1,788 research outputs found
Linear and fractal diffusion coefficients in a family of one dimensional chaotic maps
We analyse deterministic diffusion in a simple, one-dimensional setting
consisting of a family of four parameter dependent, chaotic maps defined over
the real line. When iterated under these maps, a probability density function
spreads out and one can define a diffusion coefficient. We look at how the
diffusion coefficient varies across the family of maps and under parameter
variation. Using a technique by which Taylor-Green-Kubo formulae are evaluated
in terms of generalised Takagi functions, we derive exact, fully analytical
expressions for the diffusion coefficients. Typically, for simple maps these
quantities are fractal functions of control parameters. However, our family of
four maps exhibits both fractal and linear behavior. We explain these different
structures by looking at the topology of the Markov partitions and the ergodic
properties of the maps.Comment: 21 pages, 19 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
Thermostating by Deterministic Scattering: Heat and Shear Flow
We apply a recently proposed novel thermostating mechanism to an interacting
many-particle system where the bulk particles are moving according to
Hamiltonian dynamics. At the boundaries the system is thermalized by
deterministic and time-reversible scattering. We show how this scattering
mechanism can be related to stochastic boundary conditions. We subsequently
simulate nonequilibrium steady states associated to thermal conduction and
shear flow for a hard disk fluid. The bulk behavior of the model is studied by
comparing the transport coefficients obtained from computer simulations to
theoretical results. Furthermore, thermodynamic entropy production and
exponential phase-space contraction rates in the stationary nonequilibrium
states are calculated showing that in general these quantities do not agree.Comment: 16 pages (revtex) with 9 figures (postscript
Thermostating by deterministic scattering: the periodic Lorentz gas
We present a novel mechanism for thermalizing a system of particles in
equilibrium and nonequilibrium situations, based on specifically modeling
energy transfer at the boundaries via a microscopic collision process. We apply
our method to the periodic Lorentz gas, where a point particle moves
diffusively through an ensemble of hard disks arranged on a triangular lattice.
First, collision rules are defined for this system in thermal equilibrium. They
determine the velocity of the moving particle such that the system is
deterministic, time reversible, and microcanonical. These collision rules can
systematically be adapted to the case where one associates arbitrarily many
degrees of freedom to the disk, which here acts as a boundary. Subsequently,
the system is investigated in nonequilibrium situations by applying an external
field. We show that in the limit where the disk is endowed by infinitely many
degrees of freedom it acts as a thermal reservoir yielding a well-defined
nonequilibrium steady state. The characteristic properties of this state, as
obtained from computer simulations, are finally compared to the ones of the
so-called Gaussian thermostated driven Lorentz gas.Comment: 13 pages (revtex) with 10 figures (encapsulated postscript
Thermostatting by deterministic scattering
We present a mechanism for thermalizing a moving particle by microscopic
deterministic scattering. As an example, we consider the periodic Lorentz gas.
We modify the collision rules by including energy transfer between particle and
scatterer such that the scatterer mimics a thermal reservoir with arbitrarily
many degrees of freedom. The complete system is deterministic, time-reversible,
and provides a microcanonical density in equilibrium. In the limit of the disk
representing infinitely many degrees of freedom and by applying an electric
field the system goes into a nonequilibrium steady state.Comment: 4 pages (revtex) with 4 figures (postscript
The Nose-hoover thermostated Lorentz gas
We apply the Nose-Hoover thermostat and three variations of it, which control
different combinations of velocity moments, to the periodic Lorentz gas.
Switching on an external electric field leads to nonequilibrium steady states
for the four models with a constant average kinetic energy of the moving
particle. We study the probability density, the conductivity and the attractor
in nonequilibrium and compare the results to the Gaussian thermostated Lorentz
gas and to the Lorentz gas as thermostated by deterministic scattering.Comment: 7 pages (revtex) with 10 figures (postscript), most of the figures
are bitmapped with low-resolution. The originals are many MB, they can be
obtained upon reques
Fractal dimension of transport coefficients in a deterministic dynamical system
In many low-dimensional dynamical systems transport coefficients are very
irregular, perhaps even fractal functions of control parameters. To analyse
this phenomenon we study a dynamical system defined by a piece-wise linear map
and investigate the dependence of transport coefficients on the slope of the
map. We present analytical arguments, supported by numerical calculations,
showing that both the Minkowski-Bouligand and Hausdorff fractal dimension of
the graphs of these functions is 1 with a logarithmic correction, and find that
the exponent controlling this correction is bounded from above by 1 or
2, depending on some detailed properties of the system. Using numerical
techniques we show local self-similarity of the graphs. The local
self-similarity scaling transformations turn out to depend (irregularly) on the
values of the system control parameters.Comment: 17 pages, 6 figures; ver.2: 18 pages, 7 figures (added section 5.2,
corrected typos, etc.
Understanding Anomalous Transport in Intermittent Maps: From Continuous Time Random Walks to Fractals
We show that the generalized diffusion coefficient of a subdiffusive
intermittent map is a fractal function of control parameters. A modified
continuous time random walk theory yields its coarse functional form and
correctly describes a dynamical phase transition from normal to anomalous
diffusion marked by strong suppression of diffusion. Similarly, the probability
density of moving particles is governed by a time-fractional diffusion equation
on coarse scales while exhibiting a specific fine structure. Approximations
beyond stochastic theory are derived from a generalized Taylor-Green-Kubo
formula.Comment: 4 pages, 3 eps figure
Logarithmic oscillators: ideal Hamiltonian thermostats
A logarithmic oscillator (in short, log-oscillator) behaves like an ideal
thermostat because of its infinite heat capacity: when it weakly couples to
another system, time averages of the system observables agree with ensemble
averages from a Gibbs distribution with a temperature T that is given by the
strength of the logarithmic potential. The resulting equations of motion are
Hamiltonian and may be implemented not only in a computer but also with
real-world experiments, e.g., with cold atoms.Comment: 5 pages, 3 figures. v4: version accepted in Phys. Rev. Let
Persistence effects in deterministic diffusion
In systems which exhibit deterministic diffusion, the gross parameter
dependence of the diffusion coefficient can often be understood in terms of
random walk models. Provided the decay of correlations is fast enough, one can
ignore memory effects and approximate the diffusion coefficient according to
dimensional arguments. By successively including the effects of one and two
steps of memory on this approximation, we examine the effects of
``persistence'' on the diffusion coefficients of extended two-dimensional
billiard tables and show how to properly account for these effects, using walks
in which a particle undergoes jumps in different directions with probabilities
that depend on where they came from.Comment: 7 pages, 7 figure
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