There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

In the Hamadryas baboon, males are substantially larger than females. A troop of baboons is subdivided into a number of ‘one-male groups’, consisting of one adult male and one or more females with their young. The male prevents any of ‘his’ females from moving too far from him. Kummer (1971) performed the following experiment. Two males, A and B, previously unknown to each other, were placed in a large enclosure. Male A was free to move about the enclosure, but male B was shut in a small cage, from which he could observe A but not interfere. A female, unknown to both males, was then placed in the enclosure. Within 20 minutes male A had persuaded the female to accept his ownership. Male B was then released into the open enclosure. Instead of challenging male A , B avoided any contact, accepting A’s ownership.

Evolution and the Theory of Games

This paper is based on the premises that the purpose of engineering education is to graduate engineers who can design, and that design thinking is complex. The paper begins by briefly reviewing the history and role of design in the engineering curriculum. Several dimensions of design thinking are then detailed, explaining why design is hard to learn and harder still to teach, and outlining the research available on how well design thinking skills are learned. The currently most-favored pedagogical model for teaching design, project-based learning (PBL), is explored next, along with available assessment data on its success. Two contexts for PBL are emphasized: first-year cornerstone courses and globally dispersed PBL courses. Finally, the paper lists some of the open research questions that must be answered to identify the best pedagogical practices of improving design learning, after which it closes by making recommendations for research aimed at enhancing design learning.

Engineering Design Thinking, Teaching, and Learning

An approach is presented for making short-term predictions about the trajectories of chaotic dynamical systems. The method is applied to data on measles, chickenpox, and marine phytoplankton populations, to show how apparent noise associated with deterministic chaos can be distinguished from sampling error and other sources of externally induced environmental noise.

Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series

Molecular dynamic simulations of a medium consisting of disks in a periodically tilted box yield two dynamic modes differing considerably in the total potential and kinetic energies of the disks. Depending on parameters, these modes display the following features: (i) hysteresis (coexistence of the two modes in phase space); (ii) intermingledlike basins of attraction (uncertainty exponent indistinguishable from zero); (iii) two-state on-off intermittency; and (iv) bimodal velocity distributions. Bifurcations are defined by a cross-shaped phase diagram.

On-off intermittency and intermingledlike basins in a granular medium

A Duffing oscillator is driven by a sum of $N$ chaotic time series. These time series are solutions of the undriven Duffing equation. It is shown that $N=1$ is sufficient to render the fluctuation theorem [Gallavotti and Cohen, Phys. Rev. Lett. 74, 2694 (1995); Gallavotti, J. Math. Phys. 41, 4061 (2000); Evans and Searles, Adv. Phys. 51, 1529 (2002)] for the power ${J}_{\ensuremath{\tau}}$ averaged within intervals of length $\ensuremath{\tau}$. In particular, the probabilities $p({J}_{\ensuremath{\tau}})$ follow a nearly Gaussian distribution. Also, $\mathrm{ln}[p({J}_{\ensuremath{\tau}})∕p(\ensuremath{-}{J}_{\ensuremath{\tau}})]$ versus ${J}_{\ensuremath{\tau}}$ can be fitted by strikingly linear functions, the slopes being proportional to $\ensuremath{\tau}$ for large $\ensuremath{\tau}$. These results indicate that validity of the fluctuation theorem requires neither a many-particle system nor a stochastic process, which are requirements used in previous works.

Fluctuation theorem for a deterministic one-particle system.

An excitable Belousov-Zhabotinsky reagent becomes oscillatory above a threshold of methanol concentration [Me]. The oscillation period decreases with increasing [Me]. A model describes these observations quantitatively. In a spatiotemporal setup, a [Me] gradient causes waves with spatially varying properties; this leads to wave breaks that end up in turbulence, both in experiments and in simulations with partial differential equations.

Oscillations and turbulence induced by an activating agent in an active medium.

We determine probabilities of recurrence time into finite-sized, physically meaningful subsets of phase space. We consider three different autonomous chaotic systems: (i) scattering in a three-peaked potential, (ii) connected billiards, and (iii) Lorenz equations. We find multipeaked probability distributions, similar to the distributions found in (driven) stochastically resonant systems. In nondriven systems, such as ours, only monotonic decaying distributions (exponentials, stretched exponentials, power laws, and slight variations or combinations of these) have hitherto been reported. Discrete peaks in autonomous systems have as yet escaped attention in autonomous systems and correspond to specific trajectory subsets involving an integer number of loops.

Multipeaked probability distributions of recurrence times.

EXCITABLE media, exemplified by the chemical system of the Belousov–Zhabotinsky (BZ) reaction, are often modelled theoretically through the integration of sets of partial differential equations that describe their dynamics. An alternative approach is to use cellular automata1–4, which sacrifice insight into the detailed physical mechanisms for the benefit of being able to reproduce the observed patterns of behaviour at low computational cost. But the cellular automata used in previous work have been anisotropic (mainly square or hexagonal), leading to the problem that this anisotropy tends to be propagated into the patterns produced. Here we describe a means of generating isotropic cellular automata, which are able to reproduce a wide range of observed modes of behaviour ranging from spiral-type BZ patterns to structures reminiscent of turbulence.

Mario Markus

Papers

On-off intermittency and intermingledlike basins in a granular medium

Fluctuation theorem for a deterministic one-particle system.

Oscillations and turbulence induced by an activating agent in an active medium.

Multipeaked probability distributions of recurrence times.

Isotropic cellular automaton for modelling excitable media