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A new vehicular traffic flow model based on a stochastic jump process in vehicle acceleration and braking is introduced. It is based on a master equation for the single car probability density in space, velocity and acceleration with an…

Other Condensed Matter · Physics 2009-11-10 K. Thomas Waldeer

We investigate a stochastic model hierarchy for pedestrian flow. Starting from a microscopic social force model, where the pedestrians switch randomly between the two states stop-or-go, we derive an associated macroscopic model of…

Probability · Mathematics 2019-12-13 Simone Göttlich , Stephan Knapp , Peter Schillen

We consider a kinetic model, which describes the sedimentation of rod-like particles in dilute suspensions under the influence of gravity. This model has recently been derived by Helzel and Tzavaras in \cite{HT2015}. Here we restrict our…

Fluid Dynamics · Physics 2017-03-09 Christiane Helzel , Athanasios E. Tzavaras

The goal of the paper is a rigorous derivation of a macroscopic traffic flow model with a bifurcation or a local perturbation from a microscopic one. The microscopic model is a simple follow-the-leader with random parameters. The random…

Analysis of PDEs · Mathematics 2021-11-16 P Cardaliaguet , N Forcadel

This paper examines the optimal velocity follow-the-leader dynamics, a microscopic traffic model, and explores different aspects of the dynamical model, with particular emphasis on collision analysis. More precisely, we present a rigorous…

Optimization and Control · Mathematics 2023-09-22 Hossein Nick Zinat Matin , Maria Laura Delle Monache

Simulations of condensed matter systems often focus on the dynamics of a few distinguished components but require integrating the dynamics of the full system. A prime example is a molecular dynamics simulation of a (macro)molecule in…

Computational Physics · Physics 2024-03-12 Mauricio J. del Razo , Daan Crommelin , Peter G. Bolhuis

The simulation of traffic flow on networks requires knowledge on the behavior across traffic intersections. For macroscopic models based on hyperbolic conservation laws there exist nowadays many ad-hoc models describing this behavior. Based…

Numerical Analysis · Mathematics 2023-08-21 Michael Herty , Niklas Kolbe

This paper presents a mesoscopic traffic flow model that explicitly describes the spatio-temporal evolution of the probability distributions of vehicle trajectories. The dynamics are represented by a sequence of factor graphs, which enable…

Machine Learning · Statistics 2019-09-25 Saif Eddin Jabari , Deepthi Mary Dilip , DianChao Lin , Bilal Thonnam Thodi

The aim of this work is to introduce a two-dimensional macroscopic traffic model for multiple populations of vehicles. Starting from the paper [20], where a two-dimensional model for a single class of vehicles is proposed, we extend the…

Numerical Analysis · Mathematics 2020-11-04 Caterina Balzotti , Simone Göttlich

A recently introduced particle-based model for fluid dynamics with continuous velocities is generalized to model fluids with excluded volume effects. This is achieved through the use of biased stochastic multi-particle collisions which…

Soft Condensed Matter · Physics 2007-05-23 Erkan Tuzel , Thomas Ihle , Daniel M. Kroll

This contribution summarizes and explains various principles from physics which are used for the simulation of traffic flows in large street networks, the modeling of destination, transport mode, and route choice, or the simulation of urban…

Physics and Society · Physics 2015-06-17 Dirk Helbing , Kai Nagel

Macroscopic traffic flow is stochastic, but the physics-informed deep learning methods currently used in transportation literature embed deterministic PDEs and produce point-valued outputs; the stochasticity of the governing dynamics plays…

Systems and Control · Electrical Eng. & Systems 2026-03-11 Wuping Xin

We propose and analyse a new microscopic second order Follow-the-Leader type scheme to describe traffic flows. The main novelty of this model consists in multiplying the second order term by a nonlinear function of the global density, with…

Analysis of PDEs · Mathematics 2025-03-04 Dario Mazzoleni , Emanuela Radici , Filippo Riva

We present a new class of macroscopic models for pedestrian flows. Each individual is assumed to move towards a fixed target, deviating from the best path according to the instantaneous crowd distribution. The resulting equation is a…

Analysis of PDEs · Mathematics 2011-04-21 Rinaldo M. Colombo , Mauro Garavello , Magali Lécureux-Mercier

A recently introduced particle-based model for fluid dynamics with effective excluded volume interactions is analyzed in detail. The interactions are modeled by means of stochastic multiparticle collisions which are biased and depend on…

Soft Condensed Matter · Physics 2007-05-23 Thomas Ihle , Erkan Tuzel

This paper deals with the early results of a new model of pedestrian flow, conceived within a measure-theoretical framework. The modeling approach consists in a discrete-time Eulerian macroscopic representation of the system via a family of…

Mathematical Physics · Physics 2011-03-29 Benedetto Piccoli , Andrea Tosin

We derive a conservation law on a network made of two incoming branches and a single outgoing one from a discrete traffic flow model. The continuous model is obtained from the discrete one by letting the number of vehicles tend to infinity…

Analysis of PDEs · Mathematics 2024-09-04 P Cardaliaguet

Microscopic traffic models describe how cars interact with their neighbors in an uninterrupted traffic flow and are frequently used for reference in advanced vehicle control design. In this paper, we propose a novel mechanical system…

Systems and Control · Computer Science 2019-03-12 Zhaojian Li , Firas Khasawneh , Xiang Yin , Aoxue Li , Ziyou Song

Despite the importance of urban traffic flows, there are only a few theoretical approaches to determine fundamental relationships between macroscopic traffic variables such as the traffic density, the utilization, the average velocity, and…

Physics and Society · Physics 2009-11-13 Dirk Helbing

This study addresses multilane vehicular traffic modelling, focusing on the transition between microscopic (individual vehicle-based) to macroscopic (aggregate flow-based) descriptions. While previous research on multilane traffic has…

Physics and Society · Physics 2025-09-09 Matteo Piu , Giuseppe Visconti , Gabriella Puppo