Per Dahlquist,
Royal Institute of Technology, Stockholm
Title:
Escape from intermittent repellers-
Periodic orbit theory for crossover from exponential to algebraic decay
Abstract:
We apply periodic orbit theory to study the asymptotic
distribution of escape times from an open intermittent map.
The survival probability can rigorously be bounded close to
sums over periodic orbits.
The dynamical zeta function exhibits a branch point which is associated
with
an asymptotic power law escape.
By an analytic continuation technique we compute a
pair of complex conjugate zeroes
beyond the branch point, associated with a
pre-asymptotic exponential decay.
Applications to conductance fluctuations in quantum dots
are discussed.
Thursday, November 18, 1999
Quantum Chaos Seminar
Aud C, 15:15
Stephen Creagh (with Niall Whelan),
University of Nottingham
Title:
Chaotic tunnelling: statistics and anomaly
Abstract:
In several dimensions, tunnelling rates across a potential
barrier are quite complex in nature, appearing almost random
as one moves from state to state. Certain features, however,
can often be simply characterised by the underlying classical
dynamics. Here we investigate statistical distributions
calculated on the basis of dynamical characteristics of the
complex classical orbit which crosses the barrier with
minumum imaginary action. The calculation uses
straightforward assumptions of random matrix theory
and works quite well in generic problems. Often, however,
there are strong deviations, and these seem to occur
when the complex tunnelling orbit has a real extensioon
which is periodic. It is proposed that the deviation is
connected ed to recent work by Kaplan which relates deviant
wavefunction statistics to scarring.
Abstract:
We consider the problem of correlation decay for systems
either non-uniformly hyperbolic (possessing marginally stable
structures) or not hyperbolic at all (polygonal billiards).
Numerical results, connections to transport and
non-equilibrium properties, and theoretical open questions will be
discussed.
Thursday, November 25, 1999
Quantum Chaos Seminar
Aud C, 16:30
Uzy Smilansky
Weizmann Institute, Israel
Title:
Graphs, scattering, statistics and what not
Abstract:
The universal "finger-prints" of classical chaos in quantum
dynamics are observed in the spectral statistics in bound systems, and the
cross-section (conductance) fluctuations in open systems. The same
phenomena occur also in quantum graphs (networks), which, in spite of their
simple construction, show the full complexity of fluctuations as in
generic Hamiltonian systems. Thus, quantum graphs are an excellent
paradigme for the study of the dynamical origins of the universal
finger-prints mentioned above. The Schroedinger operator on graphs will be
defined, its classical analogue will be shown to be chaotic, and the
corresponding trace formula will be derived. The fluctuations will be
analysed using these tools, and the relations to periodic orbit theory and
combinatorics will be illustrated.
Abstract:
The Path-Length Spectra (PLS) of the reflection amplitudes in mesoscopic
billiard systems attached to leads are studied. The PLS has peaks at the
length of classical trajectories starting and ending at the entrance
lead. These lengths are all positive numbers. Now we attach a
superconductor to this system. On the normal-superconducor interface
Andreev scattering occurs and electrons are retro-reflected as holes.
The holes follow the trajectories of electrons backward in time.
If in the billiard system diffractive scattering occurs the classical
trajectories are not uniquely defined. The holes retracing the electron
trajectories can follow paths different from the electron trajectory
after diffractive scattering. This makes it possible to create
diffractive periodic orbits with negative total actions, lengths and time
periods. We present a system where these strange new creatures can be
observed in the PLS.
Thursday, December 2, 1999
Quantum Chaos Seminar
Aud C, 15:15
Gregor Tanner
University of Nottingham
Title:
Graphs, combinatorics and random matrix statistics
Abstract:
Quantum graphs have recently been introduced as model systems to study the
spectral statistics of linear wave problems with chaotic classical limits.
I will generalise this approach by considering arbitrary, directed graphs
with unitary transfer matrices. A special class of graphs, so--called
binary graphs, is studied in more detail. For these, the conditions for
periodic orbit pairs to be correlated (including correlations due to the
unitarity of the transfer matrix) can be given explicitly. Using
combinatorial techniques it is possible to obtain expressions for
the form factor in terms of correlated periodic orbit pair contributions
for some low--dimensional cases. Gradual convergence towards random matrix
results is observed when increasing the number of vertices in the binary
graphs.
Chaos Seminar
Aud C, 16:30
Marc Lefranc
Université de Lille
Title:
Topological analysis of low-dimensional chaos: periodic
orbits, knot theory, and symbolic dynamics
Abstract:
It is by now well-known that unstable periodic orbits are invaluable
tools to understand and master chaotic behavior. One promising
approach is the template analysis proposed by Mindlin et al., and
based on earlier work by Birman and Williams. Unstable periodic orbits
in a 3D phase space are closed curves which can be characterized
through invariants from knot theory. Two key properties are that: (i)
the topological invariants of an UPO remain constant on the whole
domain of existence of an UPO because of the uniqueness theorem, (ii)
the topological organization of the UPO embedded in an attractor can
be globally described by means of a branched 2D manifold, a
``template''. We will review the basic concepts of template analysis
and show how it can be used to extract precise symbolic dynamical
information from a set of UPO. By construction, this approach yields
generating partitions which connect continuously to the
one-dimensional and hyperbolic symbolic encodings.