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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.

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