Search results for "Fractal"
showing 10 items of 329 documents
A physically based connection between fractional calculus and fractal geometry
2014
We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the m…
Multifractal wave functions at the Anderson transition.
1991
Electronic wave functions in disordered systems are studied within the Anderson model of localization. At the critical disorder in 3D we diagonalize very large (103 823\ifmmode\times\else\texttimes\fi{}103 823) secular matrices by means of the Lanczos algorithm. On all length scales the obtained strong spatial fluctuations of the amplitude of the eigenstates display a multifractal character, reflected in the set of generalized fractal dimensions and the singularity spectrum of the fractal measure. An analysis of 1D systems shows multifractality too, in contrast to previous claims.
Polyadic devil's lenses.
2009
Devil’s lenses (DLs) were recently proposed as a new kind of kinoform lens in which the phase structure is characterized by the “devil’s staircase” function. DLs are considered fractal lenses because they are constructed following the geometry of the triadic Cantor set and because they provide self-similar foci along the optical axis. Here, DLs are generalized allowing the inclusion of polyadic Cantor distributions in their design. The lacunarity of the selected polyadic fractal distribution is an additional design parameter. The results are coined polyadic DLs. Construction requirements and interrelations among the different parameters of these new fractal lenses are also presented. It is …
Fluctuations in mesoscopic systems
1992
Abstract Electronic wavefunctions in weakly disordered systems have been studied within the Anderson model of localization. The eigenstates calculated by means of the Lanczos diagonalization algorithm display characteristic spatial fluctuations that can be described by a multifractal analysis. For increasing disorder or energy the observed curdling of the wavefunction reflects the stronger localization, but no exponential decay can be observed. This is reflected in the set of generalized fractal dimensions and the singularity spectrum of the fractal measure.
Electronic States in Mesoscopic Systems
1992
Abstract Electronic states in disordered systems are studied within the Anderson model of localization. By means of the Green's function technique we derive the transmission coefficient for electronic states through mesoscopic samples. The transmission coefficient is shown to be not self-averaging due to strong spatial fluctuations of the amplitude of the eigenstates, which are obtained by direct diagonalization of the respective secular matrices. The wave functions display a multifractal behaviour, characterized by the set of generalized fractal dimensions and the singularity spectrum of the fractal measure.
Multiscaling Properties of Large-Scale Structure in the Universe
1995
The large-scale distribution of galaxies and galaxy clusters in the universe can be described in the mathematical language of multifractal sets. A particularly significant aspect of this description is that it furnishes a natural explanation for the observed differences in clustering properties of objects of different density in terms of multiscaling, the generic consequence of the application of a local density threshold to a multifractal set. The multiscaling hypothesis suggests ways of improving upon the traditional statistical measures of clustering pattern (correlation functions) and exploring further the connection between clustering pattern and dynamics.
Prospects of searching for (un)particles from Hidden Sectors using rapidity correlations in multiparticle production at the LHC
2008
Most signatures of new physics have been studied on the transverse plane with respect to the beam direction at the LHC where background is much reduced. In this paper we propose the analysis of inclusive longitudinal (pseudo) rapidity correlations among final-state (charged) particles in order to search for (un)particles belonging to a hidden sector beyond the Standard Model, using a selected sample of p-p minimum bias events (applying appropriate off-line cuts on events based on, e.g. minijets, high-multiplicity, event shape variables, high-p(perpendicular to) leptons and photons, etc.) collected at the early running of the LHC. To this aim, we examine inclusive and semi-inclusive two-part…
Partition function based analysis of cosmic microwave background maps
1999
We present an alternative method to analyse cosmic microwave background (CMB) maps. We base our analysis on the study of the partition function. This function is used to examine the CMB maps, making use of the different information embedded at different scales and moments. Using the partition function in a likelihood analysis in two dimensions (Qrms-PS, n), we find the best-fitting model to the best data available at present (the COBE–DMR 4 years data set). By means of this analysis we find a maximum in the likelihood function for n=1.8-0.65+0.35 and Qrms-PS = 10-2.5+3μ K (95 per cent confidence level) in agreement with the results of other similar analyses [Smoot et al. (1 yr), Bennet et a…
Asymptotic Safety in Quantum Einstein Gravity: Nonperturbative Renormalizability and Fractal Spacetime Structure
2007
The asymptotic safety scenario of Quantum Einstein Gravity, the quantum field theory of the spacetime metric, is reviewed and it is argued that the theory is likely to be nonperturbatively renormalizable. It is also shown that asymptotic safety implies that spacetime is a fractal in general, with a fractal dimension of 2 on sub-Planckian length scales.
Determination of the mobility edge in the Anderson model of localization in three dimensions by multifractal analysis.
1995
We study the Anderson model of localization in three dimensions with different probability distributions for the site energies. Using the Lanczos algorithm we calculate eigenvectors for different model parameters like disorder and energy. From these we derive the singularity spectrum typically used for the characterization of multifractal objects. We demonstrate that the singularity spectrum at the critical disorder, which determines the mobility edge at the band center, is independent of the employed probability distribution. Assuming that this singularity spectrum is universal for the metal-insulator transition regardless of specific parameters of the model we establish a straightforward …