The Buffon needle problem for Lévy distributed spacings and renewal theory

2022 ◽  
Vol 2022 (1) ◽  
pp. 013203
Author(s):  
Claude Godrèche
Keyword(s):  
Power Law ◽  
Line Segment ◽  
Statistical Physics ◽  
Renewal Theory ◽  
Power Law Distribution ◽  
Average Spacing ◽  
Hole Probability ◽  
Set Of Points ◽  
Independent Identically Distributed ◽  
Exact Scaling

Abstract What is the probability that a needle dropped at random on a set of points scattered on a line segment does not fall on any of them? We compute the exact scaling expression of this hole probability when the spacings between the points are independent identically distributed random variables with a power-law distribution of index less than unity, implying that the average spacing diverges. The theoretical framework for such a setting is renewal theory, to which the present study brings a new contribution. The question posed here is also related to the study of some correlation functions of simple models of statistical physics.

scholarly journals Avalanches and extreme value statistics in interfacial crackling dynamics

2018 ◽  
Vol 377 (2136) ◽  
pp. 20170394 ◽  
Author(s):  
S. Santucci ◽  
K. T. Tallakstad ◽  
L. Angheluta ◽  
L. Laurson ◽  
R. Toussaint ◽  
...  
Keyword(s):  
Power Law ◽  
Crack Front ◽  
Statistical Physics ◽  
Random Medium ◽  
Interfacial Crack ◽  
Maximum Amplitude ◽  
Extreme Value Statistics ◽  
Power Law Distribution ◽  
Extreme Statistics ◽  
Average Crack

We study the avalanche and extreme statistics of the global velocity of a crack front, propagating slowly along a weak heterogeneous interface of a transparent polymethyl methacrylate block. The different loading conditions used (imposed constant velocity or creep relaxation) lead to a broad range of average crack front velocities. Our high-resolution and large dataset allows one to characterize in detail the observed intermittent crackling dynamics. We specifically measure the size S , the duration D , as well as the maximum amplitude of the global avalanches, defined as bursts in the interfacial crack global velocity time series. Those quantities characterizing the crackling dynamics follow robust power-law distributions, with scaling exponents in agreement with the values predicted and obtained in numerical simulations of the critical depinning of a long-range elastic string, slowly driven in a random medium. Nevertheless, our experimental results also set the limit of such model which cannot reproduce the power-law distribution of the maximum amplitudes of avalanches of a given duration reminiscent of the underlying fat-tail statistics of the local crack front velocities. This article is part of the theme issue ‘Statistical physics of fracture and earthquakes’.

On the asymptotics of degree structure of configuration graphs with bounded number of edges

2019 ◽  
Vol 29 (4) ◽  
pp. 219-232 ◽  
Author(s):  
Yurii L. Pavlov ◽  
Irina A. Cheplyukova
Keyword(s):  
Random Graphs ◽  
Power Law ◽  
Random Variables ◽  
Random Variable ◽  
Positive Parameter ◽  
Power Law Distribution ◽  
Bounded Number ◽  
Degree Structure ◽  
Vertex Degrees ◽  
Independent Identically Distributed

Abstract We consider configuration graphs with N vertices. The degrees of vertices are independent identically distributed random variables having the power-law distribution with positive parameter $\tau .$We study properties of random graphs such that the sum of vertex degrees does not exceed n and the parameter is a random variable uniformly distributed on the interval $\left[ a,\,\,b \right],0<a<b<\infty .$We find limit distributions of the number ${{\mu }_{r}}$of vertices with degree r for various types of variation of N, n and r.

scholarly journals The structure of co-publications multilayer network

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Ghislain Romaric Meleu ◽  
Paulin Yonta Melatagia
Keyword(s):  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Small World ◽  
Generation Model ◽  
Small World Network ◽  
Power Law Distribution ◽  
Multilayer Networks ◽  
Multilayer Network ◽  
Scientific Papers

AbstractUsing the headers of scientific papers, we have built multilayer networks of entities involved in research namely: authors, laboratories, and institutions. We have analyzed some properties of such networks built from data extracted from the HAL archives and found that the network at each layer is a small-world network with power law distribution. In order to simulate such co-publication network, we propose a multilayer network generation model based on the formation of cliques at each layer and the affiliation of each new node to the higher layers. The clique is built from new and existing nodes selected using preferential attachment. We also show that, the degree distribution of generated layers follows a power law. From the simulations of our model, we show that the generated multilayer networks reproduce the studied properties of co-publication networks.

scholarly journals Power-law behavior of transcription factor dynamics at the single-molecule level implies a continuum affinity model

2021 ◽  
Author(s):  
David A Garcia ◽  
Gregory Fettweis ◽  
Diego M Presman ◽  
Ville Paakinaho ◽  
Christopher Jarzynski ◽  
...  
Keyword(s):  
Transcription Factor ◽  
Single Molecule ◽  
Power Law ◽  
Dwell Time ◽  
Specific Binding ◽  
Power Law Distribution ◽  
Single Molecule Level ◽  
Binding Behavior ◽  
Chromatin Template ◽  
Nuclear Domains

Abstract Single-molecule tracking (SMT) allows the study of transcription factor (TF) dynamics in the nucleus, giving important information regarding the diffusion and binding behavior of these proteins in the nuclear environment. Dwell time distributions obtained by SMT for most TFs appear to follow bi-exponential behavior. This has been ascribed to two discrete populations of TFs—one non-specifically bound to chromatin and another specifically bound to target sites, as implied by decades of biochemical studies. However, emerging studies suggest alternate models for dwell-time distributions, indicating the existence of more than two populations of TFs (multi-exponential distribution), or even the absence of discrete states altogether (power-law distribution). Here, we present an analytical pipeline to evaluate which model best explains SMT data. We find that a broad spectrum of TFs (including glucocorticoid receptor, oestrogen receptor, FOXA1, CTCF) follow a power-law distribution of dwell-times, blurring the temporal line between non-specific and specific binding, suggesting that productive binding may involve longer binding events than previously believed. From these observations, we propose a continuum of affinities model to explain TF dynamics, that is consistent with complex interactions of TFs with multiple nuclear domains as well as binding and searching on the chromatin template.

scholarly journals Explaining the power-law distribution of human mobility through transportationmodality decomposition

2015 ◽  
Vol 5 (1) ◽  
Author(s):  
Kai Zhao ◽  
Mirco Musolesi ◽  
Pan Hui ◽  
Weixiong Rao ◽  
Sasu Tarkoma
Keyword(s):  
Power Law ◽  
Human Mobility ◽  
Power Law Distribution

MASS FUNCTION OF HALOS: A NEW ANALYTICAL APPROACH

2004 ◽  
Vol 13 (07) ◽  
pp. 1345-1349 ◽  
Author(s):  
JOSÉ A. S. LIMA ◽  
LUCIO MARASSI
Keyword(s):  
Power Law ◽  
Free Parameter ◽  
Analytical Approach ◽  
Mass Function ◽  
New Method ◽  
Power Law Distribution ◽  
The Press ◽  
The Universe ◽  
Special Case

A generalization of the Press–Schechter (PS) formalism yielding the mass function of bound structures in the Universe is given. The extended formula is based on a power law distribution which encompasses the Gaussian PS formula as a special case. The new method keeps the original analytical simplicity of the PS approach and also solves naturally its main difficult (the missing factor 2) for a given value of the free parameter.

scholarly journals Assessing the power law distribution of TGFs

2011 ◽  
Vol 116 (A10) ◽  
pp. n/a-n/a ◽  
Author(s):  
Andrew B. Collier ◽  
Thomas Gjesteland ◽  
Nikolai Østgaard
Keyword(s):  
Power Law ◽  
Power Law Distribution

scholarly journals Predicting observational signatures of coronal heating by Alfvén waves and nanoflares

2007 ◽  
Vol 3 (S247) ◽  
pp. 279-287
Author(s):  
Patrick Antolin ◽  
Kazunari Shibata ◽  
Takahiro Kudoh ◽  
Daiko Shiota ◽  
David Brooks
Keyword(s):  
Power Law ◽  
Alfven Waves ◽  
Alfvén Waves ◽  
Power Law Distribution ◽  
Longitudinal Modes ◽  
Heating Mechanism ◽  
Set Up ◽  
Power Law Index ◽  
Power Law Distributions ◽  
Release Processes

AbstractAlfvén waves can dissipate their energy by means of nonlinear mechanisms, and constitute good candidates to heat and maintain the solar corona to the observed few million degrees. Another appealing candidate is the nanoflare-reconnection heating, in which energy is released through many small magnetic reconnection events. Distinguishing the observational features of each mechanism is an extremely difficult task. On the other hand, observations have shown that energy release processes in the corona follow a power law distribution in frequency whose index may tell us whether small heating events contribute substantially to the heating or not. In this work we show a link between the power law index and the operating heating mechanism in a loop. We set up two coronal loop models: in the first model Alfvén waves created by footpoint shuffling nonlinearly convert to longitudinal modes which dissipate their energy through shocks; in the second model numerous heating events with nanoflare-like energies are input randomly along the loop, either distributed uniformly or concentrated at the footpoints. Both models are based on a 1.5-D MHD code. The obtained coronae differ in many aspects, for instance, in the simulated intensity profile that Hinode/XRT would observe. The intensity histograms display power law distributions whose indexes differ considerably. This number is found to be related to the distribution of the shocks along the loop. We thus test the observational signatures of the power law index as a diagnostic tool for the above heating mechanisms and the influence of the location of nanoflares.

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