4,287 research outputs found
Characterizing Weak Chaos using Time Series of Lyapunov Exponents
We investigate chaos in mixed-phase-space Hamiltonian systems using time
series of the finite- time Lyapunov exponents. The methodology we propose uses
the number of Lyapunov exponents close to zero to define regimes of ordered
(stickiness), semi-ordered (or semi-chaotic), and strongly chaotic motion. The
dynamics is then investigated looking at the consecutive time spent in each
regime, the transition between different regimes, and the regions in the
phase-space associated to them. Applying our methodology to a chain of coupled
standard maps we obtain: (i) that it allows for an improved numerical
characterization of stickiness in high-dimensional Hamiltonian systems, when
compared to the previous analyses based on the distribution of recurrence
times; (ii) that the transition probabilities between different regimes are
determined by the phase-space volume associated to the corresponding regions;
(iii) the dependence of the Lyapunov exponents with the coupling strength.Comment: 8 pages, 6 figure
Non-Hamiltonian dynamics in optical microcavities resulting from wave-inspired corrections to geometric optics
We introduce and investigate billiard systems with an adjusted ray dynamics
that accounts for modifications of the conventional reflection of rays due to
universal wave effects. We show that even small modifications of the specular
reflection law have dramatic consequences on the phase space of classical
billiards. These include the creation of regions of non-Hamiltonian dynamics,
the breakdown of symmetries, and changes in the stability and morphology of
periodic orbits. Focusing on optical microcavities, we show that our adjusted
dynamics provides the missing ray counterpart to previously observed wave
phenomena and we describe how to observe its signatures in experiments. Our
findings also apply to acoustic and ultrasound waves and are important in all
situations where wavelengths are comparable to system sizes, an increasingly
likely situation considering the systematic reduction of the size of electronic
and photonic devices.Comment: 6 pages, 4 figures, final published versio
Langevin Thermostat for Rigid Body Dynamics
We present a new method for isothermal rigid body simulations using the
quaternion representation and Langevin dynamics. It can be combined with the
traditional Langevin or gradient (Brownian) dynamics for the translational
degrees of freedom to correctly sample the NVT distribution in a simulation of
rigid molecules. We propose simple, quasi-symplectic second-order numerical
integrators and test their performance on the TIP4P model of water. We also
investigate the optimal choice of thermostat parameters.Comment: 15 pages, 13 figures, 1 tabl
Poincare recurrences and transient chaos in systems with leaks
In order to simulate observational and experimental situations, we consider a
leak in the phase space of a chaotic dynamical system. We obtain an expression
for the escape rate of the survival probability applying the theory of
transient chaos. This expression improves previous estimates based on the
properties of the closed system and explains dependencies on the position and
size of the leak and on the initial ensemble. With a subtle choice of the
initial ensemble, we obtain an equivalence to the classical problem of Poincare
recurrences in closed systems, which is treated in the same framework. Finally,
we show how our results apply to weakly chaotic systems and justify a split of
the invariant saddle in hyperbolic and nonhyperbolic components, related,
respectively, to the intermediate exponential and asymptotic power-law decays
of the survival probability.Comment: Corrected version, as published. 12 pages, 9 figure
New Langevin and Gradient Thermostats for Rigid Body Dynamics
We introduce two new thermostats, one of Langevin type and one of gradient
(Brownian) type, for rigid body dynamics. We formulate rotation using the
quaternion representation of angular coordinates; both thermostats preserve the
unit length of quaternions. The Langevin thermostat also ensures that the
conjugate angular momenta stay within the tangent space of the quaternion
coordinates, as required by the Hamiltonian dynamics of rigid bodies. We have
constructed three geometric numerical integrators for the Langevin thermostat
and one for the gradient thermostat. The numerical integrators reflect key
properties of the thermostats themselves. Namely, they all preserve the unit
length of quaternions, automatically, without the need of a projection onto the
unit sphere. The Langevin integrators also ensure that the angular momenta
remain within the tangent space of the quaternion coordinates. The Langevin
integrators are quasi-symplectic and of weak order two. The numerical method
for the gradient thermostat is of weak order one. Its construction exploits
ideas of Lie-group type integrators for differential equations on manifolds. We
numerically compare the discretization errors of the Langevin integrators, as
well as the efficiency of the gradient integrator compared to the Langevin ones
when used in the simulation of rigid TIP4P water model with smoothly truncated
electrostatic interactions. We observe that the gradient integrator is
computationally less efficient than the Langevin integrators. We also compare
the relative accuracy of the Langevin integrators in evaluating various static
quantities and give recommendations as to the choice of an appropriate
integrator.Comment: 16 pages, 4 figure
Irreducible Representations of Diperiodic Groups
The irreducible representations of all of the 80 diperiodic groups, being the
symmetries of the systems translationally periodical in two directions, are
calculated. To this end, each of these groups is factorized as the product of a
generalized translational group and an axial point group. The results are
presented in the form of the tables, containing the matrices of the irreducible
representations of the generators of the groups. General properties and some
physical applications (degeneracy and topology of the energy bands, selection
rules, etc.) are discussed.Comment: 30 pages, 5 figures, 28 tables, 18 refs, LaTex2.0
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