# Probability Distributions with Singularities

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## Abstract

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## 1. Introduction

## 2. Probability Distributions: Generalities

## 3. Singular Probability Distributions: Examples

#### 3.1. Gaussian Model

#### 3.2. Large-$\mathcal{N}$ Model

#### 3.3. Urn Model

#### 3.4. Stochastic Maxwell-Lorentz Particle Model

#### 3.5. Some Other Models

## 4. General Features of Singular Probability Distributions

#### 4.1. Duality

#### 4.2. Condensation

#### 4.3. Mathematical Mechanism

#### 4.4. Fluctuation Relation

## 5. Some Peculiarities of Singular Distributions

#### 5.1. Giant Response

#### 5.2. Development of a Singular Fluctuation

#### 5.3. Observability

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The (negative) rate function $I\left(\rho \right)$ of the variance N of the order parameter field in the Gaussian model in $d=3$, with $r=1$, in equilibrium at the temperature $T=0.2$.

**Figure 2.**The (negative) rate function $I\left(\rho \right)$ of the probability distribution $P\left(N\right)$ of the energy N exchanged by the large-$\mathcal{N}$ model in $d=3$, with $g=-r=1$, with the environment after a quench to zero temperature.

**Figure 3.**The rate function $I\left(\rho \right)$ of the probability distribution $P\left(N\right)$ of the total number of particles N in the urn model with $k=3$.

**Figure 4.**The rate function $I\left(\rho \right)$ of the quantity $\rho =\Delta {s}_{\mathrm{tot}}/t$ for the Maxwell–Lorentz gas model [11], computed analytically in the limit $t\to \infty $.

**Figure 5.**The function $\pi (n,N,M)$ is plotted, for $k=3$ and $M=100$, against $n+1$ for two values of N: $N=35$, corresponding to a case without condensations, and $N=300$, corresponding to a condensed situation. The dotted green curve is the power-law ${x}^{-k}$.

**Figure 7.**P is plotted for $M=333$ and the three different choices (see text) (i) ${k}_{\ell}\equiv k=3$, continuous blue with asterisks, (ii) ${k}_{\ell}\equiv k=6$, dot-dashed green with squares and (iii) ${k}_{\ell}=3$ , $k=6$, dashed magenta with a circles.

**Figure 8.**The probability $P(N,M,t)$ with $k=3$ is plotted against N with double logarithmic scales for different times (see key), exponentially spaced. The dotted green line is the asymptotic form.

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Corberi, F.; Sarracino, A.
Probability Distributions with Singularities. *Entropy* **2019**, *21*, 312.
https://doi.org/10.3390/e21030312

**AMA Style**

Corberi F, Sarracino A.
Probability Distributions with Singularities. *Entropy*. 2019; 21(3):312.
https://doi.org/10.3390/e21030312

**Chicago/Turabian Style**

Corberi, Federico, and Alessandro Sarracino.
2019. "Probability Distributions with Singularities" *Entropy* 21, no. 3: 312.
https://doi.org/10.3390/e21030312