Toward ideal gas law for crowds with large pressures*

Human collective behavior exhibits a wide range of phenomena. The humans (particles) in the crowds are active, and one may not expect the speed distribution to follow the 2D Maxwell-Boltzmann distribution. However, in many of the cases, the measured/simulated human crowd does follow the Maxwell-Boltzmann distribution. Our study reveals the criterion for crowd speed (corresponds to fluctuation velocity) to follow the Maxwell-Boltzmann distribution. We reveal that the speed PDF becomes more similar to the Maxwell-Boltzmann distribution as the pressure increases, and there is a power law relationship between the mean square error of fit and pressure. To explain the possible reason, we claim that it is the relative magnitude between the relaxation time of individuals and the collision time scale. We reveal that our result is general regardless of the boundary conditions of the crowd. With the Maxwell-Boltzmann speed distribution, one can directly imply that high-pressure crowds follow the ideal gas law. This result will mark a clear pathway to study crowds with high pressures.