# A Continuous-Time Urn Model for a System of Activated Particles

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## Abstract

**:**

## 1. Introduction

#### 1.1. Physical Motivation

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**Activated particle systems:****Chemical reactions:**A chemical reaction where particles change into an active state and start new reactions could be described by the model.**Biological systems:**This model can be used to simulate biological system activation processes, such as the activation of enzymes or signaling pathways.

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**Stochastic behavior:****Random activation events:**Particle activation is a stochastic process that is impacted by random events in many physical systems. Randomness can be incorporated into the modeling of activation processes using the continuous-time urn model.**Noise in physical systems:**The model might be used to study how noise or fluctuations in a physical system can impact the activation and deactivation of particles.

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**Statistical physics:****Phase transitions:**The model can be connected to the study of phase transitions in statistical physics, where particles undergo a sudden change in behavior or state.**Thermodynamic equilibrium:**Understanding how systems of activated particles reach equilibrium states and the associated thermodynamics.

#### 1.2. Examples

#### 1.3. Continuous-Time Pólya Interpretation

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- Activation of a particle on standby at a given site: this concerns a ball of color $i\in \{1,\dots ,N\}$, which will inevitably be replaced by a ball of color 0.
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- Standby mode of a particle initially on the move: this concerns a ball of color 0 which will necessarily be replaced by a ball of color j chosen at random from among colors $1,\dots ,N$.

#### 1.4. Related Works

#### 1.5. Outline

## 2. Average Analysis

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- A ball of color 0 if $i\in \{1,\dots ,N\}$;
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- ball of color j chosen at random among colors $1,\dots ,N$ if $i=0$.

#### 2.1. Active Particles or Balls of Type 0

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

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- Note that the average number of balls of color 0 does not depend on the parameter N. Indeed, for balls of color 0 (active particles), the other colors do not differ in terms of behavior and are, therefore, not distinguishable.
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- When $N=1$ (two colors) and $\lambda =\mu =1$, we obtain a model that is very close to the one studied in [6], but is less general since the replacement matrix in [6] is given by$$\left[\begin{array}{cc}-\alpha & \beta \\ \alpha & -\beta \end{array}\right]$$
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- For the trivial case $M=N=1$ (one ball, two colors), we obtain for all $t>0$:$$\begin{array}{c}\hfill {X}_{0}\left(t\right)hasasdistribution\mathrm{Ber}\left(\frac{\lambda}{\lambda +\mu}\left(1-{e}^{(\lambda +\mu )}\right)\right)\end{array}$$

#### 2.2. Inactive Particles or Balls of Type 1

**Proposition**

**2.**

**Proof.**

## 3. Distribution Analysis

#### 3.1. Active Particles or Balls of Type 0

**Lemma**

**1.**

**Proof.**

**Theorem**

**1**

**Proof.**

**Remark**

**2.**

#### 3.2. Inactive Particles or Balls of Type 1

**Theorem**

**2.**

**Proof.**

**Remark**

**3.**

**Proposition**

**3.**

**Remark**

**4.**

## 4. Discussion

- By $\overline{\mu}=\frac{1}{N}{\sum}_{i=1}^{N}{\mu}_{\{\u21ddi\}}$ in the asymptotic distribution of $\left({X}_{0}\left(t\right)\right)$;
- By ${\mu}_{\{\u21ddi\}}$ in the asymptotic distribution of $\left({X}_{i}\left(t\right)\right)$.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**M particles moving between N sites. Particles in blue are inactive and those in red are active. The move time of an active particle between two sites, i and j, is random ∼$\mathcal{E}\left(\mu \right)$. The sleep time of an inactive particle ∼$\mathcal{E}\left(\lambda \right)$.

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**MDPI and ACS Style**

Aguech, R.; Mohamed, H.
A Continuous-Time Urn Model for a System of Activated Particles. *Mathematics* **2023**, *11*, 4967.
https://doi.org/10.3390/math11244967

**AMA Style**

Aguech R, Mohamed H.
A Continuous-Time Urn Model for a System of Activated Particles. *Mathematics*. 2023; 11(24):4967.
https://doi.org/10.3390/math11244967

**Chicago/Turabian Style**

Aguech, Rafik, and Hanene Mohamed.
2023. "A Continuous-Time Urn Model for a System of Activated Particles" *Mathematics* 11, no. 24: 4967.
https://doi.org/10.3390/math11244967