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 conservedenergy (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.

In any legal chess position, we define an attacked-square entropy [Formula: see text] for either Black or White pieces in terms of the square occupation probability [Formula: see text], where [Formula: see text] is the number of all possible movements to square [Formula: see text] (free or occupied by an opponent’s piece) and [Formula: see text] is the total mobility defined as the sum of all possible movements. Thus, each attacked square contributes to the entropy according to its received “firepower” concentration. A simpler nonsquare-dependant equiprobable entropy [Formula: see text] in terms of equal probabilities [Formula: see text] always yields [Formula: see text]. On average, the difference [Formula: see text] is very large in the Opening phase and [Formula: see text] decreases faster for lower ranked players after move 25. A major cause of the reduction of [Formula: see text] during a game is material loss, which is an irreversible process. By game outcome, gaps in average [Formula: see text] among winners, draws and losers are larger for Amateur players than for Elite players, both in the Middlegame and Endgame. Statistically, Elite players exhibit narrower dispersions in [Formula: see text]. Also, the entropy rates of the Elite level fluctuate much less than the entropies of other levels. Density of attacks in the four-square central zone is very high in the Opening, specially for Elite players.