2,789 research outputs found
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
Fractal Dimensions of the Hydrodynamic Modes of Diffusion
We consider the time-dependent statistical distributions of diffusive
processes in relaxation to a stationary state for simple, two dimensional
chaotic models based upon random walks on a line. We show that the cumulative
functions of the hydrodynamic modes of diffusion form fractal curves in the
complex plane, with a Hausdorff dimension larger than one. In the limit of
vanishing wavenumber, we derive a simple expression of the diffusion
coefficient in terms of this Hausdorff dimension and the positive Lyapunov
exponent of the chaotic model.Comment: 20 pages, 6 figures, submitted to Nonlinearit
Time-resolved broadband Raman spectroscopies; A unified six-wave-mixing representation
Excited-state vibrational dynamics in molecules can be studied by an
electronically off-resonant Raman process induced by a probe pulse with
variable delay with respect to an actinic pulse. We establish the connection
between several variants of the technique that involve either spontaneous or
stimulated Raman detection and different pulse configurations. By using loop
diagrams in the frequency domain we show that all signals can be described as
six wave mixing which depend on the same four point molecular correlation
functions involving two transition dipoles and two polarizabilities and
accompanied by a different gating. Simulations for the stochastic
two-state-jump model illustrate the origin of the absorptive and dispersive
features observed experimentally
Broadband infrared and Raman probes of excited-state vibrational molecular dynamics; Simulation protocols based on loop diagram
Vibrational motions in electronically excited states can be observed by
either time and frequency resolved infrared absorption or by off resonant
stimulated Raman techniques. Multipoint correlation function expressions are
derived for both signals. Three representations for the signal which suggest
different simulation protocols are developed. These are based on the forward
and the backward propagation of the wavefunction, sum over state expansion
using an effective vibration Hamiltonian and a semiclassical treatment of a
bath. We show that the effective temporal () and spectral
() resolution of the techniques is not controlled solely by
experimental knobs but also depends on the system dynamics being probed. The
Fourier uncertainty is never violated
The Fractality of the Hydrodynamic Modes of Diffusion
Transport by normal diffusion can be decomposed into the so-called
hydrodynamic modes which relax exponentially toward the equilibrium state. In
chaotic systems with two degrees of freedom, the fine scale structure of these
hydrodynamic modes is singular and fractal. We characterize them by their
Hausdorff dimension which is given in terms of Ruelle's topological pressure.
For long-wavelength modes, we derive a striking relation between the Hausdorff
dimension, the diffusion coefficient, and the positive Lyapunov exponent of the
system. This relation is tested numerically on two chaotic systems exhibiting
diffusion, both periodic Lorentz gases, one with hard repulsive forces, the
other with attractive, Yukawa forces. The agreement of the data with the theory
is excellent
Chaotic Properties of Dilute Two and Three Dimensional Random Lorentz Gases II: Open Systems
We calculate the spectrum of Lyapunov exponents for a point particle moving
in a random array of fixed hard disk or hard sphere scatterers, i.e. the
disordered Lorentz gas, in a generic nonequilibrium situation. In a large
system which is finite in at least some directions, and with absorbing boundary
conditions, the moving particle escapes the system with probability one.
However, there is a set of zero Lebesgue measure of initial phase points for
the moving particle, such that escape never occurs. Typically, this set of
points forms a fractal repeller, and the Lyapunov spectrum is calculated here
for trajectories on this repeller. For this calculation, we need the solution
of the recently introduced extended Boltzmann equation for the nonequilibrium
distribution of the radius of curvature matrix and the solution of the standard
Boltzmann equation. The escape-rate formalism then gives an explicit result for
the Kolmogorov Sinai entropy on the repeller.Comment: submitted to Phys Rev
Ergodicity Breaking in a Deterministic Dynamical System
The concept of weak ergodicity breaking is defined and studied in the context
of deterministic dynamics. We show that weak ergodicity breaking describes a
weakly chaotic dynamical system: a nonlinear map which generates subdiffusion
deterministically. In the non-ergodic phase non-trivial distribution of the
fraction of occupation times is obtained. The visitation fraction remains
uniform even in the non-ergodic phase. In this sense the non-ergodicity is
quantified, leading to a statistical mechanical description of the system even
though it is not ergodic.Comment: 11 pages, 4 figure
Capturing correlations in chaotic diffusion by approximation methods
We investigate three different methods for systematically approximating the
diffusion coefficient of a deterministic random walk on the line which contains
dynamical correlations that change irregularly under parameter variation.
Capturing these correlations by incorporating higher order terms, all schemes
converge to the analytically exact result. Two of these methods are based on
expanding the Taylor-Green-Kubo formula for diffusion, whilst the third method
approximates Markov partitions and transition matrices by using the escape rate
theory of chaotic diffusion. We check the practicability of the different
methods by working them out analytically and numerically for a simple
one-dimensional map, study their convergence and critically discuss their
usefulness in identifying a possible fractal instability of parameter-dependent
diffusion, in case of dynamics where exact results for the diffusion
coefficient are not available.Comment: 11 pages, 5 figure
Time-oscillating Lyapunov modes and auto-correlation functions for quasi-one-dimensional systems
The time-dependent structure of the Lyapunov vectors corresponding to the
steps of Lyapunov spectra and their basis set representation are discussed for
a quasi-one-dimensional many-hard-disk systems. Time-oscillating behavior is
observed in two types of Lyapunov modes, one associated with the time
translational invariance and another with the spatial translational invariance,
and their phase relation is specified. It is shown that the longest period of
the Lyapunov modes is twice as long as the period of the longitudinal momentum
auto-correlation function. A simple explanation for this relation is proposed.
This result gives the first quantitative connection between the Lyapunov modes
and an experimentally accessible quantity.Comment: 4 pages, 3 figure
Chaotic properties of systems with Markov dynamics
We present a general approach for computing the dynamic partition function of
a continuous-time Markov process. The Ruelle topological pressure is identified
with the large deviation function of a physical observable. We construct for
the first time a corresponding finite Kolmogorov-Sinai entropy for these
processes. Then, as an example, the latter is computed for a symmetric
exclusion process. We further present the first exact calculation of the
topological pressure for an N-body stochastic interacting system, namely an
infinite-range Ising model endowed with spin-flip dynamics. Expressions for the
Kolmogorov-Sinai and the topological entropies follow.Comment: 4 pages, to appear in the Physical Review Letter
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