Kinetic description of collision avoidance in pedestrian crowds by sidestepping

Abstract

In this paper we study a kinetic model for pedestrians, who are assumed to adapt their motion towards a desired direction while avoiding collisions with others by stepping aside. These minimal microscopic interaction rules lead to complex emergent macroscopic phenomena, such as velocity alignment in unidirectional flows and lane or stripe formation in bidirectional flows. We start by discussing collision avoidance mechanisms at the microscopic scale, then we study the corresponding Boltzmann-type kinetic description and its hydrodynamic meanfield approximation in the grazing collision limit. In the spatially homogeneous case we prove directional alignment under specific conditions on the sidestepping rules for both the collisional and the mean-field model. In the spatially inhomogeneous case we illustrate, by means of various numerical experiments, the rich dynamics that the proposed model is able to reproduce.2010 Mathematics Subject Classification. 35Q20, 35Q70, 90B20, 91F99.

Kinetic-Controlled Hydrodynamics for Traffic Models with Driver-Assist Vehicles

“…As already set forth in Section 3.1, in order to study the quasi-invariant interaction regime of the Boltzmann-type equation (25) we consider the limit γ, ν, σ 2 → 0 + and assume σ 2 /γ → λ > 0, ν/γ → κ > 0, implying that both diffusive and control contributions balance asymptotically with interactions. Under the time scaling τ := γ 2 t we obtain the Fokker-Planck equation…”

“…x, w) dv dw . (36) Notice that this is the spatially inhomogeneous counterpart of (25), with x ∈ R denoting the space position of the vehicles. On the whole, the microscopic state of the vehicles is now defined by the position-speed pair (x, v) ∈ R × [0, 1] and f * = f * (t, x, v) is its probability density at time t ≥ 0.…”

Hybrid Stochastic Kinetic Description of Two-Dimensional Traffic Dynamics

“…Since then it has been widely used in the literature to study the largetime trends of e.g. traffic flow models [21,48], crowd dynamics models [18], opinion formation models [3], socio-economic models [10,19].…”

“…In recent years the legacy of classical kinetic theory has found fruitful applications in the mathematical description of social phenomena [3,8,10,19,36,44], including those, such as traffic flow of both vehicles and pedestrians, which mix mechanical and behavioural aspects of the agents [2,11,12,17,18,21,26,41]. For the sake of completeness, however, we mention that the mathematical modelling of vehicular traffic by means of methods of the kinetic theory has by now a quite long history dating back to the pioneering works [38,39,40].…”

A Mean Field Game approach for multi-lane traffic management

“…The scheme is an adaptation of the one proposed in [8,7,6]. We also refer the reader to [17], where a similar scheme has been applied to a non-linear continuity equation modeling a kinetic pedestrian model. Continuity equations.…”