1,899 research outputs found
Remarks on the Spectral Properties of Tight Binding and Kronig-Penney Models with Substitution Sequences
We comment on some recent investigations on the electronic properties of
models associated to the Thue-Morse chain and point out that their conclusions
are in contradiction with rigorously proven theorems and indicate some of the
sources of these misinterpretations. We briefly review and explain the current
status of mathematical results in this field and discuss some conjectures and
open problems.Comment: 15,CPT-94/P.3003,tex,
Contact process on generalized Fibonacci chains: infinite-modulation criticality and double-log periodic oscillations
We study the nonequilibrium phase transition of the contact process with
aperiodic transition rates using a real-space renormalization group as well as
Monte-Carlo simulations. The transition rates are modulated according to the
generalized Fibonacci sequences defined by the inflation rules A AB
and B A. For and 2, the aperiodic fluctuations are irrelevant, and
the nonequilibrium transition is in the clean directed percolation universality
class. For , the aperiodic fluctuations are relevant. We develop a
complete theory of the resulting unconventional "infinite-modulation" critical
point which is characterized by activated dynamical scaling. Moreover,
observables such as the survival probability and the size of the active cloud
display pronounced double-log periodic oscillations in time which reflect the
discrete scale invariance of the aperiodic chains. We illustrate our theory by
extensive numerical results, and we discuss relations to phase transitions in
other quasiperiodic systems.Comment: 12 pages, 9 eps figures included, final version as publishe
Entanglement entropy in aperiodic singlet phases
We study the average entanglement entropy of blocks of contiguous spins in
aperiodic XXZ chains which possess an aperiodic singlet phase at least in a
certain limit of the coupling ratios. In this phase, where the ground state
constructed by a real space renormalization group method, consists
(asymptotically) of independent singlet pairs, the average entanglement entropy
is found to be a piecewise linear function of the block size. The enveloping
curve of this function is growing logarithmically with the block size, with an
effective central charge in front of the logarithm which is characteristic for
the underlying aperiodic sequence. The aperiodic sequence producing the largest
effective central charge is identified, and the latter is found to exceed the
central charge of the corresponding homogeneous model. For marginal aperiodic
modulations, numerical investigations performed for the XX model show a
logarithmic dependence, as well, with an effective central charge varying
continuously with the coupling ratio.Comment: 18 pages, 9 figure
Theory of Analogous Force on Number Sets
A general statistical thermodynamic theory that considers given sequences of
x-integers to play the role of particles of known type in an isolated elastic
system is proposed. By also considering some explicit discrete probability
distributions p_{x} for natural numbers, we claim that they lead to a better
understanding of probabilistic laws associated with number theory. Sequences of
numbers are treated as the size measure of finite sets. By considering p_{x} to
describe complex phenomena, the theory leads to derive a distinct analogous
force f_{x} on number sets proportional to at an analogous system temperature T. In particular, this yields to an
understanding of the uneven distribution of integers of random sets in terms of
analogous scale invariance and a screened inverse square force acting on the
significant digits. The theory also allows to establish recursion relations to
predict sequences of Fibonacci numbers and to give an answer to the interesting
theoretical question of the appearance of the Benford's law in Fibonacci
numbers. A possible relevance to prime numbers is also analyzed.Comment: RevTeX, PostScript Fig, To Appear Phys.
Exact Solution of an Evolutionary Model without Ageing
We introduce an age-structured asexual population model containing all the
relevant features of evolutionary ageing theories. Beneficial as well as
deleterious mutations, heredity and arbitrary fecundity are present and managed
by natural selection. An exact solution without ageing is found. We show that
fertility is associated with generalized forms of the Fibonacci sequence, while
mutations and natural selection are merged into an integral equation which is
solved by Fourier series. Average survival probabilities and Malthusian growth
exponents are calculated indicating that the system may exhibit mutational
meltdown. The relevance of the model in the context of fissile reproduction
groups as many protozoa and coelenterates is discussed.Comment: LaTeX file, 15 pages, 2 ps figures, to appear in Phys. Rev.
Growth rate for the expected value of a generalized random Fibonacci sequence
A random Fibonacci sequence is defined by the relation g_n = | g_{n-1} +/-
g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for each n.
We generalize these sequences to the case when the coin is unbalanced (denoting
by p the probability of a +), and the recurrence relation is of the form g_n =
|\lambda g_{n-1} +/- g_{n-2} |. When \lambda >=2 and 0 < p <= 1, we prove that
the expected value of g_n grows exponentially fast. When \lambda = \lambda_k =
2 cos(\pi/k) for some fixed integer k>2, we show that the expected value of g_n
grows exponentially fast for p>(2-\lambda_k)/4 and give an algebraic expression
for the growth rate. The involved methods extend (and correct) those introduced
in a previous paper by the second author
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