Abstract
A unified formalism is presented to study Hamiltonian linear systems driven by noise. With this formalism, the phase averaging approximation, valid at weak noise, is easily performed. Already known results are straightforwardly recovered and new ones are obtained. After introducing this formalism on the exactly solvable one-degree-of-freedom problem with uncorrelated noise, one studies the corresponding exponentially correlated case. The validity of the approximate results thus obtained is considered by investigating the systematic weak-disorder expansion beyond the quasilinear approximation. In particular, it is argued that this expansion behaves uniformly for weak and large correlation time. The two-degrees-of-freedom problem is completely solved at the low-disorder approximation and this result is applied to the two-channel Anderson localization problem. The invariant measure and the two positive Lyapunov exponents are computed at all coupling between the channels. For systems withn degree of freedom the phase averaging leads to a Fokker-Planck equation for the measure in action space describing the system. However, it is argued that it is not solvable except in a special case which is explicitly displayed and solved. Nevertheless, in the large-n limit, it is possible to compute the largest Lyapunov exponent. Moreover, generalized Lyapunov exponents are calculated in this limit, and they do not exhibit a dispersion: in particular, log〈ℰ〉/〈logℰ〉∼1, where ℰ is the energy of the system and where the brackets denote averaging over the noise. On the other hand, it is possible to compute at weak noise the sum of all the positive Lyapunov exponents. Taking into account all these results allows more insight on the whole spectrum of Lyapunov exponents.
Similar content being viewed by others
References
O. N. Dorokhov,Sov. Phys.-JETP 58:606 (1983).
B. Souillard, inProceedings of the summer school Chance and Matter, Les Houches 1986 (North-Holland, to appear), and references therein.
J. L. Pichard, Thesis, Université Paris-Orsay (1984).
R. Livi, A. Politi, and S. Ruffo,J. Phys. A: Math. Gen. 19:2033 (1986); J. L. Pichard and G. André,Europhys. Lett. 2:477 (1986).
C. M. Newman,Commun. Math. Phys. 103:121 (1986).
B. J. Halperin,Phys. Rev. 139A:104 (1965).
B. Douçot and R. Rammal,J. Phys. (Paris) 48:933 (1987).
R. Z. Khas'minskii,Stochastic Stability of Differential Equations (Stijthoff and Noodhof, Alphan oan den Rijn, The Netherlands), 1980).
D. Hansel and J. F. Luciani,J. Phys. A: Math. Gen. 20:L1031 (1987).
O. N. Dorokhov,Solid State Commun. 51:381 (1984).
A. D. Stone, C. D. Allan, and J. D. Joannopoulos,Phys. Rev. B 27:836 (1983).
N. G. Van Kampen,Phys. Rep. 24:171 (1976).
V. I. Melnikov,Sov. Phys. Solid State 23:444 (1981).
B. Derrida and E. Gardner,J. Phys. (Paris) 45:1283 (1984).
L. Pastur and E. P. Fel'dman,JETP 40:241 (1975).
F. Delyon and P. Foulon,J. Stat. Phys. 45:41 (1986).
U. Sivan and Y. Imry,Phys. Rev. B 35:6074 (1987).
R. Graham and D. Rockaerts,Phys. Rev. A 31:3364 (1985).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hansel, D., Luciani, J.F. On diffusion equations for dynamical systems driven by noise. J Stat Phys 54, 971–995 (1989). https://doi.org/10.1007/BF01019784
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01019784