Wealth Thermalization Hypothesis and Social Networks

TL;DR

The paper formulates the Wealth Thermalization Hypothesis (WTH), positing that wealth in closed social systems follows a Rayleigh-Jeans (RJ) distribution with two conserved quantities: total wealth $E$ and norm $\mathcal{N}$. By mapping wealth to RJ energy and exploring both equilibrium (Part I) and dynamical (Part II) aspects, it demonstrates that RJ condensation naturally yields extreme inequality, as seen in Lorenz curves and Gini coefficients for world data and stock markets. The authors extend the RJ framework to nonlinear perturbations of Random Matrix Theory and to nonlinear social networks, showing that above a chaos border the system dynamically thermalizes to RJ distributions, with entropy measures $S_q$ and $S_B$ converging to RJ predictions. They validate the theory against empirical wealth data and company capitalizations, and show that RJ extended models (RJE) offer improved quantitative fits across diverse datasets. The work thus links chaotic dynamics and spectrum structure to macroeconomic inequality, suggesting that increasing the effective energy scale $\varepsilon$ could mitigate condensation and inequality, with implications for policy and future modeling of socio-economic systems.

Abstract

In 1955 Fermi, Pasta, Ulam and Tsingou performed first numerical studies with the aim to obtain the thermalization in a chain of nonlinear oscillators from dynamical equations of motion. This model happend to have several specific features and the dynamical thermalization was established only later in other studies. In this work we study more generic models based on Random Matrix Theory and social networks with a nonlinear perturbation leading to dynamical thermalization above a certain chaos border. These systems have two integrals of motion being total energy and norm so that the theoretical Rayleigh-Jeans thermal distribution depends on temperature and chemical potential. We introduce the wealth thermalization hypothesis according to which the society wealth is associated with energy in the Rayleigh-Jeans distribution. At relatively small values of total wealth or energy there is a formation of the Rayleigh-Jeans condensate, well studied in physical systems such as multimode optical fibers. This condensation leads to a huge fraction of poor households at low wealth and a small oligarchic fraction which monopolizes a dominant fraction of total wealth thus generating a strong inequality in human society. We show that this thermalization gives a good description of real data of Lorenz curves of US, UK, the whole world and capitalization of companies at Stock Exchange of New York SE (NYSE), London and Hong Kong. It is also shown that above a chaos border the dynamical Rayleigh-Jeans thermalization takes place also in social networks with the Lorenz curves being similar to those of wealth distribution in world countries. Possible actions for inequality reduction are briefly discussed.

Figures (49)

• Figure I.1: Lorenz curves for the RJS model with the linear spectrum $E_m=m/N$ (for $N=10000$) for different values of the rescaled energy $\varepsilon=E/B$. The $x$-axis corresponds the cumulated fraction of households ($h$) and the $y$-axis to the cumulated fraction of wealth ($w$). The dashed line is the line of perfect equipartition $w=h$. The Gini coefficients $G$ for all curves are $G=0.9600,\,0.9000,\,0.8006,\,0.6250,\,0.4990,\,0.4066,\,0.3333$ (bottom to top).
• Figure I.2: Color plot of wealth $w$ from Lorenz curves of the RJS model ($N=10000$). The $x$-axis corresponds to the fraction of households $h\in[0, 1]$ and the $y$-axis to the rescaled energy $\varepsilon=E/B\in[0, 0.5]$. The ticks mark integer multiples of 0.1 for $h$ and $\varepsilon$.
• Figure I.3: Comparison of the Lorenz curves for US 2019 (black), UK 2012-2014 (blue), World 2021 (dashed green) with those of RJS model (red curves; $N=10000$); US and World curves are rather close. For the three referenced curves Gini coefficients are $G=0.852$, $0.626$, $G=0.842$ respectively and the rescaled energies $\varepsilon=E/B$ of RJS model are respectively fixed as $\varepsilon=0.07420$, $\varepsilon=0.1996$, $\varepsilon=0.07911$ so that the corresponding Gini coefficients match the referenced data.
• Figure I.4: Both panels compare the Lorenz curves for different data sets (black for US 2019, blue for UK 2012-2014 and green dashed for World 2021) with those of the RMT model (a) and the DL model (b). As in Fig. \ref{['figI_3']} the Gini coefficients $G$ of the reference curves are used to fix the rescaled energy $\varepsilon=E/B$ of the corresponding model such that the model curves (red) have the same $G$. For the RMT model (a) only two data sets are shown $\varepsilon=0.07996$ (US) and $\varepsilon=0.2027$ (UK). For the DL model (b) the parameter values are $a=16$ (US and World) and $a=3$ (UK). These values are fixed to have a best possible fit of the model data with those of the reference curves. The chosen values $\varepsilon=0.01434$ (US), $\varepsilon=0.1355$ (UK), $\varepsilon=0.01535$ (World) match the $G$ values of the reference data. In (b) the curves for US and World are rather close and a zoomed view is given in Appendix Figure \ref{['figA4']}. In (a) the RMT Lorenz curves are shown for one realization of a random matrix, other realizations give practically the same curves.
• Figure I.5: Gini coefficient versus rescaled energy $\varepsilon=(E-E_0)/(E_{N-1}-E_0)$ for the RJS model (red), RMT model (green), DL model (blue; only for the case $a=16$), and the EQI model (pink for the offset $E_0=0.1$ and cyan for $E_0=1$; same values of $E_0$ are used in Appendix Figure \ref{['figA6']}). The thin black lines show the values of $G=0.852$, $G=0.842$ and $0.626$ for the data of US 2019, World 2021 and UK 2014. The intersection of these lines with the red and green curves correspond to $\varepsilon$ values used in Figs. \ref{['figI_3']}, \ref{['figI_4']}.