# Statistics of Correlations and Fluctuations in a Stochastic Model of Wealth Exchange

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## Abstract

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## 1. Introduction

## 2. Simulations for a Fixed Value of the Initial Total Income and Gini Index

#### 2.1. Langevin Equation. Deterministic Solutions

#### 2.2. Stochastic Time-Series

## 3. Dependence of the Correlations on the Total Income and on G

## 4. Langevin Equation with Ornstein–Uhlenbeck Noise and Dependence of the Correlations on $\Gamma $ and $\tau $

## 5. Fluctuations of the Populations in Dependence from $\Gamma $

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Dependence of the correlation ${R}_{GM}$ on the Gini index with mixed noise in a range which extends far below the usual values of G. Each dot represents the average of 80 realizations with 2500 steps. There are 480 dots in total.

**Figure 2.**Histogram of a typical deterministic equilibrium configuration, with 10 income classes (see details in the text). The bars represent the populations ${x}_{1},\dots ,{x}_{10}$.

**Figure 3.**Behavior of the Gini index G and the social mobility M as functions of the total income $\mu $ for the deterministic model, at equilibrium. The asymptotic equilibrium of the deterministic equations does not depend on the detailed initial conditions ${x}_{i}(0)$ (class populations), but only on the total income $\mu =\mu (0)={\sum}_{i}{r}_{i}{x}_{i}(0)$ ($\mu $ is conserved in the deterministic evolution). This plot is obtained by taking several values of $\mu (0)$ as explained in Section 2.1, letting the system evolve deterministically and computing G and M in the equilibrium state. The relation between G and $\mu $ allows to determine a range of values of $\mu $ which corresponds to a range of realistic values of G.

**Figure 4.**Example of time-series of G and $\mu $ in the canonical case, with multiplicative white noise of amplitude $\Gamma =0.01$. The time steps in the series are 40,000, with a sampling each 100 steps.

**Figure 5.**Time auto-correlation of G in the time-series of Figure 4. T denotes the number of time steps.

**Figure 6.**Histogram of the correlation ${R}_{GM}$ in 6000 realizations of 5000 steps, with multiplicative noise of amplitude $\Gamma =0.001$.

**Figure 8.**Correlation ${R}_{GM}$ for Ornstein-Uhlenbeck noise in dependence on noise amplitude $\Gamma $ and memory time $\tau $. Each dot is the average of 50 realizations with 5000 steps.

**Figure 11.**Average deviation ${\delta}_{av}$ from deterministic equilibrium in dependence on the noise amplitude $\Gamma $, for multiplicative white noise. Each dot is the average of 500 realizations with 5000 steps.

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**MDPI and ACS Style**

Bertotti, M.L.; Chattopadhyay, A.K.; Modanese, G.
Statistics of Correlations and Fluctuations in a Stochastic Model of Wealth Exchange. *Entropy* **2018**, *20*, 166.
https://doi.org/10.3390/e20030166

**AMA Style**

Bertotti ML, Chattopadhyay AK, Modanese G.
Statistics of Correlations and Fluctuations in a Stochastic Model of Wealth Exchange. *Entropy*. 2018; 20(3):166.
https://doi.org/10.3390/e20030166

**Chicago/Turabian Style**

Bertotti, Maria Letizia, Amit K. Chattopadhyay, and Giovanni Modanese.
2018. "Statistics of Correlations and Fluctuations in a Stochastic Model of Wealth Exchange" *Entropy* 20, no. 3: 166.
https://doi.org/10.3390/e20030166