Mini courses

Prof. Kazuo Aoki

Kyoto University and National Cheng Kung University

Title: Two-temperature fluid models for a polyatomic gas derived from kinetic theory

Slides: Lecture 1, Lecture 2, Lecture 3

Abstract: Multi-temperature fluid models have played important roles in the study of high-speed nonequilibrium gas flows. However, their relation with kinetic theory has not necessarily been clarified so far. One of the reasons would be that the Boltzmann equation for polyatomic gases is very complex and can be written only in abstract forms. In this short course, we focus our attention on the formal derivation of two-temperature fluid dynamics, consisting of the two-temperature Navier-Stokes equations and their boundary conditions, from kinetic theory. The derivation is based on the Chapman-Enskog approach applied to the Boltzmann-type kinetic equations for a polyatomic gas, with special attention to an appropriate parameter setting describing the slow relaxation of internal modes correctly. After some discussions based on the Boltzmann equation, an explicit analysis is carried out for the ES-BGK model of the Boltzmann equation. The resulting two-temperature Navier-Stokes equations are then applied to the problem of shock-wave structure for gases with slow relaxation of internal modes. In addition, appropriate slip boundary conditions for the two temperature Navier-Stokes equations are derived by the analysis of the Knudsen layer for the ES-BGK model. Finally, some applications of the boundary conditions are presented.

Co-funded by the European Union (ERC CoG KiLiM, 101125162).

Prof. Eric Carlen

Rutgers University

Title: Quantum Markov Semigroups in Non-equilibrium Statistical Physics

Lecture Notes: link

Slides: Lecture 1, Lecture 2, Lecture 3

Abstract: We discuss the theory of quantum Markov semigroups, developing, the mathematical theory, the derivation from reversible quantum dynamics in the weak and singular coupling limits, and applications to thermostatted systems and kinetic theory. 

Prof. Alessia Nota

Università degli Studi dell'Aquila

Title: On the Smoluchowski coagulation equation for aggregation phenomena

Slides: Lecture 1, Lecture 2, Lecture 3

Abstract: The phenomenon of coagulation is the mechanism by which particles (clusters) grow, the underlying process being successive mergers. It can be observed in physical systems, e.g., aerosol and raindrop formation, smoke, sprays and galaxies, as well as in biological systems. 

The evolution of the particle size distribution is described at mesoscopic level by the Smoluchowski coagulation equation, an integro-differential equation of kinetic type, which provides a mean-field model of clustering dynamics. Coagulation equations are not only relevant from the point of view of applications; they are also fascinating due to the rich structure that solutions can display. It is well known that in kinetic models satisfying the so-called detailed balance condition it is possible to construct an entropy functional which can be used to obtain convergence to equilibrium results, when considering instead kinetic equations for which the detailed balance condition fails (as in the case of the coagulation equation) more complicated dynamical behaviours can arise. Indeed, depending on the structure of the rate kernels, the solutions of the equation exhibit rich behaviour, such as gelation (formation of particles with infinite mass in finite time) or self-similarity (preservation of the shape over time). 

Prof. Benjamin Schlein

University of Zurich

Title: Bogoliubov theory in many-body quantum mechanics

Lecture Notes: link1, link2

Abstract: In this mini-course, I am going to discuss some recent progress in the mathematical analysis of many-body quantum systems. I will present a recently developed rigorous version of Bogoliubov theory and I will explain how it can be used to study equilibrium and non-equilibrium properties of quantum gases. 

I will first consider a system of N bosons confined in a volume of order one and interacting through a repulsive potential with scattering length of the order 1/N (the Gross-Pitaevskii regime). In this limit, I will derive precise estimates on the ground state energy and on the low-energy excitation spectrum of the Hamilton operator. Furthermore, I will discuss how to approximate the many-body time-evolution resulting from a change of the trapping fields, through an effective dynamics (governed by the nonlinear Gross-Pitaevskii equation) for the Bose-Einstein condensate and a quadratic evolution for its excitations. 

I will then explain how similar techniques can also be applied to infer precise estimates on the ground state energy and the low-temperature free energy of dilute Bose gases in the thermodynamic limit, at fixed but small density. 

At the end, I will show that Bogoliubov theory is also an important tool in the study of Fermi gases. In particular, I will sketch how it can be used to obtain precise estimates on the correlation energy of mean-field systems. 

Prof. Paul Dario

University of Paris-Est Créteil

Title: Integer-valued height functions and the XY model

Abstract: In this course, we will discuss two related models of statistical physics: the XY (or rotator) model and the integer-valued height functions.

The XY model is a lattice spin system with continuous symmetry (the spins are valued in the circle $S^1$). One of the important features of this model is the existence of a rather specific two-dimensional phase transition known as the Berezinskii-Kosterlitz-Thouless (BKT) transition.

The integer-valued height functions are a class of models of random interfaces which also exhibit a phase transition in two dimensions, known as the localization/delocalization transition or roughening transition. This phase transition is closely related to the BKT phase transition for the XY model.

Prof. Sabine Jansen 

Ludwig Maximilian University, Munich

Title: Diagrams and cluster expansions in probability and statistical mechanics

Abstract: Many quantities in probability and statistical physics cannot be computed explicitly. Often some information can be gained from perturbation series: cluster and virial expansions, Feynman diagrams. The minicourse gives an introduction to cluster expansions - old & new results - and highlights some connections with (1) combinatorial species, (2) cumulants and diagrams as used in stochastic geometry.

Co-funded by the European Union (ERC CoG KiLiM, 101125162).

Prof. Eva Löcherbach 

University of Paris 1

Title: Probabilistic models for interacting systems of spiking neurons

Lecture Notes: link

Abstract: I will introduce a continuous time probabilistic model for systems of interacting and spiking neurons. In this process, neurons spike at a rate depending on their membrane potential value. When spiking, they have a direct influence on their post-synaptic partners, namely, a fixed value, called "synaptic weight", is added to the potential of the postsynaptic neurons. In between successive spikes, due to some leakage effects, the membrane potential process follows a deterministic flow. 

In a first part of the lecture, we discuss the construction, well-posedness and the longtime behavior of the process, for a finite number of neurons and for infinite systems of neurons, both in the case with and without reset of the spiking neuron. 

We then discuss mean field limits for the Hawkes description (without reset) of the model. In particular we will see how in the limit an ODE describing the evolution of the mean firing rate appears and how this approach allows to describe for example oscillatory behavior. The case with reset will be discussed as well, and we will see how the limit process and its longtime behavior help us to explain important phenomena in neuroscience such as "metastability". 

A second part of the lecture is devoted to the question of how to sample from the stationary time evolution of a potentially infinite system of neurons, by means of a clan of ancestors method, involving a so-called Kalikow decomposition of the transition probabilities or intensities. We discuss this approach first in discrete time models and then show how to extend such methods to continuous time. 

Prof. Fabio Toninelli 

Technical University of Vienna

Title: Large-scale behavior of (super-)critical stochastic PDEs

Slides: Lecture 1

Abstract: This course will focus on some singular stochastic PDEs (SPDEs) motivated by out-of-equilibrium statistical physics, in particular driven diffusive systems and stochastic interface growth. We will deal with equations that are "critical" or "super-critical", for which scaling or Renormalization group arguments suggest a  Gaussian limit for large space-time scales (with logarithmic corrections to diffusivity in the critical dimension). In particular, this class includes the "Anisotropic KPZ equation" in dimension d=2 and the stochastic Burgers' equation in dimension $d\geq 2$. Focusing on the specific case of the Burgers' equation, I will explain how a careful analysis of the generator of the processes allows to prove Gaussian scaling limits in the super-critical and critical dimension. Based on joint works with Giuseppe Cannizzaro, Dirk Erhard, Massimiliano Gubinelli, Quentin Moulard.