, J. Vollmer, and W. Breymann, Europhys. Lett. 35, 659 (1996).
42. R. Klages and J.R. Dorfman, Phys. Rev. Lett. 74, 387 (1995); G. Radons, ibid. 77, 4748 (1996).
43. R. Klages and J.R. Dorfman, Phys. Rev. Lett. 74, 387 (1995); G. Radons, ibid. 77, 4748 (1996).
44. Z. Kaufmann, Phys. Rev. E 59, 6552 (1999); Z. Kaufmann, A. Nemeth, and P. Szépfalusy, ibid. 61, 2543 (2000); Z. Kaufmann and P. Szépfalusy, J. Stat. Phys. (to be published).
45. Z. Kaufmann, Phys. Rev. E 59, 6552 (1999); Z. Kaufmann, A. Nemeth, and P. Szépfalusy, ibid. 61, 2543 (2000); Z. Kaufmann and P. Szépfalusy, J. Stat. Phys. (to be published).
46. Z. Kaufmann, Phys. Rev. E 59, 6552 (1999); Z. Kaufmann, A. Nemeth, and P. Szépfalusy, ibid. 61, 2543 (2000); Z. Kaufmann and P. Szépfalusy, J. Stat. Phys. (to be published).
47. J.L. Lebowitz and H. Spohn, J. Stat. Phys. 19, 633 (1978); G. Casati, J. Ford, F. Vivaldi, and W.M. Visscher, Phys. Rev. Lett. 52, 1861 (1984); L.A. Bunimovich and H. Spohn, Commun. Math. Phys. 176, 661 (1996).
48. J.L. Lebowitz and H. Spohn, J. Stat. Phys. 19, 633 (1978); G. Casati, J. Ford, F. Vivaldi, and W.M. Visscher, Phys. Rev. Lett. 52, 1861 (1984); L.A. Bunimovich and H. Spohn, Commun. Math. Phys. 176, 661 (1996).
49. J.L. Lebowitz and H. Spohn, J. Stat. Phys. 19, 633 (1978); G. Casati, J. Ford, F. Vivaldi, and W.M. Visscher, Phys. Rev. Lett. 52, 1861 (1984); L.A. Bunimovich and H. Spohn, Commun. Math. Phys. 176, 661 (1996).
50. A. Krámli, N. Simányi, and D. Szász, J. Stat. Phys. 46, 303 (1984); S. Lepri, R. Livi, and A. Politi, Phys. Rev. Lett. 78, 1896 (1997); H.A. Posch and Wm.G. Hoover, Phys. Rev. E 55, 6803 (1997); D. Alonso, R. Artuso, G. Casati, and I. Guarneri, Phys. Rev. Lett. 82, 1859 (1999); C. Wagner, R. Klages, and G. Nicolis, Phys. Rev. E 60, 1401 (1999); K. Rateitschak, R. Klages and G. Nicolis, J. Stat. Phys. (to be published).
51. A. Krámli, N. Simányi, and D. Szász, J. Stat. Phys. 46, 303 (1984); S. Lepri, R. Livi, and A. Politi, Phys. Rev. Lett. 78, 1896 (1997); H.A. Posch and Wm.G. Hoover, Phys. Rev. E 55, 6803 (1997); D. Alonso, R. Artuso, G. Casati, and I. Guarneri, Phys. Rev. Lett. 82, 1859 (1999); C. Wagner, R. Klages, and G. Nicolis, Phys. Rev. E 60, 1401 (1999); K. Rateitschak, R. Klages and G. Nicolis, J. Stat. Phys. (to be published).
52. A. Krámli, N. Simányi, and D. Szász, J. Stat. Phys. 46, 303 (1984); S. Lepri, R. Livi, and A. Politi, Phys. Rev. Lett. 78, 1896 (1997); H.A. Posch and Wm.G. Hoover, Phys. Rev. E 55, 6803 (1997); D. Alonso, R. Artuso, G. Casati, and I. Guarneri, Phys. Rev. Lett. 82, 1859 (1999); C. Wagner, R. Klages, and G. Nicolis, Phys. Rev. E 60, 1401 (1999); K. Rateitschak, R. Klages and G. Nicolis, J. Stat. Phys. (to be published).
53. A. Krámli, N. Simányi, and D. Szász, J. Stat. Phys. 46, 303 (1984); S. Lepri, R. Livi, and A. Politi, Phys. Rev. Lett. 78, 1896 (1997); H.A. Posch and Wm.G. Hoover, Phys. Rev. E 55, 6803 (1997); D. Alonso, R. Artuso, G. Casati, and I. Guarneri, Phys. Rev. Lett. 82, 1859 (1999); C. Wagner, R. Klages, and G. Nicolis, Phys. Rev. E 60, 1401 (1999); K. Rateitschak, R. Klages and G. Nicolis, J. Stat. Phys. (to be published).
54. A. Krámli, N. Simányi, and D. Szász, J. Stat. Phys. 46, 303 (1984); S. Lepri, R. Livi, and A. Politi, Phys. Rev. Lett. 78, 1896 (1997); H.A. Posch and Wm.G. Hoover, Phys. Rev. E 55, 6803 (1997); D. Alonso, R. Artuso, G. Casati, and I. Guarneri, Phys. Rev. Lett. 82, 1859 (1999); C. Wagner, R. Klages, and G. Nicolis, Phys. Rev. E 60, 1401 (1999); K. Rateitschak, R. Klages and G. Nicolis, J. Stat. Phys. (to be published).
55. A. Krámli, N. Simányi, and D. Szász, J. Stat. Phys. 46, 303 (1984); S. Lepri, R. Livi, and A. Politi, Phys. Rev. Lett. 78, 1896 (1997); H.A. Posch and Wm.G. Hoover, Phys. Rev. E 55, 6803 (1997); D. Alonso, R. Artuso, G. Casati, and I. Guarneri, Phys. Rev. Lett. 82, 1859 (1999); C. Wagner, R. Klages, and G. Nicolis, Phys. Rev. E 60, 1401 (1999); K. Rateitschak, R. Klages and G. Nicolis, J. Stat. Phys. (to be published).
56. D.J. Evans and P.T. Cummings, Mol. Phys. 72, 893 (1991); S. Sarman and D.J. Evans, Phys. Rev. A 45, 2370 (1992); G. Gallavotti, J. Stat. Phys. 84, 899 (1996).
57. D.J. Evans and P.T. Cummings, Mol. Phys. 72, 893 (1991); S. Sarman and D.J. Evans, Phys. Rev. A 45, 2370 (1992); G. Gallavotti, J. Stat. Phys. 84, 899 (1996).
58. D.J. Evans and P.T. Cummings, Mol. Phys. 72, 893 (1991); S. Sarman and D.J. Evans, Phys. Rev. A 45, 2370 (1992); G. Gallavotti, J. Stat. Phys. 84, 899 (1996).
59. L. Mátyás, T. Tél, and J. Vollmer, Phys. Rev. E 61, R3295 (2000).
60. S. Tasaki and P. Gaspard, Theor. Chem. Acc. 102, 385 (1999).
61. In kinetic theory the fields are related by the square of the momentum variable. Here we show that consistency with thermodynamics can be achieved without referring to this relation.
62. S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (Elsevier, Amsterdam, 1962) (reprinted by Dover, New York, 1984).
63. D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, London 1960).
64. I. Prigogine, Non-Equilibrium Thermodynamics (John Wiley & Sons, New York, 1961).
65. D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press, London, 1963).
66. E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993).
67. We say that the particles represented by the multibaker interact weakly, since they are all mapped by the same map at every instant of time. There is only a possibility for an indirect, mean-field-type interaction due to the dependence of the parameters of the map on the local coarse-grained densities and their (discrete) gradients.
68. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), Sec. 1.9.
69. N.W. Ashcroft and N.D. Mermin, Solid State Physics (Holt, Rinehart and Winston, Saunders College, Philadelphia, 1976).
70. relax stands for the relaxation time. This shows that the finiteness of the diffusion coefficient and the conductivity in our approach corresponds to the finiteness of the relaxation time in kinetic theory.
71. An extension in this spirit for the case of thermostated systems with constant temperature can be found in [20].