Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Traffic and related self-driven many-particle systems

We review the observations and the basic laws describing the essential aspects of collective motion -- being one of the most common and spectacular manifestation of coordinated behavior Our aim is to provide a balanced discussion of the various facets of this highly multidisciplinary field, including experiments, mathematical methods and models for simulations, so that readers with a variety of background could get both the basics and a broader, more detailed picture of the field The observations we report on include systems consisting of units ranging from macromolecules through metallic rods and robots to groups of animals and people Some emphasis is put on models that are simple and realistic enough to reproduce the numerous related observations and are useful for developing concepts for a better understanding of the complexity of systems consisting of many simultaneously moving entities As such, these models allow the establishing of a few fundamental principles of flocking In particular, it is demonstrated, that in spite of considerable differences, a number of deep analogies exist between equilibrium statistical physics systems and those made of self-propelled (in most cases living) units In both cases only a few well defined macroscopic/collective states occur and the transitions between these states follow a similar scenario, involving discontinuity and algebraic divergences

Collective Motion

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.

Active Brownian Particles. From Individual to Collective Stochastic Dynamics

Do the movements of animals, including humans, follow patterns that can be described quantitatively by simple laws of motion? If so, then why? These questions have attracted the attention of scientists in many disciplines, and stimulated debates ranging from ecological matters to queries such as 'how can there be free will if one follows a law of motion?' This is the first book on this rapidly evolving subject, introducing random searches and foraging in a way that can be understood by readers without a previous background on the subject. It reviews theory as well as experiment, addresses open problems and perspectives, and discusses applications ranging from the colonization of Madagascar by Austronesians to the diffusion of genetically modified crops. The book will interest physicists working in the field of anomalous diffusion and movement ecology as well as ecologists already familiar with the concepts and methods of statistical physics.

/pdf/the-physics-of-foraging-an-introduction-to-random-searches-3tn55gp12c.pdf

The Physics of Foraging: An Introduction to Random Searches and Biological Encounters

/pdf/state-transitions-and-the-continuum-limit-for-a-2d-25srs3r4pp.pdf

State Transitions and the Continuum Limit for a 2D Interacting, Self-Propelled Particle System

We consider a model of active Brownian agents interacting via a harmonic attractive potential in a two-dimensional system in the presence of noise. By numerical simulations, we show that this model possesses a noise-induced transition characterized by the breakdown of translational motion and the onset of swarm rotation as the noise intensity is increased. Statistical properties of swarm dynamics in the weak noise limit are further analytically investigated.

https://www.researchgate.net/profile/Udo_Erdmann/publication/7669917_Noise-induced_transition_from_translational_to_rotational_motion_of_swarms/links/0912f51079ee99f783000000.pdf

Noise-Induced Transition from Translational to Rotational Motion of Swarms

The motions of many physical particles as well as living creatures are mediated by random influences or 'noise'. One might expect that over evolutionary time scales internal random processes found in living systems display characteristics that maximize fitness. Here we focus on animal random search strategies [G.M. Viswanathan, S.V. Buldyrev, S. Havlin, M.G.E. Da Luz, E.P. Raposo, H.E. Stanley, Optimizing the success of random searches, Nature 401 (1999) 911-914; F. Bartumeus, J. Catalan, U.L. Fulco, M.L. Lyra, G.M. Viswanathan, Optimizing the encounter rate in biological interactions: Levy versus Brownian stratagies, Phys. Rev. Lett. 88 (2002) 097901 and 89 (2002) 109902], and we describe experiments with the following Daphnia species: D. magna, D. galeata, D. lumholtzi, D. pulicaria, and D. pulex. We observe that the animals, while foraging for food, choose turning angles from distributions that can be described by exponential functions with a range of widths. This observation leads us to speculate and test the notion that this characteristic distribution of turning angles evolved in order to enhance survival. In the case of theoretical agents, some form of randomness is often introduced into search algorithms, especially when information regarding the sought object(s) is incomplete or even misleading. In the case of living animals, many studies have focused on search strategies that involve randomness [H.C. Berg, Random Walks in Biology, Princeton University, Princeton, New Jersey, 1993; A. Okubo, S.A. Levin (Eds.), Diffusion and Ecological Problems: Modern Perspectives, second ed., Springer, New York, 2001]. A simple theory based on stochastic differential equations of the motion backed up by a simulation shows that the collection of material (information, energy, food, supplies, etc.) by an agent executing Brownian-type hopping motions is optimized while foraging for a finite time in a supply patch of limited spatial size if the agent chooses turning angles taken from an exponential distribution with a specific stochastic intensity or 'noise width'. Search strategies that lead to optimization is a topic of high current interest across many disciplines [D. Wolpert, W. MacReady, No free lunch theorems for optimization, IEEE Transactions on Evolutionary Computation 1 (1997) 67].

http://eaton.math.rpi.edu/CSUMS/Papers/Ecomovement/garciaoptimalforagedaphnia.pdf

Optimal foraging by zooplankton within patches: The case of Daphnia

The dynamics of a ring of masses including dissipative forces (passive or active friction) and Toda interactions between the masses is investigated. The characteristic attractor structure and the influence of noise by coupling to a heat bath are studied. The system may be driven from the thermodynamic equilibrium to far from equilibrium states by including negative friction. We show, that over-critical pumping with free energy may lead to a partition of the phase space into attractor regions corresponding to several types of collective motions including uniform rotations, one- and multiple soliton-like excitations and relative oscillations. The distribution functions in the phase space and the correlation functions of the forces and the spectra of nonlinear excitations are calculated. We show that a finite-size Toda ring with weak thermal coupling develops at intermediate temperatures a broadband colored noise spectrum with an 1/f tail at low frequencies.

Nonlinear Dynamics and Fluctuations of Dissipative Toda Chains

Complex spatio-temporal patterns of cell clusters were observed in colonies of chemotactic bacteria such as Escherichia coli or Sallmonella typhimurium. The production of a potent chemoattractant by the bacteria themselves as a reaction to certain nutrients is the essential factor for this pattern formation. Additional collective dynamics, such as collective translation and rotation of bacterial clusters were reported from experiments on bacterial colonies. Motivated by this observations we suggest a simple model for the description of bacterial colonies using the concept of active Brownian particles. Individual based models represent an interesting alternative to the usually employed mean field chemotaxis-diffusion equations (Keller-Segel model) as they allow us to study the macroscopic pattern formation of the colony, the collective dynamics of bacterial ensembles, as well as the microscopic dynamics of individual cells. In this paper we derive microscopic model equations from basic assumptions about bacterial dynamics, discuss the parameter choice by comparison with biological data and analyse the macroscopic and microscopic dynamics of the system. Finally we extend the model by a velocity-alignment (swarming) interaction which leads to novel collective dynamics in the system.

Beyond the Keller-Segel model - Microscopic modelling of bacterial colonies

We study a model of Brownian particles which are pumped with energy by means of a non-linear friction function. Analytical solutions of the corresponding Fokker-Planck equation are presented and compared with computer simulations. Different to the case of passive Brownian motion, we find several new features of the dynamics, such as the formation of limit cycles, a large mean squared displacement which increases quadratically with the energy supply, or non-equilibrium velocity distributions with crater-like form. In the four-dimensional phase space, the corresponding distribution has the form of two hoops or two tires. Further, the general features of motion in arbitrary two-dimensional potentials are discussed within a diffusion approximation.

U. Erdmann

Papers

Noise-Induced Transition from Translational to Rotational Motion of Swarms

Optimal foraging by zooplankton within patches: The case of Daphnia

Nonlinear Dynamics and Fluctuations of Dissipative Toda Chains

Beyond the Keller-Segel model - Microscopic modelling of bacterial colonies

Complex motion of Brownian particles with energy supply