scholarly journals Probability Distributions with Singularities

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 312 ◽  
Author(s):  
Federico Corberi ◽  
Alessandro Sarracino
Keyword(s):  
Phase Transitions ◽  
Statistical Mechanics ◽  
Probability Distributions ◽  
Singular Behavior ◽  
Fluctuation Relations ◽  
General Properties

In this paper we review some general properties of probability distributions which exhibit a singular behavior. After introducing the matter with several examples based on various models of statistical mechanics, we discuss, with the help of such paradigms, the underlying mathematical mechanism producing the singularity and other topics such as the condensation of fluctuations, the relationships with ordinary phase-transitions, the giant response associated to anomalous fluctuations, and the interplay with fluctuation relations.

Introduction

2017 ◽  
Author(s):  
Jochen Rau
Keyword(s):  
Probability Theory ◽  
Phase Transitions ◽  
Statistical Mechanics ◽  
Conservation Laws ◽  
Law Of Large Numbers ◽  
Macroscopic Scale ◽  
Classical Probability ◽  
Large Numbers ◽  
Microscopic World ◽  
New Phenomena

Statistical mechanics concerns the transition from the microscopic to the macroscopic realm. On a macroscopic scale new phenomena arise that have no counterpart in the microscopic world. For example, macroscopic systems have a temperature; they might undergo phase transitions; and their dynamics may involve dissipation. How can such phenomena be explained? This chapter discusses the characteristic differences between the microscopic and macroscopic realms and lays out the basic challenge of statistical mechanics. It suggests how, in principle, this challenge can be tackled with the help of conservation laws and statistics. The chapter reviews some basic notions of classical probability theory. In particular, it discusses the law of large numbers and illustrates how, despite the indeterminacy of individual events, statistics can make highly accurate predictions about totals and averages.

DISSONANCE — A MEASURE OF VARIABILITY FOR ORDINAL RANDOM VARIABLES

2001 ◽  
Vol 09 (01) ◽  
pp. 39-53 ◽  
Author(s):  
RONALD R. YAGER
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
General Properties

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

scholarly journals General properties of the gravitational wave spectrum from phase transitions

2009 ◽  
Vol 79 (8) ◽  
Author(s):  
Chiara Caprini ◽  
Ruth Durrer ◽  
Thomas Konstandin ◽  
Géraldine Servant
Keyword(s):  
Phase Transitions ◽  
Gravitational Wave ◽  
Wave Spectrum ◽  
General Properties

Chapter 22. Connections to Statistical Physics

2021 ◽  
Author(s):  
Fabrizio Altarelli ◽  
Rémi Monasson ◽  
Guilhem Semerjian ◽  
Francesco Zamponi
Keyword(s):  
Phase Transitions ◽  
Low Temperature ◽  
Statistical Mechanics ◽  
Cost Function ◽  
Computer Science ◽  
Statistical Physics ◽  
Energy Landscape ◽  
Research Activity ◽  
Detailed Picture ◽  
Intense Research

This chapter surveys a part of the intense research activity that has been devoted by theoretical physicists to the study of randomly generated k-SAT instances. It can be at first sight surprising that there is a connection between physics and computer science. However low-temperature statistical mechanics concerns precisely the behaviour of the low-lying configurations of an energy landscape, in other words the optimization of a cost function. Moreover the ensemble of random k-SAT instances exhibit phase transitions, a phenomenon mostly studied in physics (think for instance at the transition between liquid and gaseous water). Besides the introduction of general concepts of statistical mechanics and their translations in computer science language, the chapter presents results on the location of the satisfiability transition, the detailed picture of the satisfiable regime and the various phase transitions it undergoes, and algorithmic issues for random k-SAT instances.

The statistical mechanics of phase transitions

1978 ◽  
Vol 19 (3) ◽  
pp. 203-224 ◽  
Author(s):  
Colin J. Thompson
Keyword(s):  
Phase Transitions ◽  
Statistical Mechanics

Automatic estimation of large residual statics corrections

Geophysics ◽  
1986 ◽  
Vol 51 (2) ◽  
pp. 332-346 ◽  
Author(s):  
Daniel H. Rothman
Keyword(s):  
Monte Carlo ◽  
Statistical Mechanics ◽  
Probability Distributions ◽  
Synthetic Data ◽  
Linear Inversion ◽  
Traveltime Inversion ◽  
Automatic Estimation ◽  
Crucial Component ◽  
Practical Algorithm ◽  
Overthrust Belt

Conventional approaches to residual statics estimation obtain solutions by performing linear inversion of observed traveltime deviations. A crucial component of these procedures is picking time delays; gross errors in these picks are known as “cycle skips” or “leg jumps” and are the bane of linear traveltime inversion schemes. This paper augments Rothman (1985), which demonstrated that the estimation of large statics in noise‐contaminated data is posed better as a nonlinear, rather than as a linear, inverse problem. Cycle skips then appear as local (secondary) minima of the resulting nonlinear optimization problem. In the earlier paper, a Monte Carlo technique from statistical mechanics was adapted to perform global optimization, and the technique was applied to synthetic data. Here I present an application of a similar Monte Carlo method to field data from the Wyoming Overthrust belt. Key changes, however, have led to a more efficient and practical algorithm. The new technique performs explicit crosscorrelation of traces. Instead of picking the peaks of these crosscorrelation functions, the method transforms the crosscorrelation functions to probability distributions and then draws random numbers from the distributions. Estimates of statics are now iteratively updated by this procedure until convergence to the optimal stack is achieved. Here I also derive several theoretical properties of the algorithm. The method is expressed as a Markov chain, in which the equilibrium (steady‐state) distribution is the Gibbs distribution of statistical mechanics.

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