群体通过最大熵到达目标：Clausius-Duhem不等式方法
Crowds reaching targets by maximizing entropy: a Clausius-Duhem inequality approach
摘要：
在本文中，我们提出了使用热力学的概念在研究人群动力学。我们的连续模型包括连续方程的人群密度和速度场的动力学方程。后者包括一个非局部项，用于模拟个体之间的相互作用。为了支持我们的建模假设，我们引入了一个类似于热力学第二定律的不等式，其中包含一个类似熵的泛函。我们证明了它的时间导数等于一个正耗散项减去一个校正项。后一项应该很小，以便熵的时间导数为正。在各向同性相互作用的情况下，校正项不存在。对于各向异性的情况下，我们支持的索赔，校正项是小的模拟相应的粒子系统。他们揭示了这个项足够小，熵仍然增加。此外，我们表明，熵收敛在时间上的极限值。
英文摘要：
In this paper we propose the use of concepts from thermodynamics in the study of crowd dynamics. Our continuous model consists of the continuity equation for the density of the crowd and a kinetic equation for the velocity field. The latter includes a nonlocal term that models interactions between individuals. To support our modelling assumptions, we introduce an inequality that resembles the Second Law of Thermodynamics, containing an entropy-like functional. We show that its time derivative equals a positive dissipation term minus a corrector term. The latter term should be small for the time derivative of the entropy to be positive. In case of isotropic interactions the corrector term is absent. For the anisotropic case, we support the claim that the corrector term is small by simulations for the corresponding particle system. They reveal that this term is sufficiently small for the entropy still to increase. Moreover, we show that the entropy converges in time towards a limit value.
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