318 research outputs found
Sierpinski signal generates spectra
We investigate the row sum of the binary pattern generated by the Sierpinski
automaton: Interpreted as a time series we calculate the power spectrum of this
Sierpinski signal analytically and obtain a unique rugged fine structure with
underlying power law decay with an exponent of approximately 1.15. Despite the
simplicity of the model, it can serve as a model for spectra in a
certain class of experimental and natural systems like catalytic reactions and
mollusc patterns.Comment: 4 pages (4 figs included). Accepted for publication in Physical
Review
Exact scaling in the expansion-modification system
This work is devoted to the study of the scaling, and the consequent
power-law behavior, of the correlation function in a mutation-replication model
known as the expansion-modification system. The latter is a biology inspired
random substitution model for the genome evolution, which is defined on a
binary alphabet and depends on a parameter interpreted as a \emph{mutation
probability}. We prove that the time-evolution of this system is such that any
initial measure converges towards a unique stationary one exhibiting decay of
correlations not slower than a power-law. We then prove, for a significant
range of mutation probabilities, that the decay of correlations indeed follows
a power-law with scaling exponent smoothly depending on the mutation
probability. Finally we put forward an argument which allows us to give a
closed expression for the corresponding scaling exponent for all the values of
the mutation probability. Such a scaling exponent turns out to be a piecewise
smooth function of the parameter.Comment: 22 pages, 2 figure
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.
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
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.
Continuation for thin film hydrodynamics and related scalar problems
This chapter illustrates how to apply continuation techniques in the analysis
of a particular class of nonlinear kinetic equations that describe the time
evolution through transport equations for a single scalar field like a
densities or interface profiles of various types. We first systematically
introduce these equations as gradient dynamics combining mass-conserving and
nonmass-conserving fluxes followed by a discussion of nonvariational amendmends
and a brief introduction to their analysis by numerical continuation. The
approach is first applied to a number of common examples of variational
equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including
certain thin-film equations for partially wetting liquids on homogeneous and
heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal
equations. Second we consider nonvariational examples as the
Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard
equations and thin-film equations describing stationary sliding drops and a
transversal front instability in a dip-coating. Through the different examples
we illustrate how to employ the numerical tools provided by the packages
auto07p and pde2path to determine steady, stationary and time-periodic
solutions in one and two dimensions and the resulting bifurcation diagrams. The
incorporation of boundary conditions and integral side conditions is also
discussed as well as problem-specific implementation issues
- …