# Majorization and Dynamics of Continuous Distributions

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Majorization of Integrable Functions and Discrete Vectors

#### 2.1. Continuous Majorization

#### 2.2. Discrete Majorization

**Theorem**

**1.**

- (a)
- $\mathbf{x}$ is majorized by $\mathbf{y}$.
- (b)
- $\varphi \left(\mathbf{x}\right)\le \varphi \left(\mathbf{y}\right)$ for each Schur-convex function $\varphi :{I}^{n}\to \mathbb{R}$.
- (c)
- $\varphi \left(\mathbf{x}\right)\le \varphi \left(\mathbf{y}\right)$ for each symmetric quasi-convex function $\varphi :{I}^{n}\to \mathbb{R}$.
- (d)
- ${\sum}_{i=1}^{n}\varphi \left({x}_{i}\right)\le {\sum}_{i=1}^{n}\varphi \left({y}_{i}\right)$ for each convex function $g:I\to \mathbb{R}$.

#### 2.3. Probability Distributions and Majorization

## 3. Temporal Evolution of Continuous Distributions from Majorization

**Lemma**

**2.**

- (I)
- $\mathcal{P}=\{{p}_{t}:t\ge 0\}$ is an ordered chain by majorization with ${p}_{{t}_{2}}\prec {p}_{{t}_{1}}$ for all ${t}_{1}\le {t}_{2}$ (i.e., the distribution at time t is majorized by all the precedent ones).
- (II)
- The function ${\lambda}_{\varphi}\left(t\right):[0,\infty ]\to \mathbb{R}$, ${\lambda}_{\varphi}\left(t\right)={\int}_{0}^{1}\varphi \left({p}_{t}\left(x\right)\right)dx$ is decreasing in t for all $\varphi \in {\mathcal{L}}_{\mathrm{cx}}\left(I\right)$.

**Proof.**

**Lemma**

**3.**

**Proof.**

## 4. Applications

#### 4.1. H-Theorem and Majorization

**Theorem**

**4.**

**Proof.**

#### 4.2. Dynamical Systems: Mixing Property

#### 4.3. Generalized Fokker–Planck Equations

#### 4.4. Quantum Dynamics

#### 4.5. Population Dynamics: Exponential Model and Majorization

- $\lambda =1$:
- Since ${N}_{k}={N}_{0}$ for all k, the population remains the same along time.
- $\lambda <1$:
- The number of individuals decreases in each time step so it tends to zero for large times.
- $\lambda >1$:
- The number of individuals is growing in such a way that it tends to infinity asymptotically.
- $\lambda =0$:
- This case correspond to the extinction of the population since ${N}_{k}=0$ for all time k.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Evolved probability distributions ${\rho}_{n}\left(x\right)$ given by the Equation (25) for $\lambda =2$ and $\lambda =1/2$, respectively. The arrow indicates the temporal evolution. The diffusive regime represents an increasing of the ignorance about the population and the majorization ordering in Equation (1) is opposite to the temporal one (arrow). By contrast, when localization occurs, the population rapidly concentrates around $x=0$, which expresses the extinction of the population, and the majorization ordering coincides with the temporal evolution.

**Figure 2.**Some necessary and sufficient conditions for continuous majorization in different contexts illustrate the relevance of the concept of majorization in a continuous dynamics.

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Gomez, I.S.; da Costa, B.G.; dos Santos, M.A.F.
Majorization and Dynamics of Continuous Distributions. *Entropy* **2019**, *21*, 590.
https://doi.org/10.3390/e21060590

**AMA Style**

Gomez IS, da Costa BG, dos Santos MAF.
Majorization and Dynamics of Continuous Distributions. *Entropy*. 2019; 21(6):590.
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Gomez, Ignacio S., Bruno G. da Costa, and Maike A. F. dos Santos.
2019. "Majorization and Dynamics of Continuous Distributions" *Entropy* 21, no. 6: 590.
https://doi.org/10.3390/e21060590