Statistical physics has proven to be a fruitful framework to describe phenomena outside the realm of traditional physics. Recent years have witnessed an attempt by physicists to study collective phenomena emerging from the interactions of individuals as elementary units in social structures. A wide list of topics are reviewed ranging from opinion and cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, and social spreading. The connections between these problems and other, more traditional, topics of statistical physics are highlighted. Comparison of model results with empirical data from social systems are also emphasized.

Statistical physics of social dynamics

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

https://www.case.edu/affil/sigmaxi/documents/collectiveMemoryAndSpatialSortingInAnimalGroups_CouzinKrauseJamesRuxtonFranks.pdf

Collective memory and spatial sorting in animal groups.

Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Aspects of multidisciplinary statistical mechanics

Starting with a short review of the available literature in the field of pedestrian and evacuation research, an overview is given over the observed collective phenomena in pedestrian crowds. This includes lane formation in corridors and oscillations at bottlenecks in normal situations, while different kinds of blocked states are produced in panic situations. By means of molecular-dynamic-like microsimulations based on a generalized force model of interactive pedestrian dynamics, the spatio-temporal patterns in pedestrian crowds are successfully reproduced and interpreted as self-organized phenomena. In contrast to previous socio-psychological approaches, this allows a physical understanding of the observations. Despite the significantly different phenomena occuring in normal and panic situations, the main effects can be described by a unified model containing only well interpretable and plausible terms. The transition between the “rational” normal behavior and the apparently “irrational” panic behavior is controlled by a single parameter, the “nervousness”, which influences fluctuation strengths, desired speeds, and the tendency of herding. Thereby, it causes paradoxial effects like “freezing by heating”, “faster is slower”, and the ignorance of available exits. Nevertheless, there are measures to improve pedestrian flows, both in normal and panic situations. For example, the suitable placement of columns can help, although they reduce the accessible space.

Simulation of pedestrian crowds in normal and evacuation situations

We study a model of flocking in order to describe the transitions during the collective motion of organisms (e.g., birds) in three dimensions. In this model the particles representing the organisms are self-propelled, i.e., they move with the same absolute velocity. In addition, the particles locally interact by choosing at each time step the average direction of motion of their neighbours and the effects of fluctuations are taken into account as well. We present the first results for large-scale flocking in the presence of noise in three dimensions. We show that depending on the control parameters both disordered and long-range ordered phases can be observed. The corresponding phase diagram has a number of features which are qualitatively different from those typical for the analogous equilibrium models.

https://arxiv.org/pdf/physics/9902021.pdf

Collective motion of organisms in three dimensions

In this paper the roughening of interfaces moving in inhomogeneous media is investigated by examining the corresponding stochastic differential equations using (i) numerical methods and (ii) dimensional analysis. We consider interface evolution equations where disorder is represented by quenched noise which can be both additive and multiplicative. Our main finding is that quenched noise leads to a new universality class as concerning the exponents δ and β describing respectively the spatial and temporal scaling of surface roughness. In particular, additive noise close to the pinning transition results in a behaviour with δ = 0.71 ± 0.08 and β = 0.61±0.06 up to a crossover time. These estimates are in very good agreement with the theoretical prediction β = 35 and δ = 34 that we derive from a dimensional analysis of the equation. Furthermore, we argue that multiplicative noise is the appropriate choice to describe experiments where the interface between two flowing phases is considered. By numerically integrating the proposed equation we have obtained (i) surfaces remarkably similar to those observed in the experiments and (ii) a scaling of the surface width as a function of time with an exponent β = 0.65 being in an excellent agreement with the experimental value. In addition to the exponents we discuss other relevant features of the surfaces, including the scaling of the average velocity of the surface νa close to pinning and the non-trivial, power law distribution of waiting times and noise along the interface in the stationary regime.

Dynamics of surface roughening in disordered media

In this letter a new approach to the investigation of the effects of quenched randomness in the experiments on kinetic roughening is introduced by considering a stochastic differential equation for the surface development with a multiplicative noise. The authors argue that this type of noise corresponds to the experimental situation in cases when the development of the interface is dominated by pinning forces. By numerically integrating the proposed equation they have obtained (i) surfaces remarkably similar to those observed in the experiments and (ii) a scaling of the surface width as a function of time with an exponent being in an excellent agreement with the measured value. Variations of the model, crossovers and questions concerning the applicability of additive noise to wetting experiments are also discussed.

https://iopscience.iop.org/article/10.1088/0305-4470/25/13/002/pdf

Kinetic roughening with multiplicative noise

In this paper we consider two lattice models: (i) the discretized version of a simple nonlinear partial differential equation (PDE) with a singular term and (ii) a coupled map lattice (CML) with unstable coupling Both systems are defined by simple iteration rules involving the nearest-neighbour grid points of the corresponding one-dimensional lattice We show numerically that these models result in an apparently stochastic spatiotemporal behaviour and exhibit interesting regimes as the function of the parameters of the models In particular, we find that by varying the relative weight B of the singular term in the PDE the total width of the solution scales with B The CML leads to an intermittent behaviour between structured and turbulent regimes

Complex spatiotemporal patterns in two lattice models with instability

We show by numerical integration that the discretized version of simple deterministic partial differential equations with singular terms proposed by Zhang (1992) exhibit rich spatio-temporal behaviour representing a mixture of stochastic and deterministic regimes. Varying the relative weight B of the singular term we have been able to detect transitions in the global behaviour of the solutions by determining their total width w(t). In particular, we have found intermittent solutions as well as a power-law dependence of W=w(t to infinity ) on B.

M. Vicsek

Papers

Collective motion of organisms in three dimensions

Dynamics of surface roughening in disordered media

Kinetic roughening with multiplicative noise

Complex spatiotemporal patterns in two lattice models with instability

Intermittency, stochastic growth and phase transition in a simple deterministic partial differential equation with a singular term