# Contrarian Majority Rule Model with External Oscillating Propaganda and Individual Inertias

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## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Numerical Results

#### 3.1. Evolution of the Magnetization

#### 3.2. Residence Times

#### 3.3. Stochastic Resonance

## 4. Mean-Field Approach

## 5. Summary and Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Glossary

Symbol | Name |

${s}_{i}=\pm 1$ | Opinion of agent i; |

N | Number of agents; |

t | Time; |

$\Delta t=1/N$ | Time step; |

$p\left(t\right)$ | Majority probability; |

$H\left(t\right)$ | External field; |

${H}_{0}$ | Amplitude of $H\left(t\right)$; |

$\tau $ | Period of $H\left(t\right)$; |

T | Temperature; |

${T}^{*}$ | Resonance temperature; |

${T}_{c}$ | Transition temperature; |

${\sigma}_{\pm}$ | Fraction of agents with $\pm 1$ opinion; |

m | Magnetization; |

$\langle \left|m\right|\rangle $ | Ensemble average of absolute value of m; |

$\overline{m}$ | Time average of m; |

${t}_{r}$ | Residence time; |

$RTD$ | Residence time distribution; |

$\mathcal{A}$ | Response; |

GMM | Galam majority rule; |

VM | Voter Model; |

MF | Mean field; |

SR | Stochastic Resonance. |

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**Figure 1.**(

**a**) Time evolution of the average value of the absolute magnetization $\left|m\right|$ in a population of $N={10}^{3}$ agents, zero field $H=0$ and various values of majority probability $p={(1+{e}^{-2/T})}^{-1}$, as indicated in the legend. (

**b**) Stationary value of $\langle \left|m\right|\rangle $ vs. temperature T for constant fields $H=0.0$ (circles), $H=0.1$ (squares) and $H=0.5$ (diamonds). The solid line is the analytical expression from Equation (11), while the dashed lines are the numerical integration of Equation (8). The averages were performed over ${10}^{3}$ independent realizations starting from a symmetric condition ${m}_{0}=0$.

**Figure 2.**Time evolution of the magnetization m in a single realization for a population of $N=1024$ agents under the influence of an oscillating field with period $\tau =256$ and amplitudes ${H}_{0}=0.1$ and $0.5$, panels (

**a**) and (

**b**), respectively, and the temperatures indicated in the legends. Solid lines correspond to MC simulations, while dashed lines in panel (

**a**) represent the numerical integration of Equation (8).

**Figure 3.**Normalized histograms of the residence time ${t}_{r}$ in a system of $N=1025$ agents under a field of amplitude ${H}_{0}=0.1$, period $\tau =256$ (

**a**) and $\tau =1024$ (

**b**), and the temperatures indicated in the legends. The bottom-right panels are on a linear-log scale.

**Figure 4.**(

**a**) Response $\mathcal{A}$ as a function of the temperature T for a field of amplitude ${H}_{0}=0.1$ and period $\tau $ indicated in the legend. (

**b**) Resonance temperature ${T}^{*}$ [maximum of $\mathcal{A}$ vs. T curves from (

**a**)] and transition temperature ${T}_{c}$ vs. period $\tau $.

**Figure 5.**(

**a**) Time evolution of the magnetization m from Equation (8) for a field of amplitude ${H}_{0}=0.1$, period $\tau =256$ and the temperatures indicated in the legend. Horizontal dashed lines represent the time average value of m, $\overline{m}$, in the interval $t\in (0,1000\tau )$. (

**b**) Time average of the magnetization, $\overline{m}$, vs. temperature T for the field’s periods indicated in the legend. The inset shows a closer look around the transition values ${T}_{c}$.

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**MDPI and ACS Style**

Gimenez, M.C.; Reinaudi, L.; Galam, S.; Vazquez, F.
Contrarian Majority Rule Model with External Oscillating Propaganda and Individual Inertias. *Entropy* **2023**, *25*, 1402.
https://doi.org/10.3390/e25101402

**AMA Style**

Gimenez MC, Reinaudi L, Galam S, Vazquez F.
Contrarian Majority Rule Model with External Oscillating Propaganda and Individual Inertias. *Entropy*. 2023; 25(10):1402.
https://doi.org/10.3390/e25101402

**Chicago/Turabian Style**

Gimenez, Maria Cecilia, Luis Reinaudi, Serge Galam, and Federico Vazquez.
2023. "Contrarian Majority Rule Model with External Oscillating Propaganda and Individual Inertias" *Entropy* 25, no. 10: 1402.
https://doi.org/10.3390/e25101402