# Wealth Rheology

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## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Monte Carlo Simulations

- a complete equality configuration where ${W}_{a,0}=1$ for all $a=1,\dots ,N$;
- an equilibrium configuration, where ${W}_{a,0}$ are drawn independently of each other from the inverse gamma distribution (4).

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Rank Correlations

## References

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**Figure 1.**(

**a**) Gini coefficient G (8) plotted as a function of $\alpha $ (solid line) and computed numerically from samples generated in Monte Carlo simulations for $N={10}^{4}$ (symbols). Different symbols correspond to $\sigma =0.02$, $0.04$ and $0.08$. (

**b**) The auto-correlation time (1) ${\tau}_{\mathrm{ac}}$ and the exponential time (2) ${\tau}_{\mathrm{exp}}$ for the Gini coefficient G measured for consecutive configurations in the stationary state for $\alpha =2$ for different $\sigma $. When $\sigma $ decreases ${\tau}_{\mathrm{ac}}$ grows as ${\sigma}^{-x}$ with $x=2.849\left(50\right)$, and ${\tau}_{\mathrm{exp}}$ grows as ${\sigma}^{-y}$ with $y=1.617\left(15\right)$. (

**c**) Evolution of the Gini coefficient G from the ‘cold’ start, $G=0$, towards the stationary state’s value $G=0.5$ for $\alpha =2.0$. The plots correspond to $\sigma =0.02$, $0.04$ and $0.08$. Please note logarithmic scale on the time axis. (

**d**) Evolution of the Gini coefficient G from the ‘hot’ start. The values fluctuate about the stationary state value $G=0.5$ for $\alpha =2.0$. The plots correspond to $\sigma =0.02$, $0.04$ and $0.08$.

**Figure 2.**Rank correlations coefficients (

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

**b**) $\rho $ for steady state configurations separated by k time steps in the simulated systems for $N={10}^{4}$ and for various values of $\alpha =2$, 3, 4 and for various values of $\sigma =0.02$, $0.04$ and $0.08$. Please note logarithmic scale on the time axis. The first parameter in the legend is the value of $\alpha $ and the second is the value of $\sigma $.

**Figure 3.**Dependence of the overlap of top 100 lists at times ${k}_{1}$ and ${k}_{2}$ on the separation time $k={k}_{2}-{k}_{1}$. The overlap is measured as the percentage of people that are on both the lists. The data points are obtained by averaging over pairs of ${k}_{1}$, ${k}_{2}$ such that ${k}_{2}-{k}_{1}=k$. They are plotted against the universal argument $x=k{\sigma}^{2}(\alpha -1)$. The data is obtained by simulations of the model for $N={10}^{4}$, and for different combinations of $\alpha =2$, 3, 4 and $\sigma =0.02$, $0.04$, $0.08$. The first parameter in the legend is the value of $\alpha $ and the second is the value of $\sigma $. The data is fitted with the Formula (12) with $A=0.7570\left(26\right)$ and $B=0.3341\left(30\right)$. The fit is shown with a solid line.

**Figure 5.**Rank coefficients for the richest people in Germany, Poland, the US and the world based on real-world data published in Refs. [11,12,13]. (

**a**) Overlap ratios ${\overline{\mathsf{\Omega}}}_{100}\left(t\right)$ for the four systems. The best fit of the Formula (12), with $A=0.1536\left(42\right)$ and $B=0.0425\left(14\right)$, to the world data is shown with a solid line. (

**b**) Goodman–Kruskal’s $\gamma $ coefficients for the four system.

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Burda, Z.; Krawczyk, M.J.; Malarz, K.; Snarska, M. Wealth Rheology. *Entropy* **2021**, *23*, 842.
https://doi.org/10.3390/e23070842

**AMA Style**

Burda Z, Krawczyk MJ, Malarz K, Snarska M. Wealth Rheology. *Entropy*. 2021; 23(7):842.
https://doi.org/10.3390/e23070842

**Chicago/Turabian Style**

Burda, Zdzislaw, Malgorzata J. Krawczyk, Krzysztof Malarz, and Malgorzata Snarska. 2021. "Wealth Rheology" *Entropy* 23, no. 7: 842.
https://doi.org/10.3390/e23070842