3,814 research outputs found
Contour lines of the discrete scale invariant rough surfaces
We study the fractal properties of the 2d discrete scale invariant (DSI)
rough surfaces. The contour lines of these rough surfaces show clear DSI. In
the appropriate limit the DSI surfaces converge to the scale invariant rough
surfaces. The fractal properties of the 2d DSI rough surfaces apart from
possessing the discrete scale invariance property follow the properties of the
contour lines of the corresponding scale invariant rough surfaces. We check
this hypothesis by calculating numerous fractal exponents of the contour lines
by using numerical calculations. Apart from calculating the known scaling
exponents some other new fractal exponents are also calculated.Comment: 9 Pages, 12 figure
Thermodynamic interpretation of the uniformity of the phase space probability measure
Uniformity of the probability measure of phase space is considered in the
framework of classical equilibrium thermodynamics. For the canonical and the
grand canonical ensembles, relations are given between the phase space
uniformities and thermodynamic potentials, their fluctuations and correlations.
For the binary system in the vicinity of the critical point the uniformity is
interpreted in terms of temperature dependent rates of phases of well defined
uniformities. Examples of a liquid-gas system and the mass spectrum of nuclear
fragments are presented.Comment: 11 pages, 2 figure
Hurst Coefficient in long time series of population size: Model for two plant populations with different reproductive strategies
Can the fractal dimension of fluctuations in population size be used to estimate extinction risk? The problem with estimating this fractal dimension is that the lengths of the time series are usually too short for conclusive results. This study answered this question with long time series data obtained from an iterative competition model. This model produces competitive extinction at different perturbation intensities for two different germination strategies: germination of all seeds vs. dormancy in half the seeds. This provided long time series of 900 years and different extinction risks. The results support the hypothesis for the effectiveness of the Hurst coefficient for estimating extinction risk
Extreme values and fat tails of multifractal fluctuations
In this paper we discuss the problem of the estimation of extreme event
occurrence probability for data drawn from some multifractal process. We also
study the heavy (power-law) tail behavior of probability density function
associated with such data. We show that because of strong correlations,
standard extreme value approach is not valid and classical tail exponent
estimators should be interpreted cautiously. Extreme statistics associated with
multifractal random processes turn out to be characterized by non
self-averaging properties. Our considerations rely upon some analogy between
random multiplicative cascades and the physics of disordered systems and also
on recent mathematical results about the so-called multifractal formalism.
Applied to financial time series, our findings allow us to propose an unified
framemork that accounts for the observed multiscaling properties of return
fluctuations, the volatility clustering phenomenon and the observed ``inverse
cubic law'' of the return pdf tails
Wealth Condensation in Pareto Macro-Economies
We discuss a Pareto macro-economy (a) in a closed system with fixed total
wealth and (b) in an open system with average mean wealth and compare our
results to a similar analysis in a super-open system (c) with unbounded wealth.
Wealth condensation takes place in the social phase for closed and open
economies, while it occurs in the liberal phase for super-open economies. In
the first two cases, the condensation is related to a mechanism known from the
balls-in-boxes model, while in the last case to the non-integrable tails of the
Pareto distribution. For a closed macro-economy in the social phase, we point
to the emergence of a ``corruption'' phenomenon: a sizeable fraction of the
total wealth is always amassed by a single individual.Comment: 4 pages, 1 figur
Zipf's law in Nuclear Multifragmentation and Percolation Theory
We investigate the average sizes of the largest fragments in nuclear
multifragmentation events near the critical point of the nuclear matter phase
diagram. We perform analytic calculations employing Poisson statistics as well
as Monte Carlo simulations of the percolation type. We find that previous
claims of manifestations of Zipf's Law in the rank-ordered fragment size
distributions are not born out in our result, neither in finite nor infinite
systems. Instead, we find that Zipf-Mandelbrot distributions are needed to
describe the results, and we show how one can derive them in the infinite size
limit. However, we agree with previous authors that the investigation of
rank-ordered fragment size distributions is an alternative way to look for the
critical point in the nuclear matter diagram.Comment: 8 pages, 11 figures, submitted to PR
Stability Analysis of a Hybrid Cellular Automaton Model of Cell Colony Growth
Cell colonies of bacteria, tumour cells and fungi, under nutrient limited
growth conditions, exhibit complex branched growth patterns. In order to
investigate this phenomenon we present a simple hybrid cellular automaton model
of cell colony growth. In the model the growth of the colony is limited by a
nutrient that is consumed by the cells and which inhibits cell division if it
falls below a certain threshold. Using this model we have investigated how the
nutrient consumption rate of the cells affects the growth dynamics of the
colony. We found that for low consumption rates the colony takes on a Eden-like
morphology, while for higher consumption rates the morphology of the colony is
branched with a fractal geometry. These findings are in agreement with previous
results, but the simplicity of the model presented here allows for a linear
stability analysis of the system. By observing that the local growth of the
colony is proportional to the flux of the nutrient we derive an approximate
dispersion relation for the growth of the colony interface. This dispersion
relation shows that the stability of the growth depends on how far the nutrient
penetrates into the colony. For low nutrient consumption rates the penetration
distance is large, which stabilises the growth, while for high consumption
rates the penetration distance is small, which leads to unstable branched
growth. When the penetration distance vanishes the dispersion relation is
reduced to the one describing Laplacian growth without ultra-violet
regularisation. The dispersion relation was verified by measuring how the
average branch width depends on the consumption rate of the cells and shows
good agreement between theory and simulations.Comment: 8 pages, 6 figure
On the self-similarity in quantum Hall systems
The Hall-resistance curve of a two-dimensional electron system in the
presence of a strong perpendicular magnetic field is an example of
self-similarity. It reveals plateaus at low temperatures and has a fractal
structure. We show that this fractal structure emerges naturally in the
Hamiltonian formulation of composite fermions. After a set of transformations
on the electronic model, we show that the model, which describes interacting
composite fermions in a partially filled energy level, is self-similar. This
mathematical property allows for the construction of a basis of higher
generations of composite fermions. The collective-excitation dispersion of the
recently observed 4/11 fractional-quantum-Hall state is discussed within the
present formalism.Comment: 7 pages, 4 figures; version accepted for publication in Europhys.
Lett., new version contains energy calculations for collective excitations of
the 4/11 stat
A Bohmian approach to quantum fractals
A quantum fractal is a wavefunction with a real and an imaginary part
continuous everywhere, but differentiable nowhere. This lack of
differentiability has been used as an argument to deny the general validity of
Bohmian mechanics (and other trajectory--based approaches) in providing a
complete interpretation of quantum mechanics. Here, this assertion is overcome
by means of a formal extension of Bohmian mechanics based on a limiting
approach. Within this novel formulation, the particle dynamics is always
satisfactorily described by a well defined equation of motion. In particular,
in the case of guidance under quantum fractals, the corresponding trajectories
will also be fractal.Comment: 19 pages, 3 figures (revised version
Fractal geometry of critical Potts clusters
Numerical simulations on the total mass, the numbers of bonds on the hull,
external perimeter, singly connected bonds and gates into large fjords of the
Fortuin-Kasteleyn clusters for two-dimensional q-state Potts models at
criticality are presented. The data are found consistent with the recently
derived corrections-to-scaling theory. However, the approach to the asymptotic
region is slow, and the present range of the data does not allow a unique
identification of the exact correction exponentsComment: 7 pages, 8 figures, Late
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