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.

中文翻译：

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论高压人群的理想气体定律

活性粒子系统（例如人群）失去平衡，这对确定合适的状态方程构成了重大挑战。然而，之前的一些观察表明，在某些尚未确定的条件下，人群的速度分布可能符合二维麦克斯韦-玻尔兹曼分布。我们的研究发现，随着个体之间碰撞时间尺度的减小，波动速度幅度的概率密度函数与其最佳二维麦克斯韦-玻尔兹曼拟合之间的差异以幂律方式减小。这些发现得到了具有不同边界条件和力势的实验数据和模拟的有力支持。群体的均衡特征通过规范的集成框架来解释。此外，我们表明高压表明人群的平衡特征。值得注意的是，我们在不诉诸任何行为假设的情况下揭示了人类群体的理想气体定律。我们证明了理想气体定律对观测数据和建模数据的预测能力。我们的研究强调了在高压人群环境中探索和验证传统热力学量和定律的新途径，增进了我们对活性物质系统的理解。