Inequality of Chances as a Symmetry Phase Transition

Abstract

Abstract:We propose a model for Lorenz curves. It provides two-parameter fits to data on incomes, electric consumption, life expectation and rate of survival after cancer. Graphs result from the condition of maximum entropy and from the symmetry of statistical distributions. Differences in populations composing a binary system (poor and rich, young and old, etc.) bring about chance inequality. Symmetrical distributions insure equality of chances, generate Gini coefficients G i  ⅓, and imply that nobody gets more than twice the per capita benefit. Graphs generated by different symmetric distributions, but having the same Gini coefficient, intersect an even number of times. The change toward asymmetric distributions follows the pattern set by second-order phase transitions in physics, in particular universality: Lorenz plots reduce to a single universal curve after normalisation and scaling. The order parameter is the difference between cumulated benefit fractions for equal and unequal chances. The model also introduces new parameters: a cohesion range describing the extent of apparent equality in an unequal society, a poor-rich asymmetry parameter, and a new Gini-like indicator that measures unequal-chance inequality and admits a theoretical expression in closed form.

Symmetry, Entropy, Diversity and (Why Not?) Quantum Statistics in Society

“…They have been visualized as nonequilibrium dynamic networks of voters [ 13 ]. A significant difference between them results here from the assumption of a symmetry phase transition in a previous paper [ 1 ]. A comparison of predictions against real data is required, and is the object of this paper.…”

“…Symmetry is just incompatible with . As shown in [ 1 ], symmetric distributions have , while asymmetry imposes , actually the usual case in real life. Now, since asymmetric distributions and Gini values above 1/3 do exist, a symmetry change―a phase transition―must take place, which is expected to be at .…”

“…In fact, important differences between such deciles are found practically everywhere, and this requires, as shown below, Gini [ 4 ] coefficients . The assumption of a symmetry phase transition, similar to that in binary alloys [ 5 , 6 ] and superconductors [ 7 ], provides very good fits to data [ 1 ]. Four cases in this paper hint indeed at asymmetric distributions as the real-world rule.…”

“…Societies evolve, produce and consume, and therefore we describe them here as nonequilibrium, interacting, entropy producing and asymmetrically distributed statistical systems with a large number of degrees of freedom. We apply the resulting theory to the prediction of data in the previous paper: they display a rather large variety of fitting parameters and ranges of interaction, necessary to test model forecasts [ 1 ]. Two types of data are fitted, economic and demographic, and two examples of diversity are discussed in each of them: incomes in the USA [ 14 ] and per capita electricity consumption [ 15 ] in 170 countries illustrate the first case; life expectancy up to one hundred years [ 14 ] and survival after cancer [ 16 ], describe the second one.…”

“…In previous papers [1], [2], we fitted Lorenz inequality curves [3] non-thermodynamic quantities at first sightwith a simple model of social entropy. Symmetric distribution laws predict equal probabilities of being in the oldest or in the youngest decile, or to belong either to the richest or to the poorest one.…”