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Scientific interests


Emergence and self-organization in complex systems

Complex systems are characterized by the spontaneous formation of spatio-temporal structures as a result of simple local interactions between agents without leaders. Complex systems mostly appear in the biological and social contexts but can also be encountered in physics, chemistry, etc. The project aims at understanding the emergence of self-organization  through  the investigation of global, macroscopic models. ...
 
 

Micro-macro passage in models of complex systems

Complex systems are commonly investigated through microscopic, Agent (or Individual)-Based Models (ABM or IBM), which predict the evolution of each agent in time. When macroscopic models are considered in the literature, they are usually based on phenomenological considerations.This project aims at rigorously establishing macroscopic models from their microscopic (Agent-Based Model) counterpart. By doing so, macroscopic models gain in predictive character and may become an invaluable aid for experimental data analysis. ...
 
 

Self-Organized Hydrodynamics

The  'Self-Organized-Hydrodynamics' (SOH) model describes self-organization phenomena at the macroscopic level. It shows some prominent differences from usual compressible hydrodynamic models. For instance, it is non-conservative and involves a constraint on the flow speed. The SOH model is one of the very first examples of non-conservative hyperbolic systems that can be derived from kinetic theory. Its derivation highlights one of the specific difficulties in complex systems, namely that elementary particle interactions may lack conservation properties. ...
 
 

Symmetry-breaking phase transitions

In kinetic models of self-organized dynamics inspired from the Vicsek model, the distribution of particle orientations exhibits symmetry-breaking phase transitions due to multiple 'Local Thermodynamical' equilibria (LTE)....


Propagation of chaos

The derivation of macroscopic models relies on the 'propagation of chaos', i.e. the fact that the particles become statistically independent when their number becomes large. In complex systems, self-organization may prevent propagation of chaos to arise. ...

 



Emergence

Emergence refers to the ability of complex systems to spontaneously form large-scale structures which are not directly encoded in the agents' interaction rules. In models, emergence results from various mechanisms. In this page are found examples of

   

Jamming transition

The typical phenomenology is that of a traffic jam: in the jammed phase, the vehicle density is maximal, corresponding to the minimal allowed distance between two vehicles, while in the free flow phase, the vehicle density is below this threshold. Pedestrian crowds or mammal herds are two-dimensional analogs of traffic jams. A fish school or a tumor may also, to some extent, be viewed as three dimensional analogs. These systems can be modeled on the macroscopic scale by fluid models with density constraints.  ...
 
 


Applications  

Economics and social sciences

Analogy between thermodynamic equilibria and Nash equilibria: We propose a framework for non-cooperative non-atomic anonymous games with a continuum of players (also known as a Mean-Field Game) which exhibit strong scale separation.
 

Fish schools

Experimental measurements of fish trajectories (J. Gautrais et al, J. Math. Biol. (2009) 58: 429-445) suggest that individual fish behaviour follows the 'Persistent Turning Walker' (PTW) process. The PTW model assumes that individual trajectories consist of pieces of circles with randomly varying radii, which are patched together (see picture below).  ...
 
 

Models of network formation

Self-organized dynamics often leads to the organization of agents into networks. An Individual-Based Model of network generation by self-propelled particles has been developed. Each agent deposits directed pheromones and interacts with them through alignment interaction.   ...


Herds

Herds of gregarious mammals offers a natural laboratory for the observation of emergence phenomena. Like in vehicular traffic or crowds, the spatial extension of the agents gives rise to congestion phenomena. Congestion phenomena and their morphogenetic capability within herds have been studied and simulated ...


Pedestrians and crowds

Interactions between pedestrians in crowds produce emerging self-organized structures. In this work, we have studied this phenomenon under controlled laboratory conditions, in the framework of the 'PEDIGREEANR projectWe have developed both microscopic (Individual-Based) and macroscopic (fluid-like) models. ...

Collective cell dynamics

Collective cell dynamics is of key importance in such processes as embryogenesis, tissue regeneration or tumor growth. Tumors grown in vitro in 3D (or tumor spheroids, see IP3D web page and pictures) offer a fascinating laboratory system. A microscopic, individual model of cell growth, division and motion is under development...





Numerical methods for asymptotic problems


Asymptotic-Preserving (AP) methods

The concept of an Asymptotic-Preserving (AP) method makes a breakthrough in the numerical resolution of asymptotic perturbations of Partial-Differential Equations. It has first been introduced by Shi Jin for transport equations in diffusive regimes (see S. Jin, SIAM J. Sci. Comp. 21, 441-454, 1999). ...

 
Below are presented several new applications of this concept to fluid and plasma problems, with a large potential of other kinds of applications.
 

Quasineutral limit in plasmas

At large time and space scales, plasmas tend to be quasineutral, i.e. the local charge vanishes. However, at small time and space scales, quasineutrality breaks down. The typical breakdown scales are the electron plasma period and the Debye length. At large plasma densities, both are very small, compared to the usual time and space scales of interest. AP-schemes applied to plasma models in the quasi-neutral limit are described below. ...
 
 

Strongly anisotropic diffusion problems ; application to large 
magnetic fields in plas 
mas 

Strongly anisotropic diffusion problems occur in various areas of physics such as structural mechanics of plates and shells, geophysical and fast rotating flows, or strongly magnetized plasmas for instance.  AP-methods for strongly anisotropic diffusion problems have been developed and applied to plasmas under large magnetic fields. ...
 

All-speed scheme for compressible fluids, multiphase flows and fluids with density constraints

Compressible fluids under small Mach-number (i.e. when the fluid motion is slow compared to the thermal speed) are subject 
to very fast acoustic waves which contribute to instantaneously equilibrate the pressure 
over the whole domain. AP- 
methods (aka 'all-speed' schemes) are necessary when the Mach-number is small in some regions and order unity in other regions. ...
 


Multi-scale methods

Multi-scale methods are needed when detailed information about the microscopic structure of the solution (i.e. the structure of the solution at scale 'eps') are needed. This may occur for instance when nonlinear interactions give rise to macroscopic structures which persist in the limit eps -> 0 .

 

Localized Model-Upscaling (LMU) and Macro-guided micro (MGM) methods

The localized model-upscaling (LMU) method consists in coupling a perturbation model and its asymptotic limit model (when the perturbation parameter is sent to zero) through a transition zone. ...

The Macro-Guided Micro method (MGM) relies on the joint solution of the microscopic and the macroscopic scales ...
 
 

Multi-Scale Finite-Element Method (MS-FEM)


This projects investigates the applicability of the Multi-Scale Finite-Element Method (MS-FEM) of T. Hou to diffusion equations in perforated domains. One targetted application is pollutant dispersion in cities. Pollutant dispersion is extremely dependent of the geometry of the city but its full account leads to very time consuming simulations. The MS-FEM method is able to provide real-time responses to critical events, which is extremely useful in crisis management. ...
 
 
 

Other topics 


Other works concerning the following topics can be found on the 'publications' page