Dynamical thermalization and turbulence in social stratification models

Abstract

We study the nonlinear chaotic dynamics in a system of linear oscillators coupled by social network links with an additional stratification of oscillator energies, or frequencies, and supplementary nonlinear interactions. It is argued that this system can be viewed as a model of social stratification in a society with nonlinear interacting agents with energies playing a role of wealth states of society. The Hamiltonian evolution is characterized by two integrals of motion being energy and probability norm. Above a certain chaos border the chaotic dynamics leads to dynamical thermalization with the Rayleigh-Jeans (RJ) distribution over states with given energy or wealth. At low energies, this distribution has RJ condensation of norm at low energy modes. We point out a similarity of this condensation with the wealth inequality in the world countries where about a half of population owns only a couple of percent of the total wealth. In the presence of energy pumping and absorption, the system reveals features of the Kolmogorov-Zakharov turbulence of nonlinear waves.

Figures (12)

• Figure 1: Rescaled density of states $\nu(E)/N$ with $\nu(E) = d m/dE_m$ for the energy eigenvalues $E_m$ of the matrix $H$ for the SSS model at $W=2,4,8,16$ given by (\ref{['eqHdef']}) for $f=0.1$ and $\kappa=0.5$. The normalization is given by $\int dE\, \nu(E)=N$.
• Figure 2: Energy dependence of IPR $\xi$ of eigenstates of the matrix $H$ for the SSS model at $W=2, 4, 8, 16$ given by (\ref{['eqHdef']}) for $f=0.1$ and $\kappa=0.5$.
• Figure 3: The left (right) panel shows the temperature $T$ (the chemical potential $\mu$) versus energy $E$ for the SSS model and the RJ case at $W= 8$, $f=0.1$, $\kappa=0.5$, $N=379$. The dashed black lines in the right panel correspond to the values of $E_1=-4.09$ and $E_N=3.99$ (minimal and maximal energies $E_m$ for the used random realization of $H$) showing that either $\mu<E_1$ (for $T>0$) or $\mu>E_N$ (for $T<0$). The transition from $T<0$ to $T>0$ happens at $E=E_c=-0.053$ for the given random realization of $H$ used to compute the spectrum $E_m$.
• Figure 4: Time dependence of von Neumann entropy $S_q(t)$ (full red line) and (rescaled) Boltzmann entropy $S_B/N$ (full green line) for the SSS model; here $W=8$, $N=379$, $\beta=2$ (top) and $\beta=4$ (bottom); initial state is the linear eigenmode at $m_0=20$, i.e. $C_m(t=0)=\delta_{m,m_0}$. Entropy values are computed from (\ref{['eqentropy']}) using time averaged $\rho_m=\langle |C_m(t)|^2\rangle$ values for successive time intervals $2^{l-1}\le t<2^{l}$ for $l=1,2,3,\ldots,30$. The dashed lines show the theoretical thermalized values, using (\ref{['eqentropy']}) and the RJ values for $\rho_m$ given by (\ref{['eqrj']}).
• Figure 5: Dependence of von Neumann entropy $S_q$ on the average linear energy $E=\sum_m E_m\rho_m\approx {\cal H}$ for several initial eigenstates for different $m_0$ and $t$ values for $\beta=2$ (top) and $\beta=4$ (bottom); the other parameters are $W=8$, $f=0.1$, $\kappa=0.5$ and $N=379$. The blue line shows the theoretical thermalized value of $S_q$. As in Fig. \ref{['fig4']}, $S_q$ is computed from (\ref{['eqentropy']}) using either numerically averaged $\rho_m$ values (data points) or the thermalized RJ values (\ref{['eqrj']}) (blue line).