59 research outputs found
The nonequilibrium Ehrenfest gas: a chaotic model with flat obstacles?
It is known that the non-equilibrium version of the Lorentz gas (a billiard
with dispersing obstacles, electric field and Gaussian thermostat) is
hyperbolic if the field is small. Differently the hyperbolicity of the
non-equilibrium Ehrenfest gas constitutes an open problem, since its obstacles
are rhombi and the techniques so far developed rely on the dispersing nature of
the obstacles. We have developed analytical and numerical investigations which
support the idea that this model of transport of matter has both chaotic
(positive Lyapunov exponent) and non-chaotic steady states with a quite
peculiar sensitive dependence on the field and on the geometry, not observed
before. The associated transport behaviour is correspondingly highly irregular,
with features whose understanding is of both theoretical and technological
interest
Kinetic Theory Estimates for the Kolmogorov-Sinai Entropy and the Largest Lyapunov Exponents for Dilute, Hard-Ball Gases and for Dilute, Random Lorentz Gases
The kinetic theory of gases provides methods for calculating Lyapunov
exponents and other quantities, such as Kolmogorov-Sinai entropies, that
characterize the chaotic behavior of hard-ball gases. Here we illustrate the
use of these methods for calculating the Kolmogorov-Sinai entropy, and the
largest positive Lyapunov exponent, for dilute hard-ball gases in equilibrium.
The calculation of the largest Lyapunov exponent makes interesting connections
with the theory of propagation of hydrodynamic fronts. Calculations are also
presented for the Lyapunov spectrum of dilute, random Lorentz gases in two and
three dimensions, which are considerably simpler than the corresponding
calculations for hard-ball gases. The article concludes with a brief discussion
of some interesting open problems.Comment: 41 pages (REVTEX); 7 figs., 4 of which are included in LaTeX source.
(Fig.7 doesn't print well on some printers) This revised paper will appear in
"Hard Ball Systems and the Lorentz Gas", D. Szasz ed., Encyclopaedia of
Mathematical Sciences, Springe
Thermodynamics of a bouncer model: a simplified one-dimensional gas
Some dynamical properties of non interacting particles in a bouncer model are
described. They move under gravity experiencing collisions with a moving
platform. The evolution to steady state is described in two cases for
dissipative dynamics with inelastic collisions: (i) for large initial energy;
(ii) for low initial energy. For (i) we prove an exponential decay while for
(ii) a power law marked by a changeover to the steady state is observed. A
relation for collisions and time is obtained and allows us to write relevant
observables as temperature and entropy as function of either number of
collisions and time.Comment: 36 pages, 10 figures. To appear in: Communications in Nonlinear
Science and Numerical Simulation, 201
Quasistatic dynamical systems
We introduce the notion of a quasistatic dynamical system, which generalizes
that of an ordinary dynamical system. Quasistatic dynamical systems are
inspired by the namesake processes in thermodynamics, which are idealized
processes where the observed system transforms (infinitesimally) slowly due to
external influence, tracing out a continuous path of thermodynamic equilibria
over an (infinitely) long time span. Time-evolution of states under a
quasistatic dynamical system is entirely deterministic, but choosing the
initial state randomly renders the process a stochastic one. In the
prototypical setting where the time-evolution is specified by strongly chaotic
maps on the circle, we obtain a description of the statistical behaviour as a
stochastic diffusion process, under surprisingly mild conditions on the initial
distribution, by solving a well-posed martingale problem. We also consider
various admissible ways of centering the process, with the curious conclusion
that the "obvious" centering suggested by the initial distribution sometimes
fails to yield the expected diffusion.Comment: 40 page
Oscillating mushrooms: adiabatic theory for a non-ergodic system
Can elliptic islands contribute to sustained energy growth as parameters of a
Hamiltonian system slowly vary with time? In this paper we show that a mushroom
billiard with a periodically oscillating boundary accelerates the particle
inside it exponentially fast. We provide an estimate for the rate of
acceleration. Our numerical experiments confirms the theory. We suggest that a
similar mechanism applies to general systems with mixed phase space.Comment: final revisio
On the derivation of Fourier's law in stochastic energy exchange systems
We present a detailed derivation of Fourier's law in a class of stochastic
energy exchange systems that naturally characterize two-dimensional mechanical
systems of locally confined particles in interaction. The stochastic systems
consist of an array of energy variables which can be partially exchanged among
nearest neighbours at variable rates. We provide two independent derivations of
the thermal conductivity and prove this quantity is identical to the frequency
of energy exchanges. The first derivation relies on the diffusion of the
Helfand moment, which is determined solely by static averages. The second
approach relies on a gradient expansion of the probability measure around a
non-equilibrium stationary state. The linear part of the heat current is
determined by local thermal equilibrium distributions which solve a
Boltzmann-like equation. A numerical scheme is presented with computations of
the conductivity along our two methods. The results are in excellent agreement
with our theory.Comment: 19 pages, 5 figures, to appear in Journal of Statistical Mechanics
(JSTAT
Thermodynamic formalism for the Lorentz gas with open boundaries in dimensions
A Lorentz gas may be defined as a system of fixed dispersing scatterers, with
a single light particle moving among these and making specular collisions on
encounters with the scatterers. For a dilute Lorentz gas with open boundaries
in dimensions we relate the thermodynamic formalism to a random flight
problem. Using this representation we analytically calculate the central
quantity within this formalism, the topological pressure, as a function of
system size and a temperature-like parameter \ba. The topological pressure is
given as the sum of the topological pressure for the closed system and a
diffusion term with a \ba-dependent diffusion coefficient. From the
topological pressure we obtain the Kolmogorov-Sinai entropy on the repeller,
the topological entropy, and the partial information dimension.Comment: 7 pages, 5 figure
Diffusion in the Lorentz gas
The Lorentz gas, a point particle making mirror-like reflections from an
extended collection of scatterers, has been a useful model of deterministic
diffusion and related statistical properties for over a century. This survey
summarises recent results, including periodic and aperiodic models, finite and
infinite horizon, external fields, smooth or polygonal obstacles, and in the
Boltzmann-Grad limit. New results are given for several moving particles and
for obstacles with flat points. Finally, a variety of applications are
presented.Comment: 28 pages, 5 figure
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