9 research outputs found
Expanding direction of the period doubling operator
We prove that the period doubling operator has an expanding direction at the
fixed point. We use the induced operator, a ``Perron-Frobenius type operator'',
to study the linearization of the period doubling operator at its fixed point.
We then use a sequence of linear operators with finite ranks to study this
induced operator. The proof is constructive. One can calculate the expanding
direction and the rate of expansion of the period doubling operator at the
fixed point
Classical and Quantum Dynamics in a Random Magnetic Field
Using the supersymmetry approach, we study spectral statistical properties of
a two-dimensional quantum particle subject to a non-uniform magnetic field. We
focus mainly on the problem of regularisation of the field theory. Our analysis
begins with an investigation of the spectral properties of the purely classical
evolution operator. We show that, although the kinetic equation is formally
time-reversible, density relaxation is controlled by {\em irreversible}
classical dynamics. In the case of a weak magnetic field, the effective kinetic
operator corresponds to diffusion in the angle space, the diffusion constant
being determined by the spectral resolution of the inhomogeneous magnetic
field. Applying these results to the quantum problem, we demonstrate that the
low-lying modes of the field theory are related to the eigenmodes of the
irreversible classical dynamics, and the higher modes are separated from the
zero mode by a gap associated with the lowest density relaxation rate. As a
consequence, we find that the long-time properties of the system are
characterised by universal Wigner-Dyson statistics. For a weak magnetic field,
we obtain a description in terms of the quasi one-dimensional non-linear
-model.Comment: 16 pages, RevTe