Complex Systems with Trivial Dynamics

TL;DR

This paper explores how simple dynamic rules in complex systems can lead to well-known statistical distributions, using geometric and non-linear models to explain their evolution toward equilibrium.

Contribution

It demonstrates the link between system geometry and resulting statistical distributions, introducing two non-linear models for out-of-equilibrium decay.

Findings

Exponential wealth distribution from hyperplanar geometry

Gaussian velocity distribution from spherical geometry

Two non-linear models explaining decay to equilibrium

Abstract

In this communication, complex systems with a near trivial dynamics are addressed. First, under the hypothesis of equiprobability in the asymptotic equilibrium, it is shown that the (hyper) planar geometry of an $N$-dimensional multi-agent economic system implies the exponential (Boltzmann-Gibss) wealth distribution and that the spherical geometry of a gas of particles implies the Gaussian (Maxwellian) distribution of velocities. Moreover, two non-linear models are proposed to explain the decay of these statistical systems from an out-of-equilibrium situation toward their asymptotic equilibrium states.