The phase-transition in the Vicsek model through Gini index

At the critical point of a phase transition, the fourth-order Binder cumulant (U), which is a measure of departure from Gaussianity in the order parameter, becomes independent of the system size and obeys a finite-size scaling relation with the correlation length exponent. This feature has been widely used to estimate the critical point and exponents accurately from simulations. Recently, the Gini index, a measure of inequality that traditionally is used in economics to quantify wealth inequality, has been shown to be useful for studying phase transitions in physical systems at equilibrium. By studying phase transition in equilibrium, it has been numerically demonstrated that at the critical point, g in the order parameter becomes independent of the system size and follows a finite-size scaling relation similar to U. Here, we investigate the nonequilibrium phase transition in the Vicsek model of active systems in two dimensions using the index g. For this model, we find that both g and U exhibit similar behavior. For a high self-propelled velocity, both g and U are system size independent at the critical point and obey a finite-size scaling relation for the correlation length exponent. For small self-propulsion, we find the transition is unusual, as there is no single point where g (or U) for various system sizes crosses.