475 research outputs found
Fixed Point and Aperiodic Tilings
An aperiodic tile set was first constructed by R.Berger while proving the
undecidability of the domino problem. It turned out that aperiodic tile sets
appear in many topics ranging from logic (the Entscheidungsproblem) to physics
(quasicrystals) We present a new construction of an aperiodic tile set that is
based on Kleene's fixed-point construction instead of geometric arguments. This
construction is similar to J. von Neumann self-reproducing automata; similar
ideas were also used by P. Gacs in the context of error-correcting
computations. The flexibility of this construction allows us to construct a
"robust" aperiodic tile set that does not have periodic (or close to periodic)
tilings even if we allow some (sparse enough) tiling errors. This property was
not known for any of the existing aperiodic tile sets.Comment: v5: technical revision (positions of figures are shifted
Noise Induced Complexity: From Subthreshold Oscillations to Spiking in Coupled Excitable Systems
We study stochastic dynamics of an ensemble of N globally coupled excitable
elements. Each element is modeled by a FitzHugh-Nagumo oscillator and is
disturbed by independent Gaussian noise. In simulations of the Langevin
dynamics we characterize the collective behavior of the ensemble in terms of
its mean field and show that with the increase of noise the mean field displays
a transition from a steady equilibrium to global oscillations and then, for
sufficiently large noise, back to another equilibrium. Diverse regimes of
collective dynamics ranging from periodic subthreshold oscillations to
large-amplitude oscillations and chaos are observed in the course of this
transition. In order to understand details and mechanisms of noise-induced
dynamics we consider a thermodynamic limit of the ensemble, and
derive the cumulant expansion describing temporal evolution of the mean field
fluctuations. In the Gaussian approximation this allows us to perform the
bifurcation analysis; its results are in good agreement with dynamical
scenarios observed in the stochastic simulations of large ensembles
Excitable elements controlled by noise and network structure
We study collective dynamics of complex networks of stochastic excitable
elements, active rotators. In the thermodynamic limit of infinite number of
elements, we apply a mean-field theory for the network and then use a Gaussian
approximation to obtain a closed set of deterministic differential equations.
These equations govern the order parameters of the network. We find that a
uniform decrease in the number of connections per element in a homogeneous
network merely shifts the bifurcation thresholds without producing qualitative
changes in the network dynamics. In contrast, heterogeneity in the number of
connections leads to bifurcations in the excitable regime. In particular we
show that a critical value of noise intensity for the saddle-node bifurcation
decreases with growing connectivity variance. The corresponding critical values
for the onset of global oscillations (Hopf bifurcation) show a non-monotone
dependency on the structural heterogeneity, displaying a minimum at moderate
connectivity variances.Comment: 13 pages, 6 figure
Waiting time distributions for clusters of complex molecules
Waiting time distributions are in the core of theories for a large variety of subjects ranging from the analysis of patch clamp records to stochastic excitable systems. Here, we present a novel exact method for the calculation of waiting time distributions for state transitions of complex molecules with independent subunit dynamics. The absorbing state is a specific set of subunit states, i.e. is defined on the molecule level. Consequently, we formulate the problem as a random walk in the molecule state space. The subunits can possess an arbitrary number of states and any topology of transitions between them. The method circumvents problems arising from combinatorial explosion due to subunit coupling and requires solutions of the subunit master equation only
Shuffling cards, factoring numbers, and the quantum baker's map
It is pointed out that an exactly solvable permutation operator, viewed as
the quantization of cyclic shifts, is useful in constructing a basis in which
to study the quantum baker's map, a paradigm system of quantum chaos. In the
basis of this operator the eigenfunctions of the quantum baker's map are
compressed by factors of around five or more. We show explicitly its connection
to an operator that is closely related to the usual quantum baker's map. This
permutation operator has interesting connections to the art of shuffling cards
as well as to the quantum factoring algorithm of Shor via the quantum order
finding one. Hence we point out that this well-known quantum algorithm makes
crucial use of a quantum chaotic operator, or at least one that is close to the
quantization of the left-shift, a closeness that we also explore
quantitatively.Comment: 12 pgs. Substantially elaborated version, including a new route to
the quantum bakers map. To appear in J. Phys.
Detectability of non-differentiable generalized synchrony
Generalized synchronization of chaos is a type of cooperative behavior in
directionally-coupled oscillators that is characterized by existence of stable
and persistent functional dependence of response trajectories from the chaotic
trajectory of driving oscillator. In many practical cases this function is
non-differentiable and has a very complex shape. The generalized synchrony in
such cases seems to be undetectable, and only the cases, in which a
differentiable synchronization function exists, are considered to make sense in
practice. We show that this viewpoint is not always correct and the
non-differentiable generalized synchrony can be revealed in many practical
cases. Conditions for detection of generalized synchrony are derived
analytically, and illustrated numerically with a simple example of
non-differentiable generalized synchronization.Comment: 8 pages, 8 figures, submitted to PR
Derivation of the Effective Chiral Lagrangian for Pseudoscalar Mesons from QCD
We formally derive the chiral Lagrangian for low lying pseudoscalar mesons
from the first principles of QCD considering the contributions from the normal
part of the theory without taking approximations. The derivation is based on
the standard generating functional of QCD in the path integral formalism. The
gluon-field integration is formally carried out by expressing the result in
terms of physical Green's functions of the gluon. To integrate over the
quark-field, we introduce a bilocal auxiliary field Phi(x,y) representing the
mesons. We then develop a consistent way of extracting the local pseudoscalar
degree of freedom U(x) in Phi(x,y) and integrating out the rest degrees of
freedom such that the complete pseudoscalar degree of freedom resides in U(x).
With certain techniques, we work out the explicit U(x)-dependence of the
effective action up to the p^4-terms in the momentum expansion, which leads to
the desired chiral Lagrangian in which all the coefficients contributed from
the normal part of the theory are expressed in terms of certain Green's
functions in QCD. Together with the existing QCD formulae for the anomaly
contributions, the present results leads to the complete QCD definition of the
coefficients in the chiral Lagrangian. The relation between the present QCD
definition of the p^2-order coefficient F_0^2 and the well-known approximate
result given by Pagels and Stokar is discussed.Comment: 16 pages in RevTex, some typos are corrected, version for publication
in Phys. Rev.
Noise Can Reduce Disorder in Chaotic Dynamics
We evoke the idea of representation of the chaotic attractor by the set of
unstable periodic orbits and disclose a novel noise-induced ordering
phenomenon. For long unstable periodic orbits forming the strange attractor the
weights (or natural measure) is generally highly inhomogeneous over the set,
either diminishing or enhancing the contribution of these orbits into system
dynamics. We show analytically and numerically a weak noise to reduce this
inhomogeneity and, additionally to obvious perturbing impact, make a
regularizing influence on the chaotic dynamics. This universal effect is rooted
into the nature of deterministic chaos.Comment: 11 pages, 5 figure
Using the Hadamard and related transforms for simplifying the spectrum of the quantum baker's map
We rationalize the somewhat surprising efficacy of the Hadamard transform in
simplifying the eigenstates of the quantum baker's map, a paradigmatic model of
quantum chaos. This allows us to construct closely related, but new, transforms
that do significantly better, thus nearly solving for many states of the
quantum baker's map. These new transforms, which combine the standard Fourier
and Hadamard transforms in an interesting manner, are constructed from
eigenvectors of the shift permutation operator that are also simultaneous
eigenvectors of bit-flip (parity) and possess bit-reversal (time-reversal)
symmetry.Comment: Version to appear in J. Phys. A. Added discussions; modified title;
corrected minor error
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