Abstract
I show how continuous products of random transformations constrained by a generic group structure can be studied by using Iwasawa's decomposition into “angular,” “diagonal,” and “shear” degrees of freedom. In the case of a Gaussian process a set of variables, adapted to the Iwasawa decomposition and still having a Gaussian distribution, is introduced and used to compute the statistics of the finite-time Lyapunov spectrum of the process. The variables also allow to show the exponential freezing of the “shear” degrees of freedom, which contain information about the Lyapunov eigenvectors.
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REFERENCES
L. Arnold and W. Kliemann, Large deviations of linear stochastic differential equations, in Stochastic Differential Systems Proceedings, Eisenach, 1986, H. J. Engelbert and W. Schmidt, eds., Lecture Notes in Control and Information Sciences, Vol. 96 (Springer, Berlin, 1987), pp. 117–151.
For a review of results, see, L. Arnold and V. Wihstutz, eds., Lyapunov Exponents, Proceedings, Bremen, 1984, Lecture Notes in Mathematics, Vol. 1186 (Springer, Berlin, 1986).
For a review of results, see, C. W. J. Beenakker, Random-matrix theory of quantum transport, Rev. Mod. Phys. 69:731(1997).
C. W. J. Beenakker and B. Rejaei, Nonlogarithmic repulsion of transmission eigenvalues in a disordered wire, Phys. Rev. Lett. 71:3689–3692 (1993).
G. Benettin, Power law behaviour of Lyapunov exponents in some conservative dynamical systems, Phys. D 13:211–213 (1984).
G. Benettin, L. Galgani, A. Giorgilli, and J. M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them, Meccanica 15:9(1980).
D. Bernard, K. Gawędzki, and A. Kupiainen, Slow modes in passive advection, J. Stat. Phys. 90:519(1998).
A. S. Cattaneo, A. Gamba, and I. Kolokolov, Statistics of the one-electron current in a one-dimensional mesoscopic ring at arbitrary magnetic fields, J. Stat. Phys. 76:1065–1074 (1994).
M. Caselle, Distribution of transmission eigenvalues in disordered wires, Phys. Rev. Lett. 74:2776–2779 (1995).
M. Chertkov, G. Falkovich, and I. Kolokolov, Intermittent dissipation of passive scalar in turbulence, Phys. Rev. Lett. 80:2121–2124 (1998).
M. Chertkov, G. Falkovich, I. Kolokolov, and V. Lebedev, Statistics of a passive scalar advected by a large scale two-dimensional velocity field: Analytic solution, Phys. Rev. E 51:5609–5627 (1995).
M. Chertkov, G. Falkovich, I. Kolokolov, and M. Vergassola, Small-scale turbulent dynamo, Phys. Rev. Lett. 83:4065(1999).
M. Chertkov, A. Gamba, and I. Kolokolov, Exact field-theoretical description of passive scalar convection in an n-dimensional long range velocity field, Phys. Lett. A 192:435–443 (1994).
For a review of results, see, A. Crisanti, G. Paladin, and A. Vulpiani, Products of Random Matrices in Statistical Physics (Springer-Verlag, Berlin/Heidelberg, 1993).
O. N. Dorokhov, Transmission coefficient and the localization length of an electron in N bound disordered chains, JETP Lett. 36:318–321 (1982).
J.-P. Eckmann and O. Gat, Hydrodynamic Lyapunov modes in translation invariant systems, J. Stat. Phys. 98:775–798 (2000).
For a review of results, see, G. Falkovich, K. Gawędzki, and M. Vergassola, Particles and fields in fluid turbulence, Rev. Modern Phys. 73:913–975 (2001).
G. Falkovich, V. Kazakov, and V. Lebedev, Particle dispersion in a multidimensional random flow with arbitrary temporal correlations, Phys. A 249:36–46 (1998).
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
A. Gamba and I. Kolokolov, The Lyapunov spectrum of a continuous product of random matrices, J. Stat. Phys. 85:489–499 (1996).
A. Gamba and I. Kolokolov, Dissipation statistics of a passive scalar in a multidimensional smooth flow, J. Stat. Phys. 94:759–777 (1999).
I. M. Gel'fand and A. M. Yaglom, Integration in function spaces and its applications in quantum physics, J. Math. Phys. 1:48(1960).
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press, 1978).
S. Helgason, Groups and Geometric Analysis (Academic Press, 1984).
A. Hüffman, Disordered wires from a geometric viewpoint, J. Phys. A 23 :5733–5744 (1990).
K. Iwasawa, On some types of topological groups, Ann. of Math. 50:507–558 (1949).
A. W. Knapp, Representation Theory of Semisimple Groups (Princeton University Press, 1986).
A. W. Knapp, Lie Groups Beyond an Introduction (Birkhäuser, 1996).
I. V. Kolokolov, Functional representation for the partition function of the quantum Heisenberg ferromagnet, Phys. Lett. A 114:99–104 (1986).
I. V. Kolokolov, Functional integration for quantum magnets: new method and new results, Ann. Physics 202:165–185 (1990).
I. V. Kolokolov, The method of functional integration for one-dimensional localization, higher correlators, and the average current flowing in a mesoscopic ring in an arbitrary magnetic field, JETP 76:1099(1993).
V. I. Oseledec, The multiplicative ergodic theorem. The Lyapunov characteristic numbers of dynamical systems, Trans. Moscow Math. Soc. 19:197–231 (1968).
G. Paladin and A. Vulpiani, Scaling law and asymptotic distribution of Lyapunov exponents in conservative dynamical systems with many degrees of freedom, J. Phys. A 19:1881(1986).
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 1996).
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Gamba, A. Finite-Time Lyapunov Exponents for Products of Random Transformations. Journal of Statistical Physics 112, 193–218 (2003). https://doi.org/10.1023/A:1023631704909
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DOI: https://doi.org/10.1023/A:1023631704909