By J. P. Keating (auth.), Igor V. Lerner, Jonathan P. Keating, David E. Khmelnitskii (eds.)
The movement of a particle in a random power in or extra dimensions is chaotic, and the trajectories in deterministically chaotic structures are successfully random. it really is for this reason no shock that there are hyperlinks among the quantum homes of disordered structures and people of easy chaotic platforms. The query is, how deep do the connec tions cross? And to what volume do the mathematical innovations designed to appreciate one challenge result in new insights into the opposite? The canonical challenge within the thought of disordered mesoscopic platforms is that of a particle relocating in a random array of scatterers. the purpose is to calculate the statistical houses of, for instance, the quantum strength degrees, wavefunctions, and conductance fluctuations through averaging over diverse arrays; that's, by way of averaging over an ensemble of alternative realizations of the random strength. In a few regimes, comparable to strength scales which are huge in comparison to the suggest point spacing, this is often performed utilizing diagrammatic perturbation concept. In others, the place the discreteness of the quantum spectrum turns into vital, such an process fails. A extra strong technique, devel oped by way of Efetov, consists of representing correlation capabilities when it comes to a supersymmetric nonlinear sigma-model. this is applicable over a much wider diversity of strength scales, overlaying either the perturbative and non-perturbative regimes. It was once proved utilizing this system that strength point correlations in disordered structures coincide with these of random matrix conception whilst the dimensionless conductance has a tendency to infinity.
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Additional info for Supersymmetry and Trace Formulae: Chaos and Disorder
A stationary phase saddle-point of the a-model can be identified at Q = ajr ® aff . A surprising connection between the perturbation theory in the vicinity of this saddle-point and the conventional saddle-point was recently stressed by Andreev and Altshuler 56 . Zero-Mode Having examined the short-time perturbative dynamics within the framework of diagrammatic perturbation theory, we now switch attention to consider the long-time non-perturbative limit. In particular, taking 0 «: Ec = liD / L2, contributions to the effective action involving spatial fluctuations of Q are strongly suppressed.
Remarkably, as the strength of the regulator is taken to zero, the spectrum, bn}, of the resulting operator, known in the literature as the Perron-Frobenius operator, reflects intrinsic irreversible properties of the purely classical dynamics61 , 62, 63, 64, 65. In ergodic systems, the leading eigenvalue 'Yo = 0 is non-degenerate, and manifests the conservation of probability density. Thus any initial density relaxes, in the course of time, to the state associated with 'Yo. If, in addition, this relaxation is exponential (a) Figure 16.
Drdr'fJ(lr - r'Dstr [8Q~(r)8Q~(r')] . Since the Q-matrices vary slowly over the length scale l at which the average Green function decays, this facilitates a gradient expansion. Using the identity ! dr/lOrD :;v' = we find that, in the leading approximation, the action of the transverse fluctuations vanishes, while the action for the longitudinal fluctuations takes the form S 1(2) 36 _ - ! 4r str ['Q 12 7rnv u 1 • (48) Figure 13. Schematic diagram of the structure of the saddle-point manifold (a) in the absence, and (b) in the presence of the symmetry breaking source O.