Probability distributions with singularities

Abstract

In this paper we review some general properties of probability distributions which exibit 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.

13 pages, 8 figures, review paper for the Special Issue of Entropy "Fluctuation Relations and Nonequilibrium Thermodynamics in Classical and Quantum Systems"

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• Dynamics of fluctuations in the Gaussian model with conserved dynamics
• Phase transitions in atypical systems induced by a condensation transition on graphs
• Work fluctuations for a confined Brownian particle: the role of initial conditions
• Partition Function Zeros of Paths and Normalization Zeros of ASEPS
• Large-deviations approach to thermalization: the case of harmonic chains with conservative noise
• Dynamics of fluctuations in the Gaussian model with dissipative Langevin Dynamics
• Asymmetric scaling in large deviations for rare values bigger or smaller than the typical value