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
Amit Einav (Durham University) - Paths of order in a jungle of chaos
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