Crowds reaching targets by maximizing entropy : a Clausius-Duhem inequality approach

In this paper we make a connection between the study of crowd dynamics and concepts from thermodynamics. The basic component of our continuous model is the continuity equation for the density of people, which we complete by prescribing the velocity field. This velocity field includes a nonlocal term modelling interactions between individuals. To provide support for our modelling assumptions we wish to prove an inequality that resembles the Second Law of Thermodynamics. To this end we define an entropy-like functional and 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 performing simulations for the corresponding particle system. In fact, this term is sufficiently small for the entropy still to increase. Moreover, we can show that the entropy converges in time towards a limit value.