35 research outputs found
Derivation, Properties, and Simulation of a Gas-Kinetic-Based, Non-Local Traffic Model
We derive macroscopic traffic equations from specific gas-kinetic equations,
dropping some of the assumptions and approximations made in previous papers.
The resulting partial differential equations for the vehicle density and
average velocity contain a non-local interaction term which is very favorable
for a fast and robust numerical integration, so that several thousand freeway
kilometers can be simulated in real-time. The model parameters can be easily
calibrated by means of empirical data. They are directly related to the
quantities characterizing individual driver-vehicle behavior, and their optimal
values have the expected order of magnitude. Therefore, they allow to
investigate the influences of varying street and weather conditions or freeway
control measures. Simulation results for realistic model parameters are in good
agreement with the diverse non-linear dynamical phenomena observed in freeway
traffic.Comment: For related work see
http://www.theo2.physik.uni-stuttgart.de/helbing.html and
http://www.theo2.physik.uni-stuttgart.de/treiber.htm
Maxwell Model of Traffic Flows
We investigate traffic flows using the kinetic Boltzmann equations with a
Maxwell collision integral. This approach allows analytical determination of
the transient behavior and the size distributions. The relaxation of the car
and cluster velocity distributions towards steady state is characterized by a
wide range of velocity dependent relaxation scales, , with
the ratio of the passing and the collision rates. Furthermore, these
relaxation time scales decrease with the velocity, with the smallest scale
corresponding to the decay of the overall density. The steady state cluster
size distribution follows an unusual scaling form . This distribution is primarily algebraic, , for , and is exponential otherwise.Comment: revtex, 10 page
Modeling and Simulation of Multi-Lane Traffic Flow
A most important aspect in the field of traffic modeling is the simulation of
bottleneck situations. For their realistic description a macroscopic multi-lane
model for uni-directional freeways including acceleration, deceleration,
velocity fluctuations, overtaking and lane-changing maneuvers is systematically
deduced from a gas-kinetic (Boltzmann-like) approach. The resulting equations
contain corrections with respect to previous models. For efficient computer
simulations, a reduced model delineating the coarse-grained temporal behavior
is derived and applied to bottleneck situations.Comment: For related work see
http://www.theo2.physik.uni-stuttgart.de/helbing.htm
Gas-kinetic derivation of Navier-Stokes-like traffic equations
Macroscopic traffic models have recently been severely criticized to base on
lax analogies only and to have a number of deficiencies. Therefore, this paper
shows how to construct a logically consistent fluid-dynamic traffic model from
basic laws for the acceleration and interaction of vehicles. These
considerations lead to the gas-kinetic traffic equation of Paveri-Fontana. Its
stationary and spatially homogeneous solution implies equilibrium relations for
the `fundamental diagram', the variance-density relation, and other quantities
which are partly difficult to determine empirically.
Paveri-Fontana's traffic equation allows the derivation of macroscopic moment
equations which build a system of non-closed equations. This system can be
closed by the well proved method of Chapman and Enskog which leads to
Euler-like traffic equations in zeroth-order approximation and to
Navier-Stokes-like traffic equations in first-order approximation. The latter
are finally corrected for the finite space requirements of vehicles. It is
shown that the resulting model is able to withstand the above mentioned
criticism.Comment: For related work see
http://www.theo2.physik.uni-stuttgart.de/helbing.htm
Traffic and Related Self-Driven Many-Particle Systems
Since the subject of traffic dynamics has captured the interest of
physicists, many astonishing 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 self-organizing systems tend to reach an
optimal state? 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, highway, and city
traffic, (iii) 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 like a general modelling framework for self-driven many-particle
systems, including spin systems. Subjects such as 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 discussed as
well.Comment: A shortened version of this article will appear in Reviews of Modern
Physics, an extended one as a book. The 63 figures were omitted because of
storage capacity. For related work see http://www.helbing.org