15,082 research outputs found
Topics in chaotic dynamics
Various kinematical quantities associated with the statistical properties of
dynamical systems are examined: statistics of the motion, dynamical bases and
Lyapunov exponents. Markov partitons for chaotic systems, without any attempt
at describing ``optimal results''. The Ruelle principle is illustrated via its
relation with the theory of gases. An example of an application predicts the
results of an experiment along the lines of Evans, Cohen, Morriss' work on
viscosity fluctuations. A sequence of mathematically oriented problems
discusses the details of the main abstract ergodic theorems guiding to a proof
of Oseledec's theorem for the Lyapunov exponents and products of random
matricesComment: Plain TeX; compile twice; 30 pages; 140K Keywords: chaos,
nonequilibrium ensembles, Markov partitions, Ruelle principle, Lyapunov
exponents, random matrices, gaussian thermostats, ergodic theory, billiards,
conductivity, gas.
The prediction of future from the past: an old problem from a modern perspective
The idea of predicting the future from the knowledge of the past is quite
natural when dealing with systems whose equations of motion are not known. Such
a long-standing issue is revisited in the light of modern ergodic theory of
dynamical systems and becomes particularly interesting from a pedagogical
perspective due to its close link with Poincar\'e's recurrence. Using such a
connection, a very general result of ergodic theory - Kac's lemma - can be used
to establish the intrinsic limitations to the possibility of predicting the
future from the past. In spite of a naive expectation, predictability results
to be hindered rather by the effective number of degrees of freedom of a system
than by the presence of chaos. If the effective number of degrees of freedom
becomes large enough, regardless the regular or chaotic nature of the system,
predictions turn out to be practically impossible. The discussion of these
issues is illustrated with the help of the numerical study of simple models.Comment: 9 pages, 4 figure
Statistical properties of eigenvalues for an operating quantum computer with static imperfections
We investigate the transition to quantum chaos, induced by static
imperfections, for an operating quantum computer that simulates efficiently a
dynamical quantum system, the sawtooth map. For the different dynamical regimes
of the map, we discuss the quantum chaos border induced by static imperfections
by analyzing the statistical properties of the quantum computer eigenvalues.
For small imperfection strengths the level spacing statistics is close to the
case of quasi-integrable systems while above the border it is described by the
random matrix theory. We have found that the border drops exponentially with
the number of qubits, both in the ergodic and quasi-integrable dynamical
regimes of the map characterized by a complex phase space structure. On the
contrary, the regime with integrable map dynamics remains more stable against
static imperfections since in this case the border drops only algebraically
with the number of qubits.Comment: 9 pages, 10 figure
No-slip Billiards
We investigate the dynamics of no-slip billiards, a model in which small rotating disks may exchange linear and angular momentum at collisions with the boundary. A general theory of rigid body collisions in is developed, which returns the known dimension two model as a special case but generalizes to higher dimensions. We give new results on periodicity and boundedness of orbits which suggest that a class of billiards (including all polygons) is not ergodic. Computer generated phase portraits demonstrate non-ergodic features, suggesting chaotic no-slip billiards cannot easily be constructed using the common techniques for generating chaos in standard billiards. However, Sinai type dispersing billiards, which are always ergodic in the case of standard billiards, appear to be ergodic above a certain curvature threshold
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