On the ideal gas law for crowds with high pressure

Active particle systems, such as human crowds, are out of equilibrium posing a significant challenge in identifying a suitable equation of state. However, several previous observations suggest that a crowd’s speed distribution may conform to a two-dimensional Maxwell–Boltzmann distribution under certain yet-to-be-determined conditions. Our research uncovers that the divergence between the fluctuation velocity magnitude’s probability density function and its best 2D Maxwell–Boltzmann fit diminishes in a power-law fashion with decreasing collision time scale between individuals. These findings are robustly supported by both experimental data and simulations with diverse boundary conditions and force potentials. The equilibrium characteristics of crowds are interpreted through a canonical ensemble framework. Furthermore, we show that high pressure indicates equilibrium characteristics in human crowds. Remarkably, we reveal the ideal gas law for human crowds without resorting to any behavior assumption. We demonstrate the predictive capability of the ideal gas law on both observational and modeling data. Our research highlights a new pathway to explore and validate traditional thermodynamic quantities and laws in the setting of high-pressure human crowds, advancing our understanding of active matter systems.