MINI COURSES
André Schlichting (University Ulm)
Breakdown of the mean-field description of interacting particle systems: Phase transitions, metastability and coarsening
TBD
Marielle Simon (Aix-Marseille Université)
The mathematical derivation of thermodynamics laws from interacting oscillators
Abstract: The heat equation is known to be a macroscopic phenomenon, emerging after a diffusive rescaling in both space and time. However, deriving Fourier's law from a microscopic dynamics consistent with Newton's laws remains a mathematical challenge. In this mini-course I will introduce a class of models for which several mathematical results have been proved in the last decades, namely the chain of interacting oscillators with some stochastic perturbation. Thanks to sufficient mixing properties of the microscopic dynamics, one can show that the macroscopic energy density diffuses, with various boundary conditions which depend on the microscopic boundary mechanisms. If I have time I will also show how modifying some properties of the noise can drastically change the macroscopic behavior of the energy, from diffusive to superdiffusive, evolving according to some fractional Laplacian.
TALKS
Panagiota Birmpa (Heriot-Watt University) - Non-equilibrium fluctuations for the stirring process with births and deaths.
Abstract: We consider the one-dimensional stirring process on the segment {−N , . . . , N }, coupled to boundary dynamics that inject particles from the right reservoir and remove particles from the left reservoir, each acting on a window of fixed and finite size. In this talk, I will present the non-equilibrium fluctuations of the system when the initial configuration is given by a product measure associated with a smooth macroscopic profile. In this regime, the fluctuations are described by an Ornstein–Uhlenbeck process driven by the Laplacian and gradient operators, with boundary conditions determined by the hydrodynamic profile. A central step in the analysis is the derivation of sharp bounds for space and space–time v-functions of arbitrary degree associated with the centered occupation variables. In particular, we prove that the v-functions of degree 2 and 3 are of order N^{−1}, while those of degree at least 4 are of order N^{ −1−\zeta} for some \zeta> 0.
Alain Blaustein (INRIA Lille) - Numerical analysis and simulation of Vlasov-Poisson type systems
Abstract: We focus on the simulation and the numerical analysis of the Vlasov-Poisson system with and without collisions. This kinetic mean-field model is central to plasma physics as it encodes several key phenomena which we aim to recover at the discrete level. In particular, we propose a Hermite - discontinuous Galerkin scheme which preserves the exponential nonlinear stability of Gaussian distributions when collisions are taken into account. We also investigate the collisionless setting and in particular, we focus on plasma oscillations occurring in the quasineutral limit. The numerical method is proven to filter these oscillations, uncovering the underlying quasineutral dynamics. Various numerical simulations illustrate the high-order accuracy and the asymptotic preserving properties of the numerical method.
Gioia Carinci (Università di Modena e R. Emilia) - From Discrete Bidding to Continuous Flows: A Multi-Agent Auction Model
Abstract: We present an auction model in which multiple autonomous bidders attempt to sell their goods with the goal of maximizing their profits. Bidders operate without knowledge of one another and adjust their bids solely based on their most recent performance behavior (myopic adaptation). More precisely, a bidder who is awarded in a given round will increase their selling price in the next round, while one who is not awarded will decrease it. In each round, the auctioneer purchases the lowest $p$-fraction of the total energy offered by the bidders.
We find a system of differential equations governing the macroscopic dynamics and derive it as a scaling limit of the microscopic model. We find an explicit solution for the max-price evolution $q_t$ and show that in the long run bidders coordinate, i.e. they tend to bid the same value only depending on their initial distribution and the value $p$.
For Poisson-distributed initial bids, we obtain hydrodynamic limits and a central limit theorem for $q_t$. Finally we prove that when bidders have heterogeneous update speeds, the max price velocity becomes proportional to the harmonic mean of the velocities of the bidders at the max price.
Joint work with P. Ferrari, C. Franceschini, N. Manelli.
Théophile Dolmaire (Università degli Studi dell'Aquila) - Inelastic collapse and global well-posedness in dissipative particle systems
When studying systems of particles, the very first step before any qualitative analysis relying on kinetic equations is to establish the well-posedness of the dynamics of the system. In the case of inelastic hard spheres, the dissipative collisions lead to emergence of clusters, and to the occurrence of infinitely many collisions in finite time, a phenomenon known as the inelastic collapse. This phenomenon remains poorly understood, yet it represents a major obstruction to a rigorous derivation of the inelastic Boltzmann equation. We will present recent results on collapsing systems, including the identification of new families of singularities. Besides, we will consider a particular class of inelastic hard sphere systems, in which a fixed amount of kinetic energy is lost in each sufficiently energetic collision. For this class, we establish the global well-posedness of the particle dynamics.
The results were obtained in collaboration with Roberto Castorrini (Università degli Studi della Tuscia), Eleni Hübner-Rosenau (Universität Regensburg) and Juan J. L. Velázquez (Universität Bonn).
Amit Einav (Durham University) - Paths of order in a jungle of chaos
Abstract: Systems that revolve around the interactions of many elements are a constant part of our day to day lives. Yet for all their prevalence, trying to explore mathematical models of such systems, theoretically or numerically, can often be a herculean task. To try and address this difficulty, it was realised early on (back in the late 19th century) that we do not need to understand how each and every element in the system behaves. Instead, it is often enough to understand how a typical or average element does.
A revolutionary idea that birthed a new way to investigate systems of many elements was formalised in the work of Mark Kac in 1956. Kac suggested to find how an average particle in dilute gas behaves by considering a “probabilistic model” of the gas, expressed via a PDE for the probability to find the system in various configurations, together with the idea that as the number of particles in the gas increases, they become more and more independent. The latter is often known as molecular chaos, or chaos. Combining these two ingredients, Kac was able to find an equation that describes how a “limiting average particle” evolves in his settings – which ended up being a one-dimensional variant of the celebrated Boltzmann equation.
These ideas are far more general and powerful than their application in Kac’s model, and they have formed the framework of what we now call the mean field limit approach. At its heart, the mean field limit approach has two ingredients:
An average model for the system, expressed via a PDE for its probability measure.
An asymptotic correlation relation that expresses the emerging phenomena we expect to get as the number of elements goes to infinity.
In recent decades the use of the mean field limit approach has expanded beyond physical models into the realm of biology, economy, and societal studies. Yet in almost all cases considered so far, the sole correlation relation used was chaos. This seems to be inappropriate in settings that have a tendency for adherence such as biological swarming.
In our talk we will briefly discuss the background to the mean field limit approach as well as Kac’s model and its limiting equation. We will then jump to 2013 and consider the Choose the Leader model, or CL model, which is a Kac-like animal swarming model introduced by Carlen, Degond, and Wennberg where chaoticity breaks. Motivated by the desire to understand this model better, we will introduce two new asymptotic correlation relations, order and partial order, and see how they arise naturally in the CL model.
Nicolas Fournier (Sorbonne University) - Stochastic particle systems for the Keller-Segel equation
The Keller-Segel equation describes the movement of cells via chemotaxis. The cells diffuse in the plane and release a chemical. This chemical, which also diffuses, attracts the cells. This leads to a singular interaction between the cells (via the chemical). This interaction is critical: depending on the values of the constants, or a cluster of cells may or not emerge in finite time.
We will discuss finite particle systems approximating this equation, both in the elliptic case where the chemical diffuses instantaneously and in the parabolic case where the product diffuses at a finite rate.
Based on joint works with B. Jourdain, Y. Tardy and M. Tomasevic
Milica Tomasevic (CNRS and Ecole polytechnique) - On the Go or Grow particles
In this talk we study a branching particle system of diffusion processes on the real line interacting through their rank in the system. Namely, each particle follows an independent Brownian motion, but only $K\geq 1$ particles on the far right are allowed to branch with constant rate, whilst the remaining particles have an additional positive drift of intensity $\chi>0$. This is the so called Go or Grow hypothesis, which serves as an elementary hypothesis to model cells in a capillary tube moving upwards a chemical gradient. Despite the discontinuous character of the coefficients for the movement of particles and their demographic events, we first obtain the limit behavior of the population as $K\to \infty$ by weighting the individuals by $1/K$. Then, on the microscopic level when $K$ is fixed, we investigate numerically the speed of propagation of the particles and recover a threshold behavior according to the parameter χ consistent with the already known behavior of the limit. Finally, by studying numerically the ancestral lineages we will see how one could categorize the traveling fronts as pushed or pulled according to the critical parameter $\chi$.
This is a joint work with M. Demircigil (Univ. of Arizona).
The slides of the talks will be uploaded here