Abstracts
Ricardo Alonso - Properties of solutions to the Landau-Fermi-Dirac equation
In this talk we discuss the essentials of the LFD equation. In particular, we present a priori estimates for solutions to the homogeneous LFD equation that show moment propagation, Lebesgue's integrability and boundedness of solutions independent of the quantum parameter. These properties in combination with entropy techniques allow us to prove algebraic convergence towards thermal equilibrium under a weak saturation regime.
Anton Arnold - Short- and long-time behavior in evolution equations: the role of the hypocoercivity index
The "index of hypocoercivity" is defined via a coercivity-type estimate for the self-adjoint/skew-adjoint parts of the generator, and it quantifies `how degenerate' a hypocoercive evolution equation is, both for ODEs and for evolutions equations in a Hilbert space. We show that this index characterizes the polynomial decay of the propagator norm for short time and illustrate these concepts for the Lorentz kinetic equation on a torus. This talk is based on joint work with F. Achleitner, E. Carlen, E. Nigsch, and V. Mehrmann.
Eric Carlen - Approach to equilibrium in a quantum Kac model
We present a class of quantum Kac models, and prove the structure of the spectrum is largely determined by the spectrum of a graph Laplacian corresponding to a classical random walk on a graph. We present simple means for determining the spectra gap for these walks, and apply our results to the rate of approach to equilibrium. This is joint work with Michael Loss.
Reda Chhaibi - On Proposition 1.4 in Johansson's seminal paper "Shape fluctuations and random matrices" (2000): A point of view from harmonic analysis
The purpose of this talk is to convince you of the depth of Proposition 1.4 in Johansson's seminal paper "Shape fluctuations and random matrices" (2000). This proposition states that the largest eigenvalue of a Wishart matrix has the same distribution as the last passage time in directed percolation, with exponential weights, which is nothing but TASEP in disguise. As such, fluctuations according to the Tracy-Widom law are not a big surprise.The goal of this talk is to virtually evade the classical RSK correspondence and only explain this equality in law from harmonic analysis, and more precisely the rigidity of harmonic analysis for groups with varying curvatures. This is intimately Pitman's classical theorem. Based on the paper "Quantum SL2, Infinite curvature and Pitman's 2M-X theorem" (PTRF2021) with F. Chapon and on ongoing works.
Laurent Desvilettes - Entropy dissipation for Landau and Landau-Fermi-Dirac operators
We present recent works on estimates relating the entropy and entropy dissipation of Landau operator, together with their applications to smoothness and large time behavior of the corresponding equations. We discuss the extensions of those estimates to cases in which the Maxwellian statistics is replaced by a Fermi-Dirac statistics.
Emanuele Dolera - Propagation of chaos for a discrete-states non-Maxwellian Kac model
In this work we start considering a spatially homogeneous N-particle system with discrete velocities, evolving through a Markovian dynamics. We derive the Master equation under two very general assumptions, that we call non-Maxwellian: 1) the collision time process depends on the velocities through a rate function Γ that determines the rates of exponential waiting times; 2) the probability distribution of the collision angles also depends on the velocities thought a non-uniform probability kernel B. Thanks to de Finetti's representation theorem, we consider the counting (Markov) process of the various states. We first deduce the Master equation for this counting process, and then we derive from it the non-linear dynamics for the single particle by a propagation-of-chaos argument. The non-linear Boltzmann-like equation that originates from this argument turns out a bit different from the usual one. Among the new features, we show that the equilibrium can depend, in general, on the given functions Γ and B. Finally, following McKean, we discuss the derivation of a Wild formula for this Boltzmann-like equation along with its use to study convergence to equilibrium.
Amit Einav - Can we have a little Order amidst all this Chaos?
Systems that involve many elements are ubiquitous in our day to day lives. Their investigation, however, is hindered by the complexity of such systems and the amount of (usually coupled) equations that are needed to be solved. To deal with these complexities the so-called mean field limit approach was created in the late 50s. This approach attempts to circumvent the issues encountered with such systems by using a probabilistic viewpoint instead of a deterministic one. The mean field limit approach’s goal is to find the behaviour of a limiting average element in the system. Two ingredients are required to achieve this: an average model of the system (i.e. an evolution equation for the probability density of the ensemble), and an asymptotic correlation relation that expresses the emerging phenomena we expect to get as the number of elements goes to infinity. While mean field limits of average models in various settings have been explored in recent centuries, to date we use only one asymptotic correlation relation – chaos, the idea that the elements become more and more independent. This, however, doesn’t seem reasonable in models that pertain to biological and societal phenomena. In our talk we outline the problem of having chaos as the sole asymptotic correlation relation and define the new asymptotic relation of order. We show that this is the right relation for a recent animal model suggested by Carlen, Degond, and Wennberg – Choose the Leader model, and highlight the importance of appropriate scaling in its investigation.
Marina Ferreira - Localization in multicomponent coagulation systems
We study the properties of multicomponent Smoluchowsky coagulation equations. These equations describe the evolution of the number density function over the high-dimensional particle composition space. We show that, under a self-similar scaling, all solutions localize along a line defined by the initial condition. This result holds on a large class of coagulation kernels and it can be used to reduce the analysis of multicomponent systems to one-component ones. This is joint work with Jani Lukkarinen, Alessia Nota and Juan Velázquez.
Dmitri Finkelshtein - Spatio-Temporal Correlations in a Class of Interacting Particle Systems
For interacting particle systems in the continuum, the study of spatial pair correlations between particles at different spatial positions is well-established, typically using second-order (spatial) correlation functions which represent factorial spatial moments of the system's states. However, the investigation of spatio-temporal pair correlations between particles located at different spatial positions in different moments in time has received relatively little attention and lacks a comprehensive approach. We introduce a general method suitable for a broad class of non-diffusive interacting particle systems in the continuum. Our approach simplifies the analysis of spatio-temporal correlations by relating them to the study of spatial correlations in auxiliary multi-type systems. We demonstrate our approach for several population dynamics and consider their mean-field and beyond-mean-field behaviour. Finally, we validate our theoretical predictions by comparing them to the results obtained through computer simulations. Based on a joint paper with Otso Ovaskainen and Panu Somervuo.
Chiara Franceschini - Duality for a boundary driven asymmetric model of energy transport
In this talk I will introduce a model of heat conduction defined on a one dimensional chain with an open boundary.
Energy is exchanged asymmetrically between nearest neighbor sites and it is kept constant by the bulk dynamics except for the interaction of the extremal sites of the chain with two reservoirs at different temperatures (non-equilibrium). Information regarding the stationary measure and duality results can be inferred thanks to the existence of a non local map which links this model with its symmetric counterpart.
Rouven Frassek - The boundary-driven q-Hahn process
I will discuss the relation between non-compact spin chains and the zero-range processes introduced by Sasamoto-Wadati, Povolotsky and Barraquand-Corwin. The main difference compared to the prime examples of integrable particle processes, namely the SSEP and the ASEP, is that for the models discussed in this talk several particles can occupy one and the same site. Guided by the desire to maintain the integrable structure, I will introduce boundary conditions for these models that are obtained from the boundary Yang-Baxter equation. This allows to define analogues of the open SSEP and ASEP with boundary reservoirs. Some recent exact results concerning these types of integrable non-equilibrium zero-range processes will be discussed.
Davide Gabrielli - Solvable Stationary Non Equilibrium States
Boundary driven stochastic lattice gases are simple but effective models for non equilibrium statistical mechanics. Apart special cases, as for example the zero range model where the stationary state is always of product type, they exhibit long range correlations. I will discuss a class of models for which it is possible, in the boundary driven case, to give a simple representation of the invariant measure in terms of mixtures of inhomogeneous product measures. This is true for the Kipnis Marchioro Presutti model and its dual and for a class of generalized zero range dynamics.
François Golse - A Model of Gas-Surface Interaction
We discuss a model of gas surface interaction, where the solid phase is a crystal. Molecular interactions with the solid phase are described by means of collisions between the gas molecules and the phonons corresponding to the vibrations of the crystal atomic sites. To some leading order of approximation, the equivalent boundary condition so obtained is similar to the Maxwell accommodation condition at the boundary, except that the accomodation coefficient depends on the energy of impinging molecules. This dependance is explicit in terms of the relaxation time of the phonon-molecule collisions.
(joint work with K. Aoki, V. Giovangigli, S. Kosuge)
Stefan Großkinsky - Size-biased diffusion limits and the inclusion process
We study the Inclusion Process with vanishing diffusion coefficient, which is known to exhibit condensation and metastable dynamics for cluster locations. We focus on the dynamics of the empirical mass distribution and consider the process on the complete graph in the thermodynamic limit with fixed particle density. Describing a given configuration by a measure on a suitably scaled mass space, we establish convergence to a limiting measure-valued process. When the diffusion coefficient scales like the inverse of the system size, the scaling limit is equivalent to the well known Poisson-Dirichlet diffusion, offering an alternative point of view for this well-established dynamics. Our approach can be generalized to other scaling regimes, providing a natural extension of the Poisson-Dirichlet diffusion to infinite mutation rate. Considering size-biased mass distributions, our approach yields an interesting characterization of the limiting dynamics via duality.
This is joint work with Simon Gabriel and Paul Chleboun (both Warwick).
Martin Grothaus - Industrial Mathematics, Hypocoercivity and SPDEs
Motivated by problems from Industrial Mathematics we further developed the concepts of hypocoercivity. The original concepts needed Poincaré inequalities and were applied to equations in linear finite dimensional spaces. Meanwhile we can treat equations in manifolds or even infinite dimensional spaces. The condition giving micro- and macroscopic coercivity we could relax from Poincaré to weak Poincaré inequalities. In this talk an overview and many examples are given.
Amirali Hannani - Localization and Poisson statistics for the "avalanche model
In this talk, first, I roughly explain the notion of Many-Body Localization (MBL) in quantum spin systems and give a brief overview of the localization-delocalization transition. Then, I explain the so-called "avalanche model": this is a simple toy model introduced to study the "stability of MBL". It is believed (non-rigorously!) that this model features a localization-delocalization transition: varying a natural parameter $\alpha \in (0,1)$. Then, I present our results: we prove for $\alpha>0$ sufficiently small, eigenvectors of this model are localized in a certain sense. Moreover, after proper rescaling, eigenvalues converge to a Poisson point process in the vague topology. If time permits, I present some general ideas about the proof.
Kohei Hayashi - Derivation of the Kardar-Parisi-Zhang equation from the Bernardin-Stoltz model
We study the so-called Bernardin-Stoltz (BS) model which admits two conserved quantities: volume and energy. We consider the BS model driven by a general nonlinear potential and then asymptotically let the inverse temperature of the system go to zero. As a consequence, by a Taylor expansion argument, we can extract a principal part driven by the harmonic potential, and we can control the strength of a reminder part which is governed by a cubic polynomial. Under this situation, we consider two fluctuation fields defined as linear combinations of the fluctuation fields of volume and energy, and as a main result, we show that the fluctuations of one field converge to a solution of the stochastic Burgers equation.
Bertrand Lods - Prodi-Serrin like criteria for Landau equation with soft potentials
The scope of the talk is to introduce Prodi-Serrin like criteria for weak solutions under which it is possible to show existence of classical solutions to the spatially homogeneous Landau equation for all classical potentials and dimensions. We establish instantaneous appearance of Lp estimates using the Prodi-Serrin criteria. This, combined with a suitable De Giorgi's argument provides appearance of pointwise bounds for such solutions. Uniqueness and regularity of solutions will also be discussed.
Alessandra Occelli - Universality for multi-component stochastic systems
We study the equilibrium fluctuations of an interacting particle system evolving on the discrete ring with three species of particles named A,B and C. The interaction rates depend on the type of particles involved and on the size of the system. This model can be seen as a multi-species generalisation of the weakly asymmetric simple exclusion process. We analyse proper choices of the density fluctuation fields associated to the conserved quantities (the densities of particles for each species), that are given by linear combinations of the fields matching those from nonlinear fluctuating hydrodynamics theory: we show that they converge, when the size of the system goes to infinity to a system of stochastic partial differential equations, that according to the asymmetry of the jumps, can either be the Ornstein–Uhlenbeck equation or the Stochastic Burgers’ equation. Based on a joint work with G. Cannizzaro, P. Gonçalves and R. Misturini.
Stefano Olla - Heat equation from a deterministic dynamics
We derive the heat equation for the thermal energy under diffusive space-time scaling from a purely deterministic microscopic dynamics satisfying Newton equations perturbed by an external chaotic force acting like a magnetic field. Joint work with Giovanni Canestrari and Carlangelo Liverani.
Valeria Ricci - A model for slowing particles in random media
We present a simple microscopic model for slowing particles in a random medium, where the random medium is represented as a collection of spherical obstacles with Poisson distributed centres. We derive the associated kinetic equation in the limit where the radius of the obstacles vanish and their density grows to infinity in such a way that the average distance travelled by a particle before its first collision with an obstacle in the medium stays finite. The limit equation includes the contribution of particles stopping at some point in the medium. The work is in collaboration with A.J. Soares and F.Golse.
Ellen Saada - Couplings and Attractiveness for General Exclusion Processes
Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in the simple exclusion process. We consider here general exclusion processes where jump rates from an occupied site to an empty one depend not only on the location of the jump but also possibly on the whole configuration. These processes include in particular exclusion processes with speed change. In the spirit of our previous papers on particle systems of misanthrope type, we derive for such processes necessary and sufficient conditions for attractiveness, through the construction of a coupled process under which, in any coupled transition, discrepancies on the involved sites do not increase, or even decrease. We emphasize the fact that basic coupling is never attractive for this class of processes, except in the case of simple exclusion, and that the coupled processes presented here necessarily differ from it. We present various examples, This is a joint work with Thierry Gobron (Lille).
Francesco Salvarani - Homogenization of the Renewal Equation with Heterogeneous External Constraints
We study the homogenization limit of the renewal equation with heterogeneous external constraints by means of the two-scale convergence theory. We prove that the homogenized limit satisfies an equation involving non-local terms, which are the consequence of the oscillations in the birth and death terms. We moreover show that the numerical approximation of the homogenized equation via the two-scale limit gives an alternative way for the numerical study of the solution of the limiting problem. The results have been obtained in collaboration with Etienne Bernard (UNPC, France).
Cristina Toninelli - Fredrickson-Andersen 2-spin facilitated model: sharp threshold
The Fredrickson-Andersen 2-spin facilitated model (FA-2f) on the integer lattice Z^d is a well-known interacting particle system with kinetic constraints, exhibiting cooperative and glassy dynamics. In the FA-2f model, vacancies play a crucial role in facilitating particle motion, allowing a particle to be created or removed from a site only if at least 2 of its nearest neighbors are unoccupied. In this context, we will present precise results for the first time, denoted as τ, at which the origin becomes empty in the stationary process when the density of empty sites (q) is small. This result holds for any dimension d greater than or equal to 2 and is given by:
where λ(d,2) represents the threshold constant for the 2-neighbor bootstrap percolation on Z^d. This outcome represents the first precise result for a critical Kinetic Constraint Model (KCM), resolving longstanding disputes within the physics literature that have spanned the past four decades. We will also provide insights into the primary relaxation mechanism that leads to this result, provide a glimpse into the proof techniques involved, and discuss additional results that can be obtained using our approach for more generalized KCMs. These additional results may include full universality results in two dimensions. [This research is a collaborative effort with I. Hartarsky and F. Martinelli.]
Sonia Velasco - Dynamic and static large deviations for the total mass of a one dimensional SSEP in contact with slow reservoirs
We consider a speeded up symmetric simple exclusion process with slow boundaries. We prove the hydrodynamic limit of the total mass as well as a pointwise hydrodynamic limit of the process. We then establish a dynamic and static large deviations principle for the total mass. The functional obtained in the static case for the mass, gives a hint at the one expected in the non accelerated case for the whole process. This work is a collaboration with Claudio Landim (IMPA, Rio de Janeiro).
Bernt Wennberg - Kac models with inclusion and exclusion
We have previously analyzed a Kac model with a hard core exclusion principle, where only particle configurations satisfying |x_j-x_k| > ε_n, are allowed, with ε_n depending on the number of particles n. This mimics dynamics of Fermions in the quantum case, and raises interesting questions concerning the limit as n approaches infinity. In this talk I will discuss variants of this model, including a model that could be said to mimic the dynamics of Bosons. It is work in progress together with Eric Carlen.
Ali Zahra - Hydrodynamic description of multispecies TASEP
Exclusion processes in one dimension first appeared in the 70's and have since dragged much attention from communities in different domains: stochastic processes, out of equilibriums statistical physics and more recently integrable systems. While it is well known that the hydrodynamic limit of the single species totally asymmetric simple exclusion process (TASEP) is described by the Burger's equation, much less is known for multispecies generalizations, which present a much richer phenomenology. In this talk, I shall present results for a version of the TASEP, containing two species of particles and a hierarchical dynamic depending on two parameters. By using results for the stationary measure of such model on ring domains, I shall formulate the conservations laws associated to the different kind of particles. I show an explicit non-linear decoupling of those equations which allows an in-depth discussion of their solutions (shocks, Riemann problem, etc.). Good agreement is found with numerical simulations. This work is in collaboration with Luigi Cantini 2022 J. Phys. A: Math. Theor. 55 305201