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Relaxation rates in chaotic and quasiperiodic systems
1.System characteristics which are responsible for chaotic vs regular motion are discussed in, for example, A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer, New York, 1983).
2.M. L. Koszykowski, D. W. Noid, M. Tabor, and R. A. Marcus, J. Chem. Phys. 74, 2530 (1981).
3.J. D. Meiss, J. R. Cary, C. Grebogi, J. D. Crawford, A. N. Kaufman, and H. D. I. Abarbanel, Physica D 6, 375 (1983).
4.M. Tabor, Adv. Chem. Phys. 46, 73 (1981).
5.See, for example, the comparison of quantum and classical relaxation in the classically chaotic stadium: K. Christoffel and P. Brumer, Phys. Rev. A 33, 1309 (1986).
6.C. Jaffe and P. Brumer, J. Phys. Chem. 88, 4829 (1984).
7.Ya G. Sinai, Acta Phys. Austraica Suppl. X, 575 (1973).
8.(a) V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Benjamin, New York, 1968);
8.For a qualitative introduction see P. Brumer, Adv. Chem. Phys. 47, 201 (1981);
8.(b) For a mathematical discussion of Anosov Systems see R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Springer, New York, 1975).
9.G. M. Zaslavsky, Chaos in Dynamic Systems (Harwood Academic, 1985).
10.J. D. Crawford and J. R. Cary, Physica D 6, 233 (1983).
11.K. D. Hansel, Chem. Phys. 33, 35 (1978);
11.S. C. Farantos and J. N. Murrell, Chem. Phys. 55, 205 (1981)., Chem. Phys.
12.I. Hamilton and P. Brumer, J. Chem. Phys. 82, 1937 (1985).
13.I. Hamilton and P. Brumer, J. Chem. Phys. 78, 2682 (1983).
14.N. Wiener, Acta Math. 55, 117 (1930);
14.A. Khinchin, Math. Ann. 109, 604 (1934).
15.D. W. Noid, M. L. Koszykowski, and R. A. Marcus, J. Chem. Phys. 67, 404 (1977);
15.G. E. Powell and I. C. Percival, J. Phys. A 12, 2053 (1979);
15.S. Blacher and J. Perdang, Physica D 3, 512 (1981).
16.R. S. Dumont and P. Brumer (to be published).
17.V. I. Arnold, Ordinary Differential Equations (MIT, Cambridge, 1973).
18.Behavior at other values of η are also of interest. The maps in Eq. (2) are quasiperiodic for In the first of these cases there are straight line orbits in Fourier space as well as period‐two orbits. In the latter three cases orbits are periodic on ellipses. Chaotic behavior, reflected in hyperbolic orbits, occurs for and
19.Consider the Gaussian phase space distribution given by where B is symmetric and has unit determinant. The width of this distribution along some direction v is well described by the parameter The corresponding Fourier space distribution is given by which has width along an arbitrary direction of To define the conjugate direction consider ξ, the larger eigenvalue of B, and let φ and define the directions of v and with respect to the corresponding eigenvector. Then for tan we have Thus, with one vector of given, this relation between φ and defines the direction v conjugate to v.
20.The Jacobian of the transformation from to is det Although the Jacobian is greater than unity for the transformation is many valued. When is tansformed to y its range of integration expands into many copies of Contracting the range of integration back down into and using periodicity one finds that the number of repetitions exactly cancels the Jacobian, giving Eq. (14).
21.R. S. Dumont, Ph.D. dissertation, University of Toronto (1987).
22.R. S. Dumont and P. Brumer, J. Phys. Chem. 90, 3509 (1986).
23.See, for example, the OCS computation in D. Carter and P. Brumer, J. Chem. Phys. 77, 4208 (1982)
23.[Erratum: J. Chem. Phys. 78, 2104 (1983)]. We also remark that one additional distinction may be noted. That is, that the molecular system often displays a “divided phase space,” i.e., regions of chaotic behavior as well as regions of quasiperiodic motion. This difficulty is easily dealt with by treating the dynamics independently within these regions.
24.See, for example, the discussion of Pesin’s formula for in J. P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
25.See the discussion in Sec. IV of the rapid relaxation subsequent to first return for localized states and Cat Map dynamics.
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