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Register a new userGradient Navigation Model for Pedestrian Dynamics
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We present a new microscopic ODE-based model for pedestrian dynamics: the Gradient Navigation Model. The model uses a superposition of gradients of distance functions to directly change the direction of the velocity vector. The velocity is then integrated to obtain the location. The approach differs fundamentally from force based models needing only three equations to derive the ODE system, as opposed to four in, e.g., the Social Force Model. Also, as a result, pedestrians are no longer subject to inertia. Several other advantages ensue: Model induced oscillations are avoided completely since no actual forces are present. The derivatives in the equations of motion are smooth and therefore allow the use of fast and accurate high order numerical integrators. At the same time, existence and uniqueness of the solution to the ODE system follow almost directly from the smoothness properties. In addition, we introduce a method to calibrate parameters by theoretical arguments based on empirically validated assumptions rather than by numerical tests. These parameters, combined with the accurate integration, yield simulation results with no collisions of pedestrians. Several empirically observed system phenomena emerge without the need to recalibrate the parameter set for each scenario: obstacle avoidance, lane formation, stop-and-go waves and congestion at bottlenecks. The density evolution in the latter is shown to be quantitatively close to controlled experiments. Likewise, we observe a dependence of the crowd velocity on the local density that compares well with benchmark fundamental diagrams.
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An extended social force model with a dynamic navigation field is proposed to study bidirectional pedestrian movement. The dynamic navigation field is introduced to describe the desired direction of pedestrian motion resulting from the decision-making processes of pedestrians. The macroscopic fundamental diagrams obtained using the extended model are validated against camera-based observations. Numerical results show that this extended model can reproduce collective phenomena in pedestrian traffic, such as dynamic multilane flow and stable separate-lane flow. Pedestrians path choice behavior significantly affects the probability of congestion and the number of self-organized lanes.
Lane formation in bidirectional pedestrian streams is based on a stimulus-response mechanism and strategies of navigation in a fast-changing environment. Although microscopic models that only guarantee volume exclusion can qualitatively reproduce this phenomenon, they are not sufficient for a quantitative description. To quantitatively describe this phenomenon, a minimal anticipatory collision-free velocity model is introduced. Compared to the original velocity model, the new model reduces the occurrence of gridlocks and reproduces the movement of pedestrians more realistically. For a quantitative description of the phenomenon, the definition of an order parameter is used to describe the formation of lanes at transient states and to show that the proposed model compares relatively well with experimental data. Furthermore, the model is validated by the experimental fundamental diagrams of bidirectional flows.
We study the time-evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. Under a physically motivated assumption on the energy of the initial data, we show that condensation is preserved by the many-body evolution and that the dynamics of the condensate wave function can be described by the time-dependent Gross-Pitaevskii equation. With respect to previous works, we provide optimal bounds on the rate of condensation (i.e. on the number of excitations of the Bose-Einstein condensate). To reach this goal, we combine the method of cite{LNS}, where fluctuations around the Hartree dynamics for $N$-particle initial data in the mean-field regime have been analyzed, with ideas from cite{BDS}, where the evolution of Fock-space initial data in the Gross-Pitaevskii regime has been considered.
We show that all Lindblad operators (i.e. generators of quantum semigroups) on a finite-dimensional Hilbert space satisfying the detailed balance condition with respect to the thermal equilibrium state can be written as a gradient system with respect to the relative entropy. We discuss also thermodynamically consistent couplings to macroscopic systems, either as damped Hamiltonian systems with constant temperature or as GENERIC systems. In particular we discuss the coupling of a quantum dot coupled to macroscopic charge carriers.
How do pedestrians choose their paths within city street networks? Human path planning has been extensively studied at the aggregate level of mobility flows, and at the individual level with strictly designed behavioural experiments. However, a comprehensive, individual-level model of how humans select pedestrian paths in real urban environments is still lacking. Here, we analyze human path planning behaviour in a large dataset of individual pedestrians, whose GPS traces were continuously recorded as they pursued their daily goals. Through statistical analysis we reveal two robust empirical discoveries, namely that (1) people increasingly deviate from the shortest path as the distance between origin and destination increases, and (2) individual choices exhibit direction-dependent asymmetries when origin and destination are swapped. In order to address the above findings, which cannot be explained by existing models, we develop a vector-based navigation framework motivated by the neural evidence of direction-encoding cells in hippocampal brain networks, and by behavioural evidence of vector navigation in animals. Modelling pedestrian path preferences by vector-based navigation increases the models predictive power by 35%, compared to a model based on minimizing distance with stochastic effects. We show that these empirical findings and modelling results generalise across two major US cities with drastically different street networks, suggesting that vector-based navigation is a universal property of human path planning, independent of specific city environments. Our results offer a simple, unified explanation of numerous findings about human navigation, and posit a computational mechanism that may underlie the human capacity to efficiently navigate in environments at various scales.
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