# Tales of Tails

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## Abstract

**:**

## 1. Introduction

## 2. Problems with Observing Known PDF Tails Directly

## 3. Fluctating Local Thermodynamic Equilibrium and Time

## 4. No Temperature, No State Function, No Equilibrium

## 5. PDFs on Finite Intervals

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Sequence of samplings of a Gaussian distribution with standard deviation $\sigma =4$. The top row samples it for a period $\tau $, while the subsequent rows are for periods $10,100,1000$, and 10,000 times longer, respectively. The left column contains linear plots, while the right column is on a semi-log scale. Being of exponentially small probability, the wings only begin to be observed for long sampling times and even then are never entirely seen.

**Figure 2.**Suitable ranges of k and c in the relations in Equations (14) and (15) generate space-filling areas in this ${a}_{x}$–${a}_{y}$ plot, indicating that no function exists relating ${a}_{x}$ to ${a}_{y}$, since both function families ${g}_{1}$ and ${g}_{2}$ are in play. No function exists, even though a relationship always exists between y and x in Equation (11). The ${g}_{2}$ family is shown in black, while the ${g}_{1}$ family is in red. The two areas overlap in the right half of the figure, indicated by the thin black lines. The boundaries of regions are envelopes made up of segments from potentially more than one function family member.

**Figure 3.**The fat tail speed distribution in Equation (18) arising from a convolution with Gaussian varying precision is shown in green. The blue curve represents a similar convolution but with a box car PDF, resulting in a sub-Gaussian speed distribution (Equation (20)). A standard Gaussian speed distribution (Equation (2)) is included in red for comparison. Not discernible in this logarithmic plot, emphasizing the tails, is that the green curve descends more rapidly than the blue curve around $v=0$, both starting from the same value of $v=0$. The green curve then crosses the blue one around $v=\pm 2$, allowing a common normalization of 1 for all the distributions.

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Essex, C.; Andresen, B.
Tales of Tails. *Entropy* **2023**, *25*, 598.
https://doi.org/10.3390/e25040598

**AMA Style**

Essex C, Andresen B.
Tales of Tails. *Entropy*. 2023; 25(4):598.
https://doi.org/10.3390/e25040598

**Chicago/Turabian Style**

Essex, Christopher, and Bjarne Andresen.
2023. "Tales of Tails" *Entropy* 25, no. 4: 598.
https://doi.org/10.3390/e25040598