# The Fourfold Way to Gaussianity: Physical Interactions, Distributional Models and Monadic Transformations

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## Abstract

**:**

## 1. Introduction

## 2. Gaussian Distributions in Physics

- the introduction of Wiener processes $W\left(t\right)$ [35,36], often referred to as mathematical Brownian motion, mimicking the long-term proprerties of the kinematics of micrometric particles in quiescent fluids. In the definition of Wiener process, Gaussianity is built-in from the beginning, as the increments $y\left(t\right)=w(t+\tau )-w\left(t\right)$, with $t>0$, $\tau >0$, are distributed in a Gaussian way with zero mean $\langle y\rangle =0$ and squared variance $\langle {y}^{2}\rangle =\tau $ (we use the convention of indicating with upper case letters the process, such as $W\left(t\right)$, and with lower-case letters, say $w\left(t\right)$, a realization of it);
- the Ito formulation of stochastic calculus on Wiener processes, leading to the theory of stochastic integration, the definition of stochastic differential equations [37,38], i.e., to the theory of Langevin equations driven by the increments of Wiener processes [39], representing another fundamental chapter in modern statistical physics;

## 3. Unbounded Additivity

## 4. The Distributional Route to Gaussianity

- two elements, say ${\mathbf{z}}_{{i}^{*}}$ and ${\mathbf{z}}_{{j}^{*}}$, with ${j}^{*}\ne {i}^{*}$ ar randomly selected from $\mathcal{E}$;
- two random functions $\phi (\mathbf{z},\mathbf{w};\mathbf{r})$, $\psi (\mathbf{z},\mathbf{w};\mathbf{r})$, $\mathbf{z},\phantom{\rule{0.166667em}{0ex}}\mathbf{w}\in {\mathbb{R}}^{n}$ are defined, such that the mapping $\mathcal{M}$ transforms ${\mathbf{z}}_{{i}^{*}}$ and ${\mathbf{z}}_{{j}^{*}}$ into the new values ${\mathbf{z}}_{{i}^{*}}^{\prime}$, ${\mathbf{z}}_{{j}^{*}}^{\prime}$, according to the random laws$$\left(\right)$$
- the values of all the other ${\mathbf{z}}_{h}$ of the ensemble with $h\ne {i}^{*},{j}^{*}$ are left unchanged.

## 5. Particle–Photon Radiative Processes and Limit Gaussianity

#### Monadic Transformations, Thermodynamic Constraints and IFS

- an element, say ${\mathbf{z}}_{{i}^{*}}$ is randomly selected from $\mathcal{E}$;
- a random function $\varphi (\mathbf{z};\mathbf{r})$ is defined such that the mapping $\mathcal{T}$ transforms ${\mathbf{z}}_{{i}^{*}}$ into the new value ${\mathbf{z}}_{{i}^{*}}^{\prime}$ according to the random law$${\mathbf{z}}_{{i}^{*}}^{\prime}=\xi ({\mathbf{z}}_{{i}^{*}};\mathbf{r})$$
- the values of all the other ${\mathbf{z}}_{h}$ of the ensemble with $h\ne {i}^{*}$ are left unchanged.

## 6. The High-Friction Limit and the Dimension of the Physical Space

## 7. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Probability density function $p(z;N)$ for ${Z}_{N}$ associated with the independent random variables defined by Equation (12). Solid line (

**a**) represents the normal pdf $g\left(z\right)={e}^{-{z}^{2}/2}/\sqrt{2\phantom{\rule{0.166667em}{0ex}}\pi}$, symbols correspond to the results of Monte Carlo simulations: (□) $N=100$, (∘) $N=1000$. Panel (

**a**) refers to $\alpha =1/2$, panel (

**b**) to $\alpha =1$, panel (

**c**) to $\alpha =2$.

**Figure 2.**Kurtosis $\kappa $ vs. ${m}^{*}=m/N$ associated with the ensemble distribution of the first entry ${z}_{h,1}$ of ${\mathbf{z}}_{h}$ iterating a CMT ($n=2$) with the dichotomic random perturbation Equations (17) and (18), discussed in the main text. The arrow indicates decreasing values of $p=3,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}1$.

**Figure 3.**Stationary density ${p}^{*}\left(z\right)$ vs. z, where $z={z}_{h,2}$ is the second entry of the ensemble elements obtained from Monte Carlo simulations of the CMT at $n=2$ with the dichotomic random perturbation (17) and (18). Symbols ((□) refer to $p=3$, (∘) to $p=2$, (•) to $p=1$) represent the stochastic simulation results of the asymptotic CMT dynamics, solid line represents the normal distribution.

**Figure 4.**${\sigma}_{12}\left({m}^{*}\right)$ vs. ${m}^{*}=m/N$ for a two-dimensional CMT, with uniformly distributed random perturbations, starting from highly correlated initial conditions Equation (20). Symbols are the results of stochastic simulations, the solid line represents the curve ${\sigma}_{12}\left({m}^{*}\right)={e}^{-{m}^{*}}$.

**Figure 5.**${p}^{*}\left(z\right)$ vs. z obtained from Monte Carlo simulations of the IFS Equation (31) at different values of $\alpha $. Panel (

**a**) refers to $\alpha =0.3$, panel (

**b**) to $\alpha =0.6$, panel (

**c**) to $\alpha =0.7$, panel (

**d**) to a zoom-in of the density at $\alpha =0.7$, depicted in panel (c) close to $z=0$.

**Figure 6.**${p}^{*}\left(z\right)$ vs. z obtained from Monte Carlo simulations of the IFS Equation (31) at different values of $\alpha $. Panel (

**a**): line (a) refers to $\alpha =1/2$, line (b) to $\alpha =0.8$, line (c) to $\alpha =0.9$. Panel (

**b**): symbols (∘) to $\alpha =0.99$. The solid line represents the normal distribution.

**Figure 7.**${p}^{*}\left(v\right)$ vs. v at different values of n obtained from Monte Carlo simulations. Panel (

**a**): $n=2$, panel (

**b**): $n=3$, panel (

**c**): $n=4$, panel (

**d**) $n=5$.

**Figure 8.**${p}^{*}\left(v\right)$ vs. v obtained from Monte Carlo simulations for large n. The arrows indicate increasing values of $n=10,\phantom{\rule{0.166667em}{0ex}}30,\phantom{\rule{0.166667em}{0ex}}100,\phantom{\rule{0.166667em}{0ex}}1000$. Symbols (•) represent the normal distribution (with zero mean and unit variance).

Route | Mechanism | Conditions |
---|---|---|

CLT | Additive | $\begin{array}{c}\mathrm{Unbounded}\mathrm{additivity}\\ \mathrm{Lyapunov}\mathrm{condition}\end{array}$ |

CMT | Distributional | $\begin{array}{c}\mathrm{Quadratic}\mathrm{nature}\mathrm{of}\\ \mathrm{the}\mathrm{energy}\mathrm{condition}\end{array}$ |

TMoT | Thermodynamic | $\begin{array}{cc}\left(a\right)\hfill & \alpha \to 1\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall n=1,2,\dots \hfill \\ \left(b\right)\hfill & \alpha \to 0\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n\to \infty \hfill \end{array}$ |

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**MDPI and ACS Style**

Giona, M.; Pezzotti, C.; Procopio, G.
The Fourfold Way to Gaussianity: Physical Interactions, Distributional Models and Monadic Transformations. *Axioms* **2023**, *12*, 278.
https://doi.org/10.3390/axioms12030278

**AMA Style**

Giona M, Pezzotti C, Procopio G.
The Fourfold Way to Gaussianity: Physical Interactions, Distributional Models and Monadic Transformations. *Axioms*. 2023; 12(3):278.
https://doi.org/10.3390/axioms12030278

**Chicago/Turabian Style**

Giona, Massimiliano, Chiara Pezzotti, and Giuseppe Procopio.
2023. "The Fourfold Way to Gaussianity: Physical Interactions, Distributional Models and Monadic Transformations" *Axioms* 12, no. 3: 278.
https://doi.org/10.3390/axioms12030278