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
TBD
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.
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.
The slides of the talks will be uploaded here