zbMATH — the first resource for mathematics

Liapunov spectra for infinite chains of nonlinear oscillators. (English) Zbl 1084.37500
Summary: We argue that the spectrum of Lyapunov exponents for long chains of nonlinear oscillators, at large energy per mode, may be well approximated by the Lyapunov exponents of products of independent random matrices. If, in addition, statistical mechanics applies to the system, the elements of these random matrices have a distribution which may be calculated from the potential and the energy alone. Under a certain isotropy hypothesis (which is not always satisfied), we argue that the Lyapunov exponents of these random matrix products can be obtained from the density of states of a typical random matrix. This construction uses an integral equation first derived by Newman. We then derive and discuss a method to compute the spectrum of a typical random matrix. Putting the pieces together, we see that the Lyapunov spectrum can be computed from the potential between the oscillators.

37A30 Ergodic theorems, spectral theory, Markov operators
60K35 Interacting random processes; statistical mechanics type models; percolation theory
70K99 Nonlinear dynamics in mechanics
82B05 Classical equilibrium statistical mechanics (general)
PDF BibTeX Cite
Full Text: DOI
[1] M. Benderskii and L. Pastur,Mat. Sb. 82:245-256 (1970). · Zbl 0216.37301
[2] F. Delyon, H. Kunz, and B. Souillard,J. Phys. A 16:25-42 (1983). · Zbl 0512.60052
[3] N. Dunford and J. T. Schwartz,Linear Operators (Interscience, New York, 1958).
[4] J.-P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57:617-656 (1985). · Zbl 0989.37516
[5] W. Kirsch and F. Martinelli,J. Phys. A 15:2139-2156 (1982). · Zbl 0492.60055
[6] R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione, and A. Vulpiani,Phys. Rev. A 31:1039-1045 (1985).
[7] R. Livi, A. Politi, and S. Ruffo,J. Phys. A 19:2033-2040 (1986). · Zbl 0624.58030
[8] C. M. Newman,Commun. Math. Phys. 103:121-126 (1986). · Zbl 0593.58051
[9] C. M. Newman, inRandom Matrices and Their Applications, J. E. Cohen, H. Kesten, and C. M. Newman, eds. (AMS, Providence, Rhode Island, 1986), p. 121.
[10] G. Paladin and A. Vulpiani,J. Phys. A 19:1881-1888 (1986). · Zbl 0609.70026
[11] H. Schmidt,Phys. Rev. 105:425-441 (1957). · Zbl 0105.43801
[12] B. Simon and M. Taylor,Commun. Math. Phys. 101:1-19 (1985). · Zbl 0577.60067
[13] K. W. Wachter,Ann. Prob. 6:1-18 (1978). · Zbl 0374.60039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.