Traffic and Related Self-Driven Many-Particle Systems Dirk Helbing

Traffic and Related Self-Driven Many-Particle Systems

Dirk Helbing1,2,3,^[1]

¹Institute for Economics and Traffic, Dresden University of Technology,

Andreas-Schubert-Str. 23, D-01062 Dresden, Germany

²II. Institute of Theoretical Physics, University of Stuttgart, Pfaffenwaldring 57/III, D-70550 Stuttgart, Germany

³Collegium Budapest – Institute for Advanced Study,

Szenth´aroms´ag utca 2, H-1014 Budapest, Hungary

I. Introduction A. Motivation: History of traffic modelling and	2
traffic impact on society	2
B. Driven many-particle systems 1. Classical mechanics, fluids, and granular	2
media	3
C. Self-driven many-particle systems	4
1. The concept of social (behavioral) forces	4
D.  What this review is about E.  Particle hopping models, power-law scaling,	4
and SOC 1. The asymmetric simple exclusion process	5
(ASEP)	6
F. Active Brownian particles	6
1. Pumped Brownian particles	6
2. Dissipative Toda and Morse chains	6
3. Active walker models	7
4. Pattern formation of bacterial colonies	7
5. Trail formation by animals and pedestrians	7
G. Vehicle and pedestrian traffic	7
II. Empirical findings for freeway traffic	7
A. Measurement techniques	8
B. Fundamental diagram and hysteresis	8
C. Time headways, headways, and velocities	10
D. Correlations	12
E. Congested traffic	13
1. Jams, stop-and-go waves, and power laws	13
2. Extended congested traffic	14
3. Pinch effect	15
F. Cars and trucks	15
G. Some critical remarks	16
III. Modelling approaches for vehicle traffic	17
A. Microscopic follow-the-leader models	17
1. Non-integer car-following model	17
2. Newell and optimal velocity model	18
3. Intelligent driver model (IDM)	19
B. Cellular automata (CA)	19

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 so-called “phantom traffic jams”, although they 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 of the traffic volume cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize in lanes, while similar systems are “freezing by heating”? Why do panicking pedestrians produce dangerous deadlocks? All these questions have been answered by applying and extending methods from statistical physics and non-linear dynamics to self-driven many-particle systems. This review article on traffic introduces (i) empirically data, facts, and observations, (ii) the main approaches to pedestrian and vehicle traffic, (iii) microscopic (particlebased), 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 like a general modelling framework for self-driven many-particle systems, including spin systems. Subjects like the optimization of traffic flows and relations to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched as well.

1. The Nagel-Schreckenberg model and its
slow-to-start variant	20
2. Some other cellular automaton models	20
3. The discrete optimal velocity model	21
4. Comparison	21
C. Master equation 1. Solution methods, mapping to spin chains,	21
and matrix product ansatz 2. Mean field approach and Boltzmann	22
equation	23
3. TASEP and Nagel-Schreckenberg model	23
4. Nucleation and jamming transition	24
5. Fokker-Planck equation	24
D. Macroscopic traffic models	24
1. The Lighthill-Whitham model (LW model)	24
2. The Burgers equation	26
3. Payne’s model and its variants	26
4.  The models by Prigogine and Phillips 5.  The models by Whitham, Ku¨hne, Kerner,	27
Konha¨user, and Lee	27
6.  The Weidlich-Hilliges model 7.  Common structure of macroscopic traffic	28
models E. Gas-kinetic traffic models and micro-macro	28
link	28
1. Prigogine’s Boltzmann-like model	28
2. Paveri-Fontana’s model	30
3. Construction of a micro-macro link 4. The non-local, gas-kinetic-based traffic	30
model (GKT model)	31
5. Navier-Stokes-like traffic equations	32
6. Simultaneous micro- and macrosimulation	33
7. Mesoscopic traffic models	34
IV. Properties of traffic models	34
A. Identical vehicles on homogeneous freeways 1. “Phantom traffic jams” and stop-and-go	34
traffic 2. (Auto-)Solitons, Korteweg-de-Vries	34
equation, and Ginzburg-Landau equation 3. Jam formation: Local breakdown and cluster effects, segregation, and	35
self-organized criticality (SOC)	36

CONTENTS

4.  Instability diagram, stop-and-go waves,

                     metastability, and hysteresis                                          37

5.  Self-organized, “natural” constants of

                    traffic flow                                                                             38

6.  Kerner’s hypotheses               39

B.  Transition to congested traffic at bottlenecks

               and ramps                                                                                    40

1.  Phase diagram of traffic states at

                    inhomogeneities                                                                  42

2.  Coexistence of states and “pinch effect”                    44

C.  Heterogeneous traffic           45

1.  Power-law platoon formation and quenched

                    disorder                                                                                  45

2.  Scattering               46

D.  Multi-lane traffic and synchronization 46

1.  Gas-kinetic and macroscopic models     46

2.  Microscopic models and cellular automata              47

E.  Bi-directional and city traffic                  47

F.  Effects beyond physics         48

G.  Traffic control and optimization            48

V.  Pedestrian traffic                49

A.  Observations       49

B.  Behavioral force model of pedestrian dynamics   50

C.  Self-organization phenomena and

                noise-induced ordering                                                           51

1.  Segregation            51

2.  Oscillations           52

3.  Rotation                  52

D.  Collective phenomena in panic situations               53

1.  “Freezing by heating”              53

2.  “Faster-is-slower effect”        53

3.  “Phantom panics”                    54

4.  Herding behavior 54

VI.  Flocking and spin systems                   54

A.  Stock markets: Herding and oscillations                 55

VII.  Summary and outlook       56

Acknowledgments                                                                                         57

References                                                                                                       57

I. INTRODUCTION

A. Motivation: History of traffic modelling and traffic impact on society

The interest in the subject of traffic dynamics is surprisingly old. In 1935, Greenshields has carried out early studies of vehicular traffic, and already in the 1950ies, a considerable publication activity is documented by journals on operations research and engineering. These papers have already addressed topics like the fundamental diagram between traffic flow and vehicle density or the instability of traffic flow, which are still relevant. The reason becomes understandable through the following quotation from H. Greenberg in 1959:

“The volume of vehicular traffic in the past several years has rapidly outstripped the capacities of the nation’s highways. It has become increasingly necessary to understand the dynamics of traffic flow and obtain a mathematical description of the process.”

More than fourty years later, the situation has deteriorated a lot. Some cities like Los Angeles and San Francisco suffer from heavy traffic congestion around the clock. In Europe, as well, the time that drivers spend standing in traffic jams amounts to several days each year. During holiday seasons, jams may grow up to more than 100 kilometers in size. Vehicle emissions like SO₂, NO_x, CO, CO₂, dust particles, smog, and noise have reached or even exceeded a level comparable to those by industrial production and those by private households, being harmful to the environment and human health. On average, every driver is involved in one accident during his lifetime. In Germany alone, the financial damage by traffic due to accidents and environmental impacts is estimated 100 billion dollars each year. The economic loss due to congested traffic is of the same order of magnitude. However, the traffic volume is still growing because of increased demands for mobility and modern logistics, including e-commerce.

Without any doubt, an efficient transportation system is essential for the functioning and success of modern, industrialized societies. But the days, when freeways were free ways, are over. Facing the increasing problems of individual traffic, the following questions came up: Is it still affordable and publicly acceptable to expand the infrastructure? Will drivers still buy cars in view of streets that effectively turn into parking lots? It is for these reasons that automobile companies started to worry about their own future and decided to spend considerable amounts of money for research on traffic dynamics and the question how the available infrastructure could be used more efficiently by new technologies (telematics).

At this time, physicists were motivated to think about what physics could contribute to the field of traffic dynamics. Although, among mathematicians, physicists, and chemists, there were already some early pioneers like Whitham, Prigogine, Montroll, and Ku¨hne, the main activities started in 1992 and 1993 with papers by Biham et al. (1992), Nagel and Schreckenberg (1992), as well as Kerner and Konh¨auser (1993). These papers initiated an avalanche of publications in various