Brownian motion, diffusion, entropy and econophysics

To model wealth distributions there exist models based on the Boltzmann-Gibbs distribution (BGD), which is obtained by simulating binary economic interactions or exchanges that are similar to particle collisions in physics with conserved energy (or money in econophysics). Also, BGD can be reproduced by numerical simulations of diffusion for many particles which experience energy fluctuations. This latter case is analogous to non-interacting pollen particles performing Brownian motion. In order to decrease inequality, we also modify the energy-conserved diffusion by taxing the richest agent. In all cases, we calculate the corresponding Gini inequality index and the time evolution of the entropy to show the stability of the statistical distributions.