Abstract
The relation between chaotic dynamics of nonlinear Hamiltonian systems and equilibrium statistical mechanics in its canonical ensemble formulation has been investigated for two different nonlinear Hamiltonian systems. We have compared time averages obtained by means of numerical simulations of molecular dynamics type with analytically computed ensemble averages. The numerical simulation of the dynamic counterpart of the canonical ensemble is obtained by considering the behavior of a small part of a given system, described by a microcanonical ensemble, in order to have fluctuations of the energy of the subsystem. The results for the Fermi-Pasta-Ulam model (i.e., a one-dimensional anharmonic solid) show a substantial agreement between time and ensemble averages independent of the degree of stochasticity of the dynamics. On the other hand, a very different behavior is observed for a chain of weakly coupled rotators, where linear exchange effects are absent. In the high-temperature limit (weak coupling) we have a strong disagreement between time and ensemble averages for the specific heat even if the dynamics is chaotic. This behavior is related to the presence of spatially localized chaos, which prevents the complete filling of the accessible phase space of the system. Localized chaos is detected by the distribution of all the characteristic Liapunov exponents.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. C. Tolman,The Principles of Statistical Mechanics (Oxford University Press, 1938).
A. I. Khinchin,Mathematical Foundations of Statistical Mechanics (Dover New York, 1949).
C. Truesdell, inErgodic Theory, Proceedings of the International School of Physics ‘Enrico Fermi’, P. Caldirola, ed. (Varenna, 1960), p. 21.
F. M. Izrailev and B. V. Chirikov,Sov. Phys. Doklady 11:30 (1966).
P. Bocchieri, A. Scotti, B. Bearzi, and A. Loinger,Phys. Rev. A 2:2013 (1970); M. Casartelli, G. Casati, E. Diana, L. Galgani, and A. Scotti,Theor. Math. Phys. 29:205 (1976) (in Russian); M. Casartelli, E. Diana, L. Galgani, and A. Scotti,Phys. Rev. A 13:1921 (1976).
R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione, and A. Vulpiani,Phys. Rev. 28:3544 (1983);Phys. Rev. A 31:1039 (1985); R. Livi, M. Pettini, S. Ruffo, and A. Vulpiani,Phys. Rev. A 31:2740 (1985).
J. Bellissard and M. Vittot, Marseille preprint CPT-MARSEILLE (1985), to appear inErgodic Theory and Dynamical Systems; J. Fröhlich, T. Spencer, and C. E. Wayne,J. Stat. Phys. 42:247 (1986).
E. Schrödinger,Ann. Physik 44:916 (1914); H. Wergeland, inIrreversibility in Many-Body Problems, J. Biel and J. Rae, eds. (Plenum Press, New York, 1972), p. 105; G. Casati,Found. Phys. 16:51 (1986).
F. Mokross and H. Büttner,J. Phys. C 16:4539 (1983); G. Casati, J. Ford, F. Vivaldi, and W. M. Visscher,Phys. Rev. Lett. 52:1861 (1984).
M. Casartelli,Phys. Rev. A 19:1741 (1979); M. C. Carotta, C. Ferrario, G. Lo Vecchio, B. Carazza, and L. Galgani,Phys. Lett. 57A:399 (1976); R.W.Hall and B. J. Berne,J. Chem. Phys. 81:3641 (1984).
E. Fermi, J. Pasta, and S. Ulam, inCollected papers of E. Fermi, (University of Chicago, Chicago, 1965), Vol. 2, p. 978.
L. Verlet,Phys. Rev. 159:89 (1967).
R. Livi, A. Politi, and S. Ruffo,J. Phys. A: Math. Gen. 19:2033 (1986).
G. Benettin, L. Galgani, and J. M. Strelcyn,Phys. Rev. A 14:2338 (1976); G. Benettin, L. Galgani, A. Giorgilli, and J. M. Strelcyn,Meccanica 15:9 (1980);15:21 (1980).
G. Benettin, L. Galgani, and A. Giorgilli,Nuovo Cimento B 89:103 (1985).
S. K. Ma,Statistical Mechanics (World Scientific, Singapore, 1985), p. 142.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Livi, R., Pettini, M., Ruffo, S. et al. Chaotic behavior in nonlinear Hamiltonian systems and equilibrium statistical mechanics. J Stat Phys 48, 539–559 (1987). https://doi.org/10.1007/BF01019687
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01019687