3,837 research outputs found
On the parametric dependences of a class of non-linear singular maps
We discuss a two-parameter family of maps that generalize piecewise linear,
expanding maps of the circle. One parameter measures the effect of a
non-linearity which bends the branches of the linear map. The second parameter
rotates points by a fixed angle. For small values of the nonlinearity
parameter, we compute the invariant measure and show that it has a singular
density to first order in the nonlinearity parameter. Its Fourier modes have
forms similar to the Weierstrass function. We discuss the consequences of this
singularity on the Lyapunov exponents and on the transport properties of the
corresponding multibaker map. For larger non-linearities, the map becomes
non-hyperbolic and exhibits a series of period-adding bifurcations.Comment: 17 pages, 13 figures, to appear in Discrete and Continuous Dynamical
Systems, series B Higher resolution versions of Figures 5 downloadable at
http://www.glue.umd.edu/~jrd
Chaotic Scattering Theory of Transport and Reaction-Rate Coefficients
The chaotic scattering theory is here extended to obtain escape-rate
expressions for the transport coefficients appropriate for a simple classical
fluid, or for a chemically reacting system. This theory allows various
transport coefficients such as the coefficients of viscosity, thermal
conductivity, etc., to be expressed in terms of the positive Lyapunov exponents
and Kolmogorov-Sinai entropy of a set of phase space trajectories that take
place on an appropriate fractal repeller. This work generalizes the previous
results of Gaspard and Nicolis for the coefficient of diffusion of a particle
moving in a fixed array of scatterers.Comment: 27 pages LaTeX, no figure
Lyapunov spreading of semi-classical wave packets for the Lorentz Gas: theory and applications
We consider the quantum mechanical propagator for a particle moving in a -dimensional Lorentz gas, with fixed, hard sphere scatterers. To evaluate this propagator in the semi-classical region, and for times less than the Ehrenfest time, we express its effect on an initial Gaussian wave packet in terms of quantities analogous to those used to describe the exponential separation of trajectories in the classical version of this system. This result relates the spread of the wave packet to the rate of separation of classical trajectories, characterized by positive Lyapunov exponents. We consider applications of these results, first to illustrate the behavior of the wave-packet auto-correlation functions for wave packets on periodic orbits. The auto-correlation function can be related to the fidelity, or Loschmidt echo, for the special case that the perturbation is a small change in the mass of the particle. An exact expression for the fidelity, appropriate for this perturbation, leads to an analytical result valid over very long time intervals, inversely proportional to the size of the mass perturbation. For such perturbations, we then calculate the long-time echo for semi-classical wave packets on periodic orbits. This paper also corrects an earlier calculation for a quantum echo, included in a previous version of this paper. We explain the reasons for this correction
Crossover from Diffusive to Ballistic Transport in Periodic Quantum Maps
We derive an expression for the mean square displacement of a particle whose
motion is governed by a uniform, periodic, quantum multi-baker map. The
expression is a function of both time, , and Planck's constant, , and
allows a study of both the long time, , and semi-classical,
, limits taken in either order. We evaluate the expression using
random matrix theory as well as numerically, and observe good agreement between
both sets of results. The long time limit shows that particle transport is
generically ballistic, for any fixed value of Planck's constant. However, for
fixed times, the semi-classical limit leads to diffusion. The mean square
displacement for non-zero Planck's constant, and finite time, exhibits a
crossover from diffusive to ballistic motion, with crossover time on the order
of the inverse of Planck's constant. We argue, that these results are generic
for a large class of 1D quantum random walks, similar to the quantum
multi-baker, and that a sufficient condition for diffusion in the
semi-classical limit is classically chaotic dynamics in each cell. Some
connections between our work and the other literature on quantum random walks
are discussed. These walks are of some interest in the theory of quantum
computation.Comment: Final version to appear in Physica D, Proceedings of the
International Workshop and Seminar on Microscopic Chaos and Transport in
Many-Particle Systems, Dresden, 2002; corrected a minor error in section 3.1,
new section 4.
A Note on the Ruelle Pressure for a Dilute Disordered Sinai Billiard
The topological pressure is evaluated for a dilute random Lorentz gas, in the
approximation that takes into account only uncorrelated collisions between the
moving particle and fixed, hard sphere scatterers. The pressure is obtained
analytically as a function of a temperature-like parameter, beta, and of the
density of scatterers. The effects of correlated collisions on the topological
pressure can be described qualitatively, at least, and they significantly
modify the results obtained by considering only uncorrelated collision
sequences. As a consequence, for large systems, the range of beta-values over
which our expressions for the topological pressure are valid becomes very
small, approaching zero, in most cases, as the inverse of the logarithm of
system size.Comment: 15 pages RevTeX with 2 figures. Final version with some typo's
correcte
On thermostats and entropy production
The connection between the rate of entropy production and the rate of phase
space contraction for thermostatted systems in nonequilibrium steady states is
discussed for a simple model of heat flow in a Lorentz gas, previously
described by Spohn and Lebowitz. It is easy to show that for the model
discussed here the two rates are not connected, since the rate of entropy
production is non-zero and positive, while the overall rate of phase space
contraction is zero. This is consistent with conclusions reached by other
workers. Fractal structures appear in the phase space for this model and their
properties are discussed. We conclude with a discussion of the implications of
this and related work for understanding the role of chaotic dynamics and
special initial conditions for an explanation of the Second Law of
Thermodynamics.Comment: 14 pages, 1 figur
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