# Statistics of Binary Exchange of Energy or Money

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Energy Transfer in an Elastic Collision

#### 2.1. General Expression

#### 2.2. Numerical Solution of the Conservation Laws

#### 2.3. Monte Carlo Evaluation of the Integral on Final States

- A numerical value of $p/m$ is chosen, in this example $p/m=0.01$, and Equation (12) is solved numerically, obtaining ${p}_{1y}$ as a function g of ${p}_{1x}$: ${p}_{1y}=g\left({p}_{1x}\right)$.
- A large number of random values of ${p}_{1x}$ (typically ${10}^{6}$ or ${10}^{7}$) is generated in the interval ${I}_{1}$.
- The corresponding value of R is calculated from the definition (4) using the function $g\left({p}_{1x}\right)$ and Equations (1) and (2).
- Each value of R is multiplied by the weight $\sqrt{{({p}_{1x}^{\prime})}^{2}+{({p}_{1y}^{\prime})}^{2}+{({p}_{2x}^{\prime})}^{2}+{({p}_{2y}^{\prime})}^{2}}$.
- In the graph, which is actually a histogram, the vertical axis represents, for 50 discretized values of R in the interval $0\le R\le 1/2$, the weighted number of counts, i.e., the number of times R has taken that value, multiplied by the weight. The part of the histogram for $1/2\le R<1$ is symmetrical, and not shown.

## 3. Agent-Based Simulations and Saving Propensity

`Mathematica`in our case) a vector $\mathbf{E}$ with ${10}^{5}$ components. The component ${E}_{i}$ represents the wealth of the i-th agent of our idealized society. At the beginning all agents have the same wealth, e.g., ${E}_{i}=20$ units for any i in our simulation. Then the simulation algorithm choses at random two integer indices i, j between 1 and ${10}^{5}$. These are the agents who are currently interacting. Their wealths are changed according to Equation (15). The random choice of agents and the modification of their wealths is repeated for ${10}^{8}$ times. This means that, on average, each agent has interacted with others for ${10}^{3}$ times. (Thermalization times larger than ${10}^{8}$ steps are found to give no changes in the final results.) A histogram of the wealth distribution is generated by rounding the wealth of each agent to the nearest integer and then using a counting function to define the frequency, for instance in

`Mathematica`in our case with the code

`frequency = Table[Count[X,n],{n,0,120}]`, where

`X`is the rounded income vector and 120 is a safe maximum value for the income. The

`frequency`vector is the one whose components are shown in the histograms in Figure 4 and Figure 5. Next we average the frequency vector over many realizations, in order to reduce the fluctuations, we normalize it by dividing each component by the sum of all components, and we use a

`FindFit`function to fit it to the $\kappa $-distribution (16). This function employs a least-squares method and returns the best fit values of the three parameters in the $\kappa $-distribution. When the fit fails to show a power-law tail, it is because it returns negative values for $\alpha $ and $\beta $, which are not admissible for a $\kappa $-distribution.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Relativistic Energy and Momentum

## Appendix B. Wealth Exchange and “Yard Sale” Models in Econophysics

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**Figure 1.**The function $g\left({p}_{1x}\right)$ obtained by numerical solution of the differential Equation (12), for a strongly relativistic case ($p/m=\gamma v=100\simeq \gamma $; $v\simeq 1$). We use units $c=1$ throughout the paper, and in this plot we also take $m=1$ for graphical reasons.

**Figure 2.**Plot of the distribution of R in a Monte Carlo simulation of $5\times {10}^{6}$ almost classical collisions ($p/m=\gamma v=0.01$, $\gamma \simeq 1$, $v\simeq {10}^{-2}$). R is expressed in % and is the fraction of kinetic energy transferred from Particle 1 to Particle 2. When $p/m$ becomes even smaller, in the fully classical case, the peak for $R\to 0$ (corresponding to a symmetric peak for $R\to 1$) becomes higher. The dot at $R=51\%$ is not meaningful.

**Figure 3.**Plot of the distribution of R in a Monte Carlo simulation of $5\times {10}^{6}$ collisions, in a fully relativistic case ($p/m=\gamma v=1000\simeq \gamma $, $v\simeq 1$). R is expressed in % and is the fraction of kinetic energy transferred from Particle 1 to Particle 2. The dot at $R=51\%$ is not meaningful.

**Figure 4.**Income distribution from an agent-based simulation (see details in text) where agents exchange money with a fixed saving propensity. A fit with the $\kappa $-distribution confirms that there is no Pareto tail.

**Figure 5.**Income distribution from an agent-based simulation (see details in text) with agents which exchange money in analogy with the behavior of relativistic particles, i.e., all exchange rates being equiprobable. The distribution has a Pareto tail proportional to $1/{E}^{2.75}$, where E is the income.

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**MDPI and ACS Style**

Bertotti, M.L.; Modanese, G.
Statistics of Binary Exchange of Energy or Money. *Entropy* **2017**, *19*, 465.
https://doi.org/10.3390/e19090465

**AMA Style**

Bertotti ML, Modanese G.
Statistics of Binary Exchange of Energy or Money. *Entropy*. 2017; 19(9):465.
https://doi.org/10.3390/e19090465

**Chicago/Turabian Style**

Bertotti, Maria Letizia, and Giovanni Modanese.
2017. "Statistics of Binary Exchange of Energy or Money" *Entropy* 19, no. 9: 465.
https://doi.org/10.3390/e19090465