Chaos, ergodicity, and the thermodynamics of lower-dimensional time-independent Hamiltonian systems

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

Volumn 65, Issue 1, 2002, Pages

  Kandrup, Henry E ^a   Sideris, Ioannis V ^a   Bohn, Courtlandt L ^b  

^a University of Florida ^*  (United States)

^b FERMI NATIONAL ACCELERATOR LABORATORY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CHOLESTERIC ELASTOMERS (CE); CHOLESTERIC LIQUID CRYSTALS (CLC); PHOTONIC BAND STRUCTURES; PHOTONIC BAND-GAP (PBG) MATERIALS;

BAND STRUCTURE; DIELECTRIC MATERIALS; ELASTOMERS; FORMABILITY; LIQUID CRYSTALS; PERMITTIVITY; PHOTONS; POLARIZATION; STRESS ANALYSIS;

SELF ASSEMBLY;

EID: 41349112466     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.65.016214     Document Type: Article

References (27)

1. See, e.g., A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (Springer, Berlin, 1992).
2. L. P. Eisenhart, Ann. Math. 30, 591 (1929).
3. E. Hopf, Trans. Am. Math. Soc. 39, 229 (1936).
4. D. V. Anosov, Trudy Mat. Inst. Steklov 90, 1 (1967).
5. M. Pettini, Phys. Rev. E 47, 828 (1993).
6. M. Toda, J. Phys. Soc. Jpn. 22, 431 (1967).
7. H. E. Kandrup, Phys. Rev. E 56, 2722 (1997).
8. L. Casetti, C. Clementi, and M. Pettini, Phys. Rev. E 54, 5969 (1996).
9. H. E. Kandrup and I. V. Sideris, Celest. Mech. (to be published).
10. D. Armbruster, J. Guckenheimer, and S. Kim, Phys. Lett. A 140, 416 (1989).
11. Cf., J. Ford, Phys. Rep. 213, 271 (1992).
12. Cf., P. Grassberger, P. Badii, and A. Poloti, J. Stat. Phys. 51, 135 (1988).
13. Cf., L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, Oxford, 1969), Sec. 44.
14. N. G. Van Kampen, Phys. Rep. 24, 71 (1976).
15. G. Contopoulos, Astrophys. J. 76, 147 (1971).
16. Cf., I. V. Pogorelov and H. E. Kandrup, Phys. Rev. E 60, 1567 (1999). Figure 1 in that paper exhibits examples of “sticky” orbits in the potentials (3.1) and (3.2).
17. M. Cerruti-Sola and M. Pettini, Phys. Rev. E 53, 179 (1996).
18. Unfortunately, an analytic solution using the approach discussed in Sec. IV seems impossible. For a single oscillator equation, one is led to three coupled moment equations and, hence, a cubic equation for χ. For two coupled oscillators, one has eight coupled equations and, hence, an eighth-order equation for χ.
19. H. E. Kandrup, Astrophys. J. 500, 120 (1998).
20. C. L. Bohn, in The Physics of High Brightness Beams, edited by J. Rosenzweig and L. Serafini (World Scientific, Singapore, 2000), pp. 358–368.
21. M. Cerruti-Sola and M. Pettini, Phys. Rev. E 53, 179 (1996).
22. T. deZeeuw, Mon. Not. R. Astron. Soc. 216, 273 (1985).
23. C. Siopis and H. E. Kandrup, Mon. Not. R. Astron. Soc. 319, 43 (2000).
24. H. E. Kandrup and I. V. Sideris, Phys. Rev. E (to be published). Discreteness effects in a chaotic potential, modeled as friction and noise, typically trigger an exponential divergence.
25. M. Di Bari, D. Boccaletti, P. Cipriani, and G. Puccacio, Phys. Rev. E 55, 6448 (1997).
26. P. Cipriani and M. Di Bari, Phys. Rev. Lett. 81, 5532 (1998).
27. G. Gallavotti, J. Math. Phys. 41, 4061 (2000).