Dimensionless numbers reveal distinct regimes in the structure and dynamics of pedestrian crowds

In fluid mechanics, dimensionless numbers like the Reynolds number help classify flows. We argue that such a classification is also relevant for crowd flows by putting forward the dimensionless Intrusion and Avoidance numbers, which quantify the intrusions into the pedestrians' personal spaces and the imminency of the collisions that they face, respectively. Using an extensive dataset, we show that these numbers delineate regimes where distinct variables characterize the crowd's arrangement, namely, Euclidean distances at low Avoidance number and times-to-collision at low Intrusion number. On the basis of these findings, a perturbative expansion of the individual pedestrian dynamics is carried out around the noninteracting state, in quite general terms. Simulations confirm that this expansion performs well in its expected regime of applicability.

Structure of road networks and the shape of the macroscopic fundamental diagram

The macroscopic fundamental diagram (MFD) is a large-scale description of the traffic in an urban area and relates the average car flow to the average car density. This MFD has been observed empirically in several cities but how its properties are related to the structure of the road network has remained unclear so far. The MFD displays in general a maximum flow q^{*} for an optimal car density k^{*} which are crucial quantities for practical applications. Here, using numerical modeling and dimensional arguments, we propose scaling laws for these quantities q^{*} and k^{*} in terms of the road density, the intersection density, the average car size and the maximum velocity. This framework is able to explain the scaling observed empirically for several cities in the world, such as the scaling of k^{*} with the road density, the relation between q^{*} and k^{*} and the impact of buses on the overall capacity q^{*}. This work opens the way to a better understanding of the traffic on a road network at a large urban scale.

An attempt to distinguish physical and socio-psychological influences on pedestrian bottleneck

It has been realized that the distinction between social-psychological effects and physical effects in pedestrian crowds is complex, and so the relevance of social psychology for the properties of pedestrian streams is still discussed controversially. Although physics-based models appear to capture many properties rather accurately, it was argued that simple systems of self-driven particles could not explain certain emergent phenomena. In particular, results from a recent empirical study of pedestrian flow at bottlenecks have been interpreted as indicating the relevance of social psychology even in relatively simple scenarios of crowd dynamics. The study showed a surprising dependence of the density near the bottleneck on the width of the corridor leading to it. The density increased with increasing corridor width, although a wider corridor provides more space for pedestrians. It has been argued that this observation is a consequence of social norms, which trigger the effect by a preference for queuing in such situations. However, convincing evidence for this hypothesis is still missing. Here, we reconsider this scenario from a physics perspective using computer simulations of a simple microscopic velocity-based model.

Heterogeneity-induced lane and band formation in self-driven particle systems

The collective motion of interacting self-driven particles describes many types of coordinated dynamics and self-organisation. Prominent examples are alignment or lane formation which can be observed alongside other ordered structures and nonuniform patterns. In this article, we investigate the effects of different types of heterogeneity in a two-species self-driven particle system. We show that heterogeneity can generically initiate segregation in the motion and identify two heterogeneity mechanisms. Longitudinal lanes parallel to the direction of motion emerge when the heterogeneity statically lies in the agent characteristics (quenched disorder). While transverse bands orthogonal to the motion direction arise from dynamic heterogeneity in the interactions (annealed disorder). In both cases, non-linear transitions occur as the heterogeneity increases, from disorder to ordered states with lane or band patterns. These generic features are observed for a first and a second order motion model and different characteristic parameters related to particle speed and size. Simulation results show that the collective dynamics occur in relatively short time intervals, persist stationary, and are partly robust against random perturbations.

Kardar-Parisi-Zhang universality of the Nagel-Schreckenberg model

Dynamical universality classes are distinguished by their dynamical exponent z and unique scaling functions encoding space-time asymmetry for, e.g., slow-relaxation modes or the distribution of time-integrated currents. So far the universality class of the Nagel-Schreckenberg (NaSch) model, which is a paradigmatic model for traffic flow on highways, was not known. Only the special case v_{max}=1, where the model corresponds to the totally asymmetric simple exclusion process, is known to belong to the superdiffusive Kardar-Parisi-Zhang (KPZ) class with z=3/2. In this paper, we show that the NaSch model also belongs to the KPZ class for general maximum velocities v_{max}>1. Using nonlinear fluctuating hydrodynamics theory we calculate the nonuniversal coefficients, fixing the exact asymptotic solutions for the dynamical structure function and the distribution of time-integrated currents. The results of large-scale Monte Carlo simulations match the exact asymptotic KPZ solutions without any fitting parameter left. Additionally, we find that nonuniversal early-time effects or the choice of initial conditions might have a strong impact on the numerical determination of the dynamical exponent and therefore lead to inconclusive results. We also show that the universality class is not changed by extending the model to a two-lane NaSch model with lane-changing rules.

Diffusion with resetting inside a circle

We study the Brownian motion of a particle in a bounded circular two-dimensional domain in search for a stationary target on the boundary of the domain. The process switches between two modes: one where it performs a two-dimensional diffusion inside the circle and one where it diffuses along the one-dimensional boundary. During the process, the Brownian particle resets to its initial position with a constant rate r. The Fokker-Planck formalism allows us to calculate the mean time to absorption (MTA) as well as the optimal resetting rate for which the MTA is minimized. From the derived analytical results the parameter regions where resetting reduces the search time can be specified. We also provide a numerical method for the verification of our results.

Braess paradox in a network of totally asymmetric exclusion processes

We study the Braess paradox in the transport network as originally proposed by Braess with totally asymmetric exclusion processes (TASEPs) on the edges. The Braess paradox describes the counterintuitive situation in which adding an edge to a road network leads to a user optimum with higher travel times for all network users. Travel times on the TASEPs are nonlinear in the density, and jammed states can occur due to the microscopic exclusion principle, leading to a more realistic description of trafficlike transport on the network than in previously studied linear macroscopic mathematical models. Furthermore, the stochastic dynamics allows us to explore the effects of fluctuations on network performance. We observe that for low densities, the added edge leads to lower travel times. For slightly higher densities, the Braess paradox occurs in its classical sense. At intermediate densities, strong fluctuations in the travel times dominate the system's behavior due to links that are in a domain-wall state. At high densities, the added link leads to lower travel times. We present a phase diagram that predicts the system's state depending on the global density and crucial path-length ratios.

Fibonacci family of dynamical universality classes

Universality is a well-established central concept of equilibrium physics. However, in systems far away from equilibrium, a deeper understanding of its underlying principles is still lacking. Up to now, a few classes have been identified. Besides the diffusive universality class with dynamical exponent [Formula: see text], another prominent example is the superdiffusive Kardar-Parisi-Zhang (KPZ) class with [Formula: see text]. It appears, e.g., in low-dimensional dynamical phenomena far from thermal equilibrium that exhibit some conservation law. Here we show that both classes are only part of an infinite discrete family of nonequilibrium universality classes. Remarkably, their dynamical exponents [Formula: see text] are given by ratios of neighboring Fibonacci numbers, starting with either [Formula: see text] (if a KPZ mode exist) or [Formula: see text] (if a diffusive mode is present). If neither a diffusive nor a KPZ mode is present, all dynamical modes have the Golden Mean [Formula: see text] as dynamical exponent. The universal scaling functions of these Fibonacci modes are asymmetric Lévy distributions that are completely fixed by the macroscopic current density relation and compressibility matrix of the system and hence accessible to experimental measurement.

Jamming transitions in force-based models for pedestrian dynamics

Force-based models describe pedestrian dynamics in analogy to classical mechanics by a system of second order ordinary differential equations. By investigating the linear stability of two main classes of forces, parameter regions with unstable homogeneous states are identified. In this unstable regime it is then checked whether phase transitions or stop-and-go waves occur. Results based on numerical simulations show, however, that the investigated models lead to unrealistic behavior in the form of backwards moving pedestrians and overlapping. This is one reason why stop-and-go waves have not been observed in these models. The unrealistic behavior is not related to the numerical treatment of the dynamic equations but rather indicates an intrinsic problem of this model class. Identifying the underlying generic problems gives indications how to define models that do not show such unrealistic behavior. As an example we introduce a force-based model which produces realistic jam dynamics without the appearance of unrealistic negative speeds for empirical desired walking speeds.

Critical bottleneck size for jamless particle flows in two dimensions

We propose a simple microscopic model for arching phenomena at bottlenecks. The dynamics of particles in front of a bottleneck is described by a one-dimensional stochastic cellular automaton on a semicircular geometry. The model reproduces oscillation phenomena due to the formation and collapsing of arches. It predicts the existence of a critical bottleneck size for continuous particle flows. The dependence of the jamming probability on the system size is approximated by the Gompertz function. The analytical results are in good agreement with simulations.

Quantitative analysis of pedestrian counterflow in a cellular automaton model

Pedestrian dynamics exhibits various collective phenomena. Here, we study bidirectional pedestrian flow in a floor field cellular automaton model. Under certain conditions, lane formation is observed. Although it has often been studied qualitatively, e.g., as a test for the realism of a model, there are almost no quantitative results, either empirically or theoretically. As basis for a quantitative analysis, we introduce an order parameter which is adopted from the analysis of colloidal suspensions. This allows us to determine a phase diagram for the system where four different states (free flow, disorder, lanes, gridlock) can be distinguished. Although the number of lanes formed is fluctuating, lanes are characterized by a typical density. It is found that the basic floor field model overestimates the tendency towards a gridlock compared to experimental bounds. Therefore, an anticipation mechanism is introduced which reduces the jamming probability.

Exact dynamical state of the exclusive queueing process with deterministic hopping

The exclusive queueing process (EQP) has recently been introduced as a model for the dynamics of queues that takes into account the spatial structure of the queue. It can be interpreted as a totally asymmetric exclusion process of varying length. Here we investigate the case of deterministic bulk hopping p=1 that turns out to be one of the rare cases where exact nontrivial results for the dynamical properties can be obtained. Using a time-dependent matrix product form we calculate several dynamical properties, e.g., the density profile of the system.

Dynamical analysis of the exclusive queueing process

Recently, the stationary state of a parallel-update totally asymmetric simple exclusion process with varying system length, which can be regarded as a queueing process with excluded-volume effect (exclusive queueing process), was obtained [C Arita and D Yanagisawa, J. Stat. Phys. 141, 829 (2010)]. In this paper, we analyze the dynamical properties of the number of particles [N(t)] and the position of the last particle (the system length) [L(t)], using an analytical method (generating function technique) as well as a phenomenological description based on domain-wall dynamics and Monte Carlo simulations. The system exhibits two phases corresponding to linear convergence or divergence of [N(t)] and [L(t)]. These phases can both further be subdivided into high-density and maximal-current subphases. The predictions of the domain-wall theory are found to be in very good agreement quantitively with results from Monte Carlo simulations in the convergent phase. On the other hand, in the divergent phase, only the prediction for [N(t)] agrees with simulations.

Generalized centrifugal-force model for pedestrian dynamics

A spatially continuous force-based model for simulating pedestrian dynamics is introduced which includes an elliptical volume exclusion of pedestrians. We discuss the phenomena of oscillations and overlapping which occur for certain choices of the forces. The main intention of this work is the quantitative description of pedestrian movement in several geometries. Measurements of the fundamental diagram in narrow and wide corridors are performed. The results of the proposed model show good agreement with empirical data obtained in controlled experiments.

Trafficlike collective movement of ants on trails: absence of a jammed phase

We report experimental results on unidirectional trafficlike collective movement of ants on trails. Our work is primarily motivated by fundamental questions on the collective spatiotemporal organization in systems of interacting motile constituents driven far from equilibrium. Making use of the analogies with vehicular traffic, we analyze our experimental data for the spatiotemporal organization of ants on a trail. From this analysis, we extract the flow-density relation as well as the distributions of velocities of the ants and distance headways. Some of our observations are consistent with our earlier models of ant traffic, which are appropriate extensions of the asymmetric simple exclusion process. In sharp contrast to highway traffic and most other transport processes, the average velocity of the ants is almost independent of their density on the trail. Consequently, no jammed phase is observed.

Disordered driven lattice gases with boundary reservoirs and Langmuir kinetics

The asymmetric simple exclusion process with additional Langmuir kinetics, i.e., attachment and detachment in the bulk, is a paradigmatic model for intracellular transport. Here we study this model in the presence of randomly distributed inhomogeneities ("defects"). Using Monte Carlo simulations, we find a multitude of coexisting high- and low-density domains. The results are generic for one-dimensional driven diffusive systems with short-range interactions and can be understood in terms of a local extremal principle for the current profile. This principle is used to determine current profiles and phase diagrams as well as statistical properties of ensembles of defect samples.

Fundamentals of Pedestrian and Evacuation Dynamics

Multi-Agent Simulation is a general and powerful framework for understanding and predicting the behaviour of social systems. Here the authors investigate the behaviour of pedestrians and human crowds, especially their physical movement. Their aim is to build a bridge between the multi-agent and pedestrian dynamics communities that facilitates the validation and calibration of modelling approaches which is essential for any application in sensitive areas like safety analysis. Understanding the dynamical properties of large crowds is of obvious practical importance. Emergency situations require efficient evacuation strategies to avoid casualties and reduce the number of injured persons. In many cases legal requirements have to be fulfilled, for example, for aircraft or cruise ships. For tests already in the planning stage reliable simulation models are required to avoid additional costs for changes in the construction. First, the empirically observed phenomena are described, emphasizing the challenges they pose for any modelling approach and their relevance for the validation and calibration. Then the authors review the basic modelling approaches used for the simulation of pedestrian dynamics in normal and emergency situations, focussing on cellular automata models. Their achievements as well as their limitations are discussed in view of the empirical results. Finally, two applications to safety analysis are briefly described.

Intracellular transport by single-headed kinesin KIF1A: effects of single-motor mechanochemistry and steric interactions

In eukaryotic cells, many motor proteins can move simultaneously on a single microtubule track. This leads to interesting collective phenomena such as jamming. Recently we reported [Phys. Rev. Lett. 95, 118101 (2005)] a lattice-gas model which describes traffic of unconventional (single-headed) kinesins KIF1A. Here we generalize this model, introducing an interaction parameter c, to account for an interesting mechanochemical process. We have been able to extract all the parameters of the model, except c, from experimentally measured quantities. In contrast to earlier models of intracellular molecular motor traffic, our model assigns distinct "chemical" (or, conformational) states to each kinesin to account for the hydrolysis of adenosine triphosphate (ATP), the chemical fuel of the motor. Our model makes experimentally testable theoretical predictions. We determine the phase diagram of the model in planes spanned by experimentally controllable parameters, namely, the concentrations of kinesins and ATP. Furthermore, the phase-separated regime is studied in some detail using analytical methods and simulations to determine, e.g., the position of shocks. Comparison of our theoretical predictions with experimental results is expected to elucidate the nature of the mechanochemical process captured by the parameter c.

Intracellular transport of single-headed molecular motors KIF1A

Motivated by experiments on single-headed kinesin KIF1A, we develop a model of intracellular transport by interacting molecular motors. It captures explicitly not only the effects of adenosine triphosphate hydrolysis, but also the ratchet mechanism which drives individual motors. Our model accounts for the experimentally observed single-molecule properties in the low-density limit and also predicts a phase diagram that shows the influence of hydrolysis and Langmuir kinetics on the collective spatiotemporal organization of the motors. Finally, we provide experimental evidence for the existence of domain walls in our in vitro experiment with fluorescently labeled KIF1A.

Collective effects in traffic on bi-directional ant trails

Motivated by recent experimental work of Burd et al., we propose a model of bi-directional ant traffic on pre-existing ant trails. It captures in a simple way some of the generic collective features of movements of real ants on a trail. Analysing this model, we demonstrate that there are crucial qualitative differences between vehicular- and ant-traffics. In particular, we predict some unusual features of the flow rate that can be tested experimentally. As in the uni-directional model a non-monotonic density-dependence of the average velocity can be observed in certain parameter regimes. As a consequence of the interaction between oppositely moving ants the flow rate can become approximately constant over some density interval.

Empirical test for cellular automaton models of traffic flow

Based on a detailed microscopic test scenario motivated by recent empirical studies of single-vehicle data, several cellular automaton models for traffic flow are compared. We find three levels of agreement with the empirical data: (1) models that do not reproduce even qualitatively the most important empirical observations, (2) models that are on a macroscopic level in reasonable agreement with the empirics, and (3) models that reproduce the empirical data on a microscopic level as well. Our results are not only relevant for applications, but also shed light on the relevant interactions in traffic flow.

Friction effects and clogging in a cellular automaton model for pedestrian dynamics

We investigate the role of conflicts in pedestrian traffic, i.e., situations where two or more people try to enter the same space. Therefore a recently introduced cellular automaton model for pedestrian dynamics is extended by a friction parameter mu. This parameter controls the probability that the movement of all particles involved in a conflict is denied at one time step. It is shown that these conflicts are not an undesirable artifact of the parallel update scheme, but are important for a correct description of the dynamics. The friction parameter mu can be interpreted as a kind of an internal local pressure between the pedestrians which becomes important in regions of high density, occurring, e.g., in panic situations. We present simulations of the evacuation of a large room with one door. It is found that friction has not only quantitative effects, but can also lead to qualitative changes, e.g., of the dependence of the evacuation time on the system parameters. We also observe similarities to the flow of granular materials, e.g., arching effects.

Cluster formation and anomalous fundamental diagram in an ant-trail model

A recently proposed stochastic cellular automaton model [J. Phys. A 35, L573 (2002)], motivated by the motions of ants in a trail, is investigated in detail in this paper. The flux of ants in this model is sensitive to the probability of evaporation of pheromone, and the average speed of the ants varies nonmonotonically with their density. This remarkable property is analyzed here using phenomenological and microscopic approximations thereby elucidating the nature of the spatiotemporal organization of the ants. We find that the observations can be understood by the formation of loose clusters, i.e., space regions of enhanced, but not maximal, density.

Open boundaries in a cellular automaton model for traffic flow with metastable states

The effects of open boundaries in the velocity-dependent randomization (VDR) model, a modified version of the well-known Nagel-Schreckenberg (NaSch) cellular automaton model for traffic flow, are investigated. In contrast to the NaSch model, the VDR model exhibits metastable states and phase separation in a certain density regime. A proper insertion strategy allows us to investigate the whole spectrum of possible system states and the structure of the phase diagram by Monte Carlo simulations. We observe an interesting microscopic structure of the jammed phases, which is different from the one of the NaSch model. For finite systems, the existence of high flow states in a certain parameter regime leads to a special structure of the fundamental diagram measured in the open system. Apart from that, the results are in agreement with an extremal principle for the flow, which has been introduced for models with a unique flow-density relation. Finally, we discuss the application of our findings for a systematic flow optimization. Here some surprising results are obtained, e.g., a restriction of the inflow can lead to an improvement of the total flow through a bottleneck.

Broken ergodicity in a stochastic model with condensation

We introduce a variant of the asymmetric random average process with continuous state variables where the maximal mass transport is restricted by a cutoff. For periodic boundary conditions, we show the existence of a phase transition between a pure high flow phase and a mixed phase, whereby the latter consists of a homogeneous high flow and a condensed low flow substate without translation invariance. The finite system alternates between these substates which both have diverging lifetimes in the thermodynamic limit, so ergodicity is broken in the infinite system. However, the scaling behavior of the lifetimes in dependence of the system size is different due to different underlying flipping mechanisms.

Single-vehicle data of highway traffic: microscopic description of traffic phases

We present a detailed analysis of single-vehicle data, which sheds some light on the microscopic interaction of the vehicles. Besides the analysis of free flow and synchronized traffic the data sets especially provide information about wide jams that persist for a long time. The data have been collected at a location far away from ramps and in the absence of speed limits, which allows a comparison with idealized traffic simulations. We also resolve some open questions concerning the time-headway distribution.

Human behavior as origin of traffic phases

It is shown that the desire for smooth and comfortable driving is directly responsible for the occurrence of synchronized traffic in highway traffic. This desire goes beyond the avoidance of accidents, which so far has been the main focus of microscopic modeling and that is mainly responsible for the other two phases observed empirically, free flow and wide moving jams. These features have been incorporated into a microscopic model based on stochastic cellular automata by means of event-driven anticipation. The results of computer simulations are compared with empirical data. It turns out that anticipation effects are responsible for the stabilization of the traffic phases and even reproduce the empirically observed coexistence of wide moving jams with both free flow and synchronized traffic.

Optimizing traffic lights in a cellular automaton model for city traffic

We study the impact of global traffic light control strategies in a recently proposed cellular automaton model for vehicular traffic in city networks. The model combines basic ideas of the Biham-Middleton-Levine model for city traffic and the Nagel-Schreckenberg model for highway traffic. The city network has a simple square lattice geometry. All streets and intersections are treated equally, i.e., there are no dominant streets. Starting from a simple synchronized strategy, we show that the capacity of the network strongly depends on the cycle times of the traffic lights. Moreover, we point out that the optimal time periods are determined by the geometric characteristics of the network, i.e., the distance between the intersections. In the case of synchronized traffic lights, the derivation of the optimal cycle times in the network can be reduced to a simpler problem, the flow optimization of a single street with one traffic light operating as a bottleneck. In order to obtain an enhanced throughput in the model, improved global strategies are tested, e.g., green wave and random switching strategies, which lead to surprising results.

Single-vehicle data of highway traffic: a statistical analysis

In the present paper, single-vehicle data of highway traffic are analyzed in great detail. By using the single-vehicle data directly, empirical time headway distributions and speed-distance relations can be established. Both quantities yield relevant information about the microscopic states. Several fundamental diagrams are also presented, which are based on time-averaged quantities and compared with earlier empirical investigations. In the remaining part, time-series analyses of the averaged as well as the single-vehicle data are carried out. The results will be used in order to propose objective criteria for an identification of the different traffic states, e.g., synchronized traffic.