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Probability Distributions with Singularities.

Federico Corberi1,2, Alessandro Sarracino3

  • 1Dipartimento di Fisica "E. R. Caianiello", Università di Salerno, via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy.

Entropy (Basel, Switzerland)
|December 3, 2020
PubMed
Summary
This summary is machine-generated.

This paper explores probability distributions with singular behavior, using statistical mechanics models to explain mathematical mechanisms behind these singularities and related phenomena like fluctuation condensation and giant responses.

Keywords:
condensation of fluctuationsfluctuation relationslarge deviationsphase transitions

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Area of Science:

  • Statistical Mechanics
  • Probability Theory
  • Mathematical Physics

Background:

  • Probability distributions can exhibit singular behavior, deviating from standard continuous or discrete forms.
  • Singularities in probability distributions are observed in various models within statistical mechanics.
  • Understanding these singularities is crucial for comprehending complex systems' behavior.

Purpose of the Study:

  • To review general properties of probability distributions exhibiting singular behavior.
  • To elucidate the underlying mathematical mechanisms responsible for these singularities.
  • To explore related phenomena such as fluctuation condensation and giant responses.

Main Methods:

  • Analysis of various statistical mechanics models serving as paradigms.
  • Mathematical investigation of the mechanisms producing singularities.
  • Discussion of concepts including condensation of fluctuations and giant responses.

Main Results:

  • Identification of general properties characterizing singular probability distributions.
  • Explanation of the mathematical framework underlying singularities in these distributions.
  • Characterization of phenomena such as condensation of fluctuations and giant responses.

Conclusions:

  • Singular probability distributions possess unique mathematical properties.
  • These properties are linked to phenomena like phase transitions and anomalous fluctuations.
  • The study provides insights into the interplay between fluctuations and macroscopic behavior.