8 research outputs found
Gauss map and Lyapunov exponents of interacting particles in a billiard
We show that the Lyapunov exponent (LE) of periodic orbits with Lebesgue
measure zero from the Gauss map can be used to determine the main qualitative
behavior of the LE of a Hamiltonian system. The Hamiltonian system is a
one-dimensional box with two particles interacting via a Yukawa potential and
does not possess Kolmogorov-Arnold-Moser (KAM) curves. In our case the Gauss
map is applied to the mass ratio between particles. Besides
the main qualitative behavior, some unexpected peaks in the dependence
of the mean LE and the appearance of 'stickness' in phase space can also be
understand via LE from the Gauss map. This shows a nice example of the relation
between the "instability" of the continued fraction representation of a number
with the stability of non-periodic curves (no KAM curves) from the physical
model. Our results also confirm the intuition that pseudo-integrable systems
with more complicated invariant surfaces of the flow (higher genus) should be
more unstable under perturbation.Comment: 13 pages, 2 figure
Semiclassical Theory of Coulomb Blockade Peak Heights in Chaotic Quantum Dots
We develop a semiclassical theory of Coulomb blockade peak heights in chaotic
quantum dots. Using Berry's conjecture, we calculate the peak height
distributions and the correlation functions. We demonstrate that the
corrections to the corresponding results of the standard statistical theory are
non-universal and can be expressed in terms of the classical periodic orbits of
the dot that are well coupled to the leads. The main effect is an oscillatory
dependence of the peak heights on any parameter which is varied; it is
substantial for both symmetric and asymmetric lead placement. Surprisingly,
these dynamical effects do not influence the full distribution of peak heights,
but are clearly seen in the correlation function or power spectrum. For
non-zero temperature, the correlation function obtained theoretically is in
good agreement with that measured experimentally.Comment: 5 color eps figure
Approach to ergodicity in quantum wave functions
According to theorems of Shnirelman and followers, in the semiclassical limit
the quantum wavefunctions of classically ergodic systems tend to the
microcanonical density on the energy shell. We here develop a semiclassical
theory that relates the rate of approach to the decay of certain classical
fluctuations. For uniformly hyperbolic systems we find that the variance of the
quantum matrix elements is proportional to the variance of the integral of the
associated classical operator over trajectory segments of length , and
inversely proportional to , where is the Heisenberg
time, being the mean density of states. Since for these systems the
classical variance increases linearly with , the variance of the matrix
elements decays like . For non-hyperbolic systems, like Hamiltonians
with a mixed phase space and the stadium billiard, our results predict a slower
decay due to sticking in marginally unstable regions. Numerical computations
supporting these conclusions are presented for the bakers map and the hydrogen
atom in a magnetic field.Comment: 11 pages postscript and 4 figures in two files, tar-compressed and
uuencoded using uufiles, to appear in Phys Rev E. For related papers, see
http://www.icbm.uni-oldenburg.de/icbm/kosy/ag.htm