Loi de puissance | EPFL Graph Search

La loi de puissance est une relation mathématique entre deux quantités. Si une quantité est la fréquence d'un évènement et l'autre est la taille d'un évènement, alors la relation est une distribution de la loi de puissance si les fréquences diminuent très lentement lorsque la taille de l'évènement augmente. En science, une loi de puissance est une relation entre deux quantités x et y qui peut s'écrire de la façon suivante : :y = ax^k où a est une constante dite constante de proportionnalité, k, valeur négative, est une autre constante, dite exposant, puissance, indice ou encore degré de la loi et x nombre réel strictement positif. On observe des lois de puissance dans beaucoup de domaines scientifiques (physique, biologie, psychologie, sociologie, économie, linguistique). Elles permettent en effet de décrire tous les phénomènes qui présentent une invariance d'échelle. Le terme anglais power law est parfois aussi utilisé en français. Propriétés Invaria

Quantum phase transitions in low-dimensional weakly interacting disordered Bose gases

Joseph Saliba

In this thesis, we study the quantum phase transition triggered by an external random po- tential in ultra-cold low-dimensional weakly-interacting Bose gases at zero temperature. In one-dimensional systems, the quantum phases are characterized by the decay at long-range of the one-body density matrix. This decay exhibits an algebraic behaviour in the superfluid quasi-condensed phase while in the insulating Bose glass phase this it becomes exponen- tial, signaling the complete loss of coherence in the system. In two-dimensional systems, a characterization based on the long-range behaviour of the one-body density matrix appears to be highly demanding from a computational point of view. We therefore characterized the superfluid-insulator transition in two-dimensional systems by the low-energy behaviour of the cumulative density of states of the Bogoliubov excitations. While in the superfluid condensed phase this quantity grows as the square of the excitation energy in agreement with the existence of a finite velocity of sound, in the insulating Bose glass phase this power law is less than quadratic. This study is performed in the framework of an extended Bogoliubov approach properly adapted to treat low-dimensional Bose gases. Using a systematic numerical study, we draw the interaction-disorder phase diagrams of the superfluid-insulator transition in 1D and 2D. The phase boundary follows two different power- laws depending on the length scales characterizing the spatial correlations of the disorder and the strength of interactions. The power-law exponents were found to be in agreement with the ones predicted by scaling arguments, both in the white noise and the Thomas-Fermi regimes. In the two-dimensional system, the classical percolation threshold in the Thomas-Fermi limit was found to overestimate the critical disorder below which the onset of superfluidity should be observed for a given interaction strength.

EPFL2016

Scaling as a key conceptual tool for the interpretation of empirical and experimental patterns in ecology and biology

Silvia Zaoli

This thesis is concerned with regular patterns found in ecology and biology, their linkages and the statistical description of their fluctuations around average trends. Among such patterns, often conforming to power laws, well-known examples include the Species-Area relationship (SAR), quantifying the increase of the number of species S inhabiting an ecosystem with ecosystem area, and the scale-invariant body size spectrum, routinely observed, e.g., in aquatic ecosystems. In biology, Kleiber's law is an allometric relationship describing how metabolic rates scale with an organism's body size. While ecological laws have often been studied independently, simple heuristic reasonings show that they are linked. The need for a unifying effort in ecology, coherently synthesizing the vast and diverse set of empirical observations across scales, has been often voiced. However, a theoretical framework answering this need was still lacking. Furthermore, ecological variables are the result of the interplay between several stochastic ecological processes, and are therefore stochastic variables fluctuating around average values. Ecological and biological scaling laws typically make predictions for such averages, but the issue of fluctuations received scarce attention in the literature. Similarly, biological fluctuations have been typically neglected in the study of the size-scaling of metabolic rates, even though body sizes and metabolic rates may have a significant variability within a species. Fluctuations may be relevant to interpret empirical observations, judge the reliability of predictions and understand ecosystem dynamics. An hypothesis for the distribution of abundances and body sizes of species inhabiting an ecosystem of finite area is proposed here. The hypothesis is inspired by finite-size scaling and is used to derive macroecological patterns and their linkages within a coherent theoretical framework. Stochastic models of community dynamics are used to support the hypothesis, and the derived linkages are tested on empirical datasets. Several stochastic models of community dynamics are also used here to study the fluctuations of S and how they scale with the average S. The intra-specific variability of metabolic rates and body sizes is investigated experimentally using freshwater phytoplankton species by nanoscale secondary ion mass spectrometry (NanoSIMS). The linkages among ecological scaling laws predicted by the theoretical framework are verified in several empirical datasets. Theoretical generalizations including deviations from pure power-law behavior and heavy-tailed intra-specific size distributions are also addressed. The theoretical study of the relative scaling of the fluctuations of S with the mean

\langle S \rangle

in various community dynamics models shows that different ecological processes predict radically different fluctuations scalings, highlighting the need of empirical investigations to sort out which scenario applies to real ecosystems. Experiments on phytoplankton metabolic rate scaling with body size suggest that intra-specific metabolic rate distributions are described by a universal scaling form across different taxa and over three orders of magnitude in body size. This thesis, along with previous works, suggests that scaling concepts derived for inanimate systems can provide new insights into the dynamics of ecosystems and help unraveling regularities across scales of biological complexity.

EPFL2018

Metastable densities for the contact process on power law random graphs

Thomas Mountford, Daniel Rodrigues Valesin

We consider the contact process on a random graph with a fixed degree distribution given by a power law. We follow the work of Chatterjee and Durrett [2], who showed that for arbitrarily small infection parameter lambda, the survival time of the process is larger than a stretched exponential function of the number of vertices. For lambda close to 0 (that is, "near criticality"), we obtain sharp bounds for the typical density of infected sites in the graph, as the number of vertices tends to infinity. We exhibit three different regimes for this density, depending on the tail of the degree law.

Univ Washington, Dept Mathematics2013