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2019 | Book

Active Particles, Volume 2

Advances in Theory, Models, and Applications

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About this book

This volume compiles eight recent surveys that present state-of-the-art results in the field of active matter at different scales, modeled by agent-based, kinetic, and hydrodynamic descriptions. Following the previously published volume, these chapters were written by leading experts in the field and accurately reflect the diversity of subject matter in theory and applications. Several mathematical tools are employed throughout the volume, including analysis of nonlinear PDEs, network theory, mean field approximations, control theory, and flocking analysis. The book also covers a wide range of applications, including:
Biological network formationSocial systemsControl theory of sparse systemsDynamics of swarming and flocking systemsStochastic particles and mean field approximations
Mathematicians and other members of the scientific community interested in active matter and its many applications will find this volume to be a timely, authoritative, and valuable resource.

Table of Contents

Frontmatter
Kinetic and Moment Models for Cell Motion in Fiber Structures
Abstract
This review focuses on kinetic and macroscopic models for the migration of cells in fiber structures. Typical applications of cell migration models in such geometries are tumor cell invasion into tissue, or tissue-engineering and the movement of fibroblasts on artificial scaffolds during wound healing.
Raul Borsche, Axel Klar, Florian Schneider
Kinetic Models for Pattern Formation in Animal Aggregations: A Symmetry and Bifurcation Approach
Abstract
In this study we start by reviewing a class of 1D hyperbolic/kinetic models (with two velocities) used to investigate the collective behaviour of cells, bacteria or animals. We then focus on a restricted class of nonlocal models that incorporate various inter-individual communication mechanisms, and discuss how the symmetries of these models impact the various types of spatially heterogeneous and spatially homogeneous equilibria exhibited by these nonlocal models. In particular, we characterise a new type of equilibria that was not discussed before for this class of models, namely a relative equilibria. Then we simulate numerically these models and show a variety of spatio-temporal patterns (including classic equilibria and relative equilibria) exhibited by these models. We conclude by introducing a continuation algorithm (which takes into account the models symmetries) that allows us to track the solutions bifurcating from these different equilibria. Finally, we apply this algorithm to identify a D3-symmetric steady-state solution.
Pietro-Luciano Buono, Raluca Eftimie, Mitchell Kovacic, Lennaert van Veen
Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits
Abstract
Given a large ensemble of interacting particles, driven by nonlocal interactions and localized repulsion, the mean-field limit leads to a class of nonlocal, nonlinear partial differential equations known as aggregation-diffusion equations. Over the past 15 years, aggregation-diffusion equations have become widespread in biological applications and have also attracted significant mathematical interest, due to their competing forces at different length scales. These competing forces lead to rich dynamics, including symmetrization, stabilization, and metastability, as well as sharp dichotomies separating well-posedness from finite time blow-up. In the present work, we review known analytical results for aggregation-diffusion equations and consider singular limits of these equations, including the slow diffusion limit, which leads to the constrained aggregation equation, and localized aggregation and vanishing diffusion limits, which lead to metastability behavior. We also review the range of numerical methods available for simulating solutions, with special attention devoted to recent advances in deterministic particle methods. We close by applying such a method—the blob method for diffusion—to showcase key properties of the dynamics of aggregation-diffusion equations and related singular limits.
José A. Carrillo, Katy Craig, Yao Yao
High-Resolution Positivity and Asymptotic Preserving Numerical Methods for Chemotaxis and Related Models
Abstract
Many microorganisms exhibit a special pattern formation at the presence of a chemoattractant, food, light, or areas with high oxygen concentration. Collective cell movement can be described by a system of nonlinear PDEs on both macroscopic and microscopic levels. The classical PDE chemotaxis model is the Patlak-Keller-Segel system, which consists of a convection-diffusion equation for the cell density and a reaction-diffusion equation for the chemoattractant concentration. At the cellular (microscopic) level, a multiscale chemotaxis models can be used. These models are based on a combination of the macroscopic evolution equation for chemoattractant and microscopic models for cell evolution. The latter is governed by a Boltzmann-type kinetic equation with a local turning kernel operator that describes the velocity change of the cells.
A common property of the chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in solutions rapidly growing in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. In either case, capturing such singular solutions numerically is a challenging problem and the use of higher-order methods and/or adaptive strategies is often necessary. In addition, positivity preserving is an absolutely crucial property a good numerical method used to simulate chemotaxis should satisfy: this is the only way to guarantee a nonlinear stability of the method. For kinetic chemotaxis systems, it is also essential that numerical methods provide a consistent and stable discretization in certain asymptotic regimes.
In this paper, we review some of the recent advances in developing of high-resolution finite-volume and finite-difference numerical methods that possess the aforementioned properties of the chemotaxis-type systems.
Alina Chertock, Alexander Kurganov
Control Strategies for the Dynamics of Large Particle Systems
Abstract
We survey some recent approaches to control problems for large particle systems. Particle systems are transversal to many applications, ranging from classical physics to social sciences. The temporal evolution of the particles is determined by deterministic or stochastic dynamics and they are additionally able to optimize their trajectory over a large time. In particular, we investigate the limit of infinitely many particles leading to control of kinetic partial differential equations. To this goal a different notion of differentiability of the meanfield equation is introduced. Different mathematical methods based on meanfield games, model predictive control, and optimal control techniques will be discussed.
Michael Herty, Lorenzo Pareschi, Sonja Steffensen
Kinetic Equations and Self-organized Band Formations
Abstract
Self-organization is a ubiquitous phenomenon in nature which can be observed in a variety of different contexts and scales, with examples ranging from schools of fish, swarms of birds or locusts to flocks of bacteria. The observation of such global patterns can often be reproduced in models based on simple interactions between neighboring particles. In this paper we focus on two particular interaction dynamics closely related to the one described in the seminal paper of Vicsek and collaborators. After reviewing the current state of the art in the subject, we study a numerical scheme for the kinetic equation associated with the Vicsek models which has the specificity of reproducing many physical properties of the continuous models, like the preservation of energy and positivity and the diminution of an entropy functional. We describe a stable pattern of bands emerging in the dynamics proposed by Degond–Frouvelle–Liu dynamics and give some insights about their formation.
Quentin Griette, Sebastien Motsch
Singular Cucker–Smale Dynamics
Abstract
This chapter is dedicated to the singular models of flocking. We give an overview of the existing literature starting from microscopic Cucker–Smale (CS) model with singular communication weight, through its mesoscopic mean-field limit, up to the corresponding macroscopic regime. For the microscopic CS model and its selected variants, the collision-avoidance phenomenon is discussed. For the kinetic mean-field model, we sketch the existence of global-in-time measure-valued solutions, paying special attention to weak-atomic uniqueness of solutions. Ultimately, for the macroscopic singular model, we provide a summary of existence results for the Euler-type alignment system. This includes the existence of strong solutions on a one-dimensional torus, and the extension of this result to higher dimensions by restricting the size of the initial data. Additionally, we present the pressureless Navier–Stokes-type system corresponding to particular choice of alignment kernel. This system is then compared—analytically and numerically—to the porous medium equation.
Piotr Minakowski, Piotr B. Mucha, Jan Peszek, Ewelina Zatorska
A Stochastic-Statistical Residential Burglary Model with Finite Size Effects
Abstract
Transience of spatio-temporal clusters of residential burglary is well documented in empirical observations, and could be due to finite size effects anecdotally. However a theoretical understanding has been lacking. The existing agent-based statistical models of criminal behavior for residential burglary assume deterministic-time steps for arrivals of events. To incorporate random arrivals, this article introduces a Poisson clock into the model of residential burglaries, which could set time increments as independently exponentially distributed random variables. We apply the Poisson clock into the seminal deterministic-time-step model in Short et al. (Math Models Methods Appl Sci 18:1249–1267, 2008). Introduction of the Poisson clock not only produces similar simulation output, but also brings in theoretically the mathematical framework of the Markov pure jump processes, e.g., a martingale approach. The martingale formula leads to a continuum equation that coincides with a well-known mean-field continuum limit. Moreover, the martingale formulation together with statistics quantifying the relevant pattern formation leads to a theoretical explanation of the finite size effects. Our conjecture is supported by numerical simulations.
Chuntian Wang, Yuan Zhang, Andrea L. Bertozzi, Martin B. Short
Correction to: A Stochastic-Statistical Residential Burglary Model with Finite Size Effects
Chuntian Wang, Yuan Zhang, Andrea L. Bertozzi, Martin B. Short
Metadata
Title
Active Particles, Volume 2
Editors
Nicola Bellomo
Pierre Degond
Eitan Tadmor
Copyright Year
2019
Electronic ISBN
978-3-030-20297-2
Print ISBN
978-3-030-20296-5
DOI
https://doi.org/10.1007/978-3-030-20297-2

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