835 research outputs found
Proposal of a risk model for vehicular traffic: A Boltzmann-type kinetic approach
This paper deals with a Boltzmann-type kinetic model describing the interplay
between vehicle dynamics and safety aspects in vehicular traffic. Sticking to
the idea that the macroscopic characteristics of traffic flow, including the
distribution of the driving risk along a road, are ultimately generated by
one-to-one interactions among drivers, the model links the personal (i.e.,
individual) risk to the changes of speeds of single vehicles and implements a
probabilistic description of such microscopic interactions in a Boltzmann-type
collisional operator. By means of suitable statistical moments of the kinetic
distribution function, it is finally possible to recover macroscopic
relationships between the average risk and the road congestion, which show an
interesting and reasonable correlation with the well-known free and congested
phases of the flow of vehicles.Comment: 23 pages, 3 figures, Commun. Math. Sci., 201
A fully-discrete-state kinetic theory approach to modeling vehicular traffic
This paper presents a new mathematical model of vehicular traffic, based on
the methods of the generalized kinetic theory, in which the space of
microscopic states (position and velocity) of the vehicles is genuinely
discrete. While in the recent literature discrete-velocity kinetic models of
car traffic have already been successfully proposed, this is, to our knowledge,
the first attempt to account for all aspects of the physical granularity of car
flow within the formalism of the aforesaid mathematical theory. Thanks to a
rich but handy structure, the resulting model allows one to easily implement
and simulate various realistic scenarios giving rise to characteristic traffic
phenomena of practical interest (e.g., queue formation due to roadworks or to a
traffic light). Moreover, it is analytically tractable under quite general
assumptions, whereby fundamental properties of the solutions can be rigorously
proved.Comment: 22 pages, 3 figure
A fully-discrete-state kinetic theory approach to traffic flow on road networks
This paper presents a new approach to the modeling of vehicular traffic flows
on road networks based on kinetic equations. While in the literature the
problem has been extensively studied by means of macroscopic hydrodynamic
models, to date there are still not, to the authors' knowledge, contributions
tackling it from a genuine statistical mechanics point of view. Probably one of
the reasons is the higher technical complexity of kinetic traffic models,
further increased in case of several interconnected roads. Here such
difficulties of the theory are overcome by taking advantage of a discrete
structure of the space of microscopic states of the vehicles, which is also
significant in view of including the intrinsic microscopic granularity of the
system in the mesoscopic representation.Comment: 36 pages, 10 figure
The BGK approximation of kinetic models for traffic
We study spatially non-homogeneous kinetic models for vehicular traffic flow.
Classical formulations, as for instance the BGK equation, lead to
unconditionally unstable solutions in the congested regime of traffic. We
address this issue by deriving a modified formulation of the BGK-type equation.
The new kinetic model allows to reproduce conditionally stable non-equilibrium
phenomena in traffic flow. In particular, stop and go waves appear as bounded
backward propagating signals occurring in bounded regimes of the density where
the model is unstable. The BGK-type model introduced here also offers the
mesoscopic description between the microscopic follow-the-leader model and the
macroscopic Aw-Rascle and Zhang model
Hybrid stochastic kinetic description of two-dimensional traffic dynamics
In this work we present a two-dimensional kinetic traffic model which takes
into account speed changes both when vehicles interact along the road lanes and
when they change lane. Assuming that lane changes are less frequent than
interactions along the same lane and considering that their mathematical
description can be done up to some uncertainty in the model parameters, we
derive a hybrid stochastic Fokker-Planck-Boltzmann equation in the
quasi-invariant interaction limit. By means of suitable numerical methods,
precisely structure preserving and direct Monte Carlo schemes, we use this
equation to compute theoretical speed-density diagrams of traffic both along
and across the lanes, including estimates of the data dispersion, and validate
them against real data
Towards a Mathematical Theory of Behavioral Human Crowds
Nicola Bellomo acknowledges the support of the University of Granada, Project Modeling in Nature MNat from micro to macro, https://www.modelingnature.org.This paper has been partially supported by the MINECO-Feder (Spain) research Grant Number RTI2018-098850-B-I00, the Junta de Andalucia (Spain) Project PY18-RT-2422, A-FQM-311-UGR18, and B-FQM-580-UGR20.
Livio Gibelli, gratefully acknowledges the financial support of the Engineering and Physical Sciences Research Council (EPSRC) Under Grants EP/N016602/1, EP/R007438/1.
Annalisa Quaini acknowledges support from the Radcliffe Institute for Advanced Study at Harvard University where she has been a 2021-2022 William and Flora Hewlett Foundation Fellow.
Alessandro Reali acknowledges the partial support of the MIUR-PRIN Project XFAST-SIMS (No. 20173C478N).The first part of our paper presents a general survey on the modeling, analytic problems,
and applications of the dynamics of human crowds, where the specific features of living
systems are taken into account in the modeling approach. This critical analysis leads to
the second part which is devoted to research perspectives on modeling, analytic problems,
multiscale topics which are followed by hints towards possible achievements. Perspectives include the modeling of social dynamics, multiscale problems and a detailed study of
the link between crowds and swarms modeling.University of Granada, Project Modeling in Nature MNat from micro to macroSpanish Government RTI2018-098850-B-I00Junta de AndaluciaEuropean Commission PY18-RT-2422
A-FQM-311-UGR18
B-FQM-580-UGR20UK Research & Innovation (UKRI)Engineering & Physical Sciences Research Council (EPSRC) EP/N016602/1
EP/R007438/1Radcliffe Institute for Advanced Study at Harvard UniversityMinistry of Education, Universities and Research (MIUR) 20173C478
A two-dimensional data-driven model for traffic flow on highways
Based on experimental traffic data obtained from German and US highways, we
propose a novel two-dimensional first-order macroscopic traffic flow model. The
goal is to reproduce a detailed description of traffic dynamics for the real
road geometry. In our approach both the dynamic along the road and across the
lanes is continuous. The closure relations, being necessary to complete the
hydrodynamic equation, are obtained by regression on fundamental diagram data.
Comparison with prediction of one-dimensional models shows the improvement in
performance of the novel model.Comment: 27 page
Kinetic Theory and Swarming Tools to Modeling Complex Systems—Symmetry problems in the Science of Living Systems
This MPDI book comprises a number of selected contributions to a Special Issue devoted to the modeling and simulation of living systems based on developments in kinetic mathematical tools. The focus is on a fascinating research field which cannot be tackled by the approach of the so-called hard sciences—specifically mathematics—without the invention of new methods in view of a new mathematical theory. The contents proposed by eight contributions witness the growing interest of scientists this field. The first contribution is an editorial paper which presents the motivations for studying the mathematics and physics of living systems within the framework an interdisciplinary approach, where mathematics and physics interact with specific fields of the class of systems object of modeling and simulations. The different contributions refer to economy, collective learning, cell motion, vehicular traffic, crowd dynamics, and social swarms. The key problem towards modeling consists in capturing the complexity features of living systems. All articles refer to large systems of interaction living entities and follow, towards modeling, a common rationale which consists firstly in representing the system by a probability distribution over the microscopic state of the said entities, secondly, in deriving a general mathematical structure deemed to provide the conceptual basis for the derivation of models and, finally, in implementing the said structure by models of interactions at the microscopic scale. Therefore, the modeling approach transfers the dynamics at the low scale to collective behaviors. Interactions are modeled by theoretical tools of stochastic game theory. Overall, the interested reader will find, in the contents, a forward look comprising various research perspectives and issues, followed by hints on to tackle these
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