Entropic Geometry of Crowd Dynamics
  We propose an entropic geometrical model of psycho-physical crowd dynamics (with dissipative crowd kinematics), using Feynman action-amplitude formalism that operates on three synergetic levels: macro, meso and micro. The intent is to explain the dynamics of crowds simultaneously and consistently across these three levels, in order to characterize their geometrical properties particularly with respect to behavior regimes and the state changes between them. Its most natural statistical descriptor is crowd entropy $S$ that satisfies the Prigogine's extended second law of thermodynamics, $\partial_tS\geq 0$ (for any nonisolated multi-component system). Qualitative similarities and superpositions between individual and crowd configuration manifolds motivate our claim that goal-directed crowd movement operates under entropy conservation, $\partial_tS = 0$, while natural crowd dynamics operates under (monotonically) increasing entropy function, $\partial_tS > 0$. Between these two distinct topological phases lies a phase transition with a chaotic inter-phase. Both inertial crowd dynamics and its dissipative kinematics represent diffusion processes on the crowd manifold governed by the Ricci flow, with the associated Perelman entropy-action.   Keywords: Crowd psycho-physical dynamics, action-amplitude formalism, crowd manifold, Ricci flow, Perelman entropy, topological phase transition 
Vladimir G. Ivancevic and Darryn J. Reid