MINI COURSES
Pietro Caputo (Roma Tre University)
Nonlinear dynamics for spin systems: some convergence results
We introduce a natural class of nonlinear dynamics for combinatorial structures and spin systems, such as the Ising model. This class is based on the framework of mass action kinetics, which models the evolution of particle systems under pairwise interactions, capturing several important nonlinear models from various fields, including Boltzmann's equation, recombination in population genetics, and genetic algorithms. In the context of spin systems, this approach provides a nonlinear Monte Carlo dynamics, which appears to be much harder to analyze than its linear Markov chain counterpart.
In these lectures, we prove a general theorem on convergence to stationarity and present quantitative results in the high-temperature regime, recently obtained in collaboration with A. Sinclair. We begin by discussing the non-interacting case, where precise control over total variation distance can be achieved by analyzing a suitable fragmentation process. We also demonstrate how a complete implementation of the Kac program is possible in this scenario, allowing us to estimate the rate of relative entropy decay.
Next, we address the interacting case and show how to obtain sharp convergence results at high temperatures. This is accomplished by combining the analysis of fragmentation and branching processes with a novel coupling of high-temperature Ising models and sub-critical Erdős-Rényi random graphs. Several open problems will also be discussed.
Milana Pavić-Čolić (University of Novi Sad)
Polyatomic Boltzmann equation: well-posedness, higher integrability and physical relevance
This course will focus on the Boltzmann equation describing a polyatomic gas. The model captures peculiarities related to the internal structure of a polyatomic molecule by introducing one additional internal energy variable of continuous type, and by using the Borgnakke-Larsen procedure for the parametrization of collision conservation laws. One of the main ingredients of the Boltzmann collision operator is a collision kernel, which describes physics of particles interaction and whose form within the continuous framework is not known in advance.
Recent study of the space homogeneous problem imposes an assumption on the collision kernel, which can be understood as a generalization of the hard potentials with an integrable angular part. For such a collision kernel, we will discuss moments theory and global well-posedness of the space homogeneous Boltzmann equation. Moreover, the analysis will be complemented by the study of higher integrability properties for such solutions.
These promising results of the analysis of the space homogeneous Boltzmann equation motive a question of physical relevance of the proposed collision kernel. A possible way to connect the collision kernel to macroscopic observables that are measured experimentally and thus to have an insight into physical interpretation is to build continuum models from the Boltzmann equation. We will discuss a formal evaluation of the Boltzmann collision operator that allows to extract models for transport coefficients, such as shear and bulk viscosities and heat conductivity, containing parameters of the collision kernel, which can be matched with the experimental data.
TALKS
Luisa Andreis (PoliMI)
A variational approach to the spatial Smoluchowski equation
Since Smoluchowski introduced his well-known coagulation equation in 1917, there has been an active line of research focused on understanding the properties of the solutions to this equation and related models for coagulation. In particular, in 2000, Norris introduced a generalised version of the model, which he named the cluster coagulation model. This model was intended to extend the framework established by Smoluchowski, allowing particles to have additional properties beyond their mass, such as shape or spatial location.
We focus on particles having a mass and a spatial location (in a general Polish space S). We consider the Markovian particle system associated with this model: after independent exponential random times, pairs of particles merge into a single particle, with their masses being summed. The location of the resulting new particle in S is chosen according to a certain kernel. Using an approach from statistical mechanics, we express the distribution of particles in terms of a reference Poisson point process and a pairwise interaction term. Based on this formula, we derive a (conditional) large-deviation principle for the joint distribution of the particles in the limit of many particles, with an explicit identification of the rate function. We characterize its minimizer(s) through a variational problem. Finally, we prove that, in certain cases (specifically in the absence of the phase transition commonly referred to as gelation), these minimizers indeed solve the spatial version of the Smoluchowski coagulation equation.
This talk is based on a joint work with W. K ̈onig, H. Langhammer and R.I.A. Patterson (WIAS Berlin).
Inés Armendáriz (University of Buenos Aires)
Continuous Box Ball System
The Box Ball System was introduced by Takahashi and Satsuma in 1990 as a cellular automaton that exhibits solitons (travelling waves).
We study a continuous version, where blocks of consecutive occupied boxes are replaced by occupied intervals of the real numbers, separated by empty intervals. The walk representation of a configuration is given by a zig-zag function, and the dynamics by Pitman's transformation. We describe how to identify solitons, and show that they are
conserved under the dynamics. We also show that the soliton decomposition of some random zig-zag walks can be mapped to a bidimensional Poisson process, a representation that linearizes the dynamics. This extends discrete space results by Ferrari, Gabrielli, Nguyen, Rolla and Wang.
Joint work with Pablo Blanc, Pablo Ferrari and Davide Gabrielli
Anton Arnold (TU Wien)
All relative entropies for general nonlinear Fokker-Planck equations
We shall revisit the entropy method for quasilinear Fokker-Planck equations with confinement to deduce exponential convergence to the equilibrium. Even for prototypical examples like the porous-medium equation, only one relative entropy has been known so far - the Ralston-Newman entropy, which is the analog of the logarithmic entropy in the linear case. We shall give a complete characterization of all admissible relative entropies for each quasilinear Fokker-Planck equation. In particular we find that fast-diffusion equations with power-law nonlinearities admit only one entropy, while porous medium equations give rise to a whole family of admissible relative entropies (similar to linear Fokker-Planck equations). These additional entropies then imply also new moment-control estimates on the porous-medium solution. Joint work with Jose Carrillo, Daniel Matthes.
Giada Basile (Sapienza University of Rome)
Large deviations for binary collision models: energy non-conserving paths
I will present some large deviation results for binary collision stochastic models and I will exhibit some atypical paths that violate energy conservation. I will discuss the connection with the gradient flow formulation of the homogeneous Boltzmann equation. Founded by the European Union - Next Generation EU.
Alessandra Bianchi (University of Padova)
Limit Theorems for Exponential Random Graphs
Exponential Random Graphs are defined through probabilistic ensembles with one or more adjustable parameters. They can be seen as a generalization of the classical Erdos Renyi random graph, obtained by defining a tilted probability measure that is proportional to the densities of certain given finite subgraphs. In this talk we will focus on the edge-triangle model, that is a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. Borrowing tools from statistical mechanics, together with large deviations techniques, we will characterize the limiting behavior of the edge density for all parameters in the so-called replica symmetric regime, where a complete characterization of the phase diagram of the model is accessible. Joint work with Francesca Collet and Elena Magnanini.
Oriane Blondel (University of Lyon)
Ofer Busani (University of Edinburgh)
On the number of infinite geodesics at exceptional directions in the KPZ class
The KPZ class consists of many models of random growth interface. In many cases, the dynamics of these models can be studied via variational forms that give rise to metric-like spaces, which in turn, can be studied through geodesics. The study of infinite geodesics in the KPZ class has been studied intensively in the past 30 years. One central question is the following: Given a direction, how many infinite geodesics that are asymptotically going in that direction are there? In this talk I shall discuss what we know about infinite in the KPZ class and some recent developments regarding the question above. Based on several works with Marton Balazs, Timo Seppalainen and Evan Sorensen.
Rossella Della Marca (SISSA)
Investigating the role of agents’ viral load in the spread of epidemics
In classical epidemic models, a neglected aspect is the heterogeneity of disease transmis- sion and progression linked to the viral load of each infectious individual. Here, we attempt to investigate the interplay between the evolution of individuals’ viral load and the epidemic dynamics from a theoretical point of view. In the framework of multi–agent systems [3], we propose a particle stochastic model describing the infection transmission through interactions among agents and the individual physiological course of the disease. Agents have a double microscopic state: a discrete label, that denotes the epidemiological compartment to which they belong, and a microscopic trait, representing a normalized measure of their viral load. Specifically, we consider Susceptible–Infected–Removed–like dynamics where the disease transmission rate [2] or the isolation rate [1] of infectious individuals may depend on their viral load. We derive kinetic evolution equations for the distribution functions of the viral load of the individuals in each compartment, whence, via suitable upscaling procedures, we obtain a macroscopic model for the densities and viral load momentum. We perform then a qualitative analysis of the ensuing macroscopic model, and we present numerical tests in the case of both constant and viral load–dependent model parameters. This is a joint work with N. Loy and A. Tosin (Politecnico di Torino).
[1] Della Marca R., Loy N., Tosin A.: An SIR–like kinetic model tracking individuals’ viral load. Netw. Heterog. Media 17 n.3, 467–494 (2022)
[2] Della Marca R., Loy N., Tosin A.: An SIR model with viral load–dependent transmission. J. Math. Biol. 86, 61 (2023)
[3] Loy N., Tosin A.: Boltzmann–type equations for multi–agent systems with label switching. Kinet. Relat. Models 14 n.5, 867–894 (2021)
Tertuliano Franco (Universidade Federal da Bahia)
A functional CLT for the most general Brownian motion on the half-line
We prove a general functional CLT for continuous-time random walks with Berry-Esseen estimates based on the convergence of its generators. As the main consequence of this result, we deduce a functional CLT for the homogenous random walk on the nonnegative integers, under general boundary conditions at the origin. Depending on the chosen parameters, the random walk converges to any possible Brownian-type process with boundary conditions at zero, namely: absorbed BM, reflected BM, killed BM, stick BM, elastic BM, exponential holding BM, or mixed BM. Joint work with D. Erhard, M. Jara and E. Pimenta.
François Golse (Ecole Polytechnique)
Maria Groppi (University of Parma)
Action potential dynamics on heterogenous neural networks: from kinetic to macroscopic description
In the context of multi-agent systems of binary interacting particles, a kinetic model for action potential dynamics on a neural network is proposed, accounting for heterogeneity in the neuron-to-neuron connections, as well as in the brain structure. Two levels of description are coupled: in a single area, pairwise neuron interactions for the exchange of membrane potential are statistically described; among different areas, a graph description of the brain network topology is included. This allows us to obtain, from the kinetic level, classical macroscopic neural network models, which instead are usually postulated directly at the macroscopic scale of the brain regions. Equilibria of the kinetic and macroscopic settings are analyzed and numerical simulations of the system dynamics are performed in different network scenarios, with the aim of studying the influence of the network heterogeneities on the membrane potential propagation and synchronization. This is a joint work with Marzia Bisi and Martina Conte, University of Parma (Italy).
Seonwoo Kim (KIAS, Seoul)
Metastable Hierarchy in Abstract Low-Temperature Lattice Models
The phenomenon of metastability, and especially its hierarchical decomposition, is ubiquitous in a large class of dynamical systems, both real-life and theoretic, with two or more locally stable states. In this talk, I will briefly review this phenomenon in the setting of abstract lattice models at low temperatures. I will also talk about a few examples which include Glauber and Kawasaki dynamics for Ising/Potts models. The talk is partially based on a joint work with Insuk Seo (SNU).
Angeliki Menegaki (Imperial College London)
Alessia Nota (University of L'Aquila)
Recent advances on the Smoluchowski coagulation equation under non-equilibrium conditions
The Smoluchowski’s coagulation equation is an integro-differential equation of kinetic type which provides a mean-field description for mass aggregation phenomena. In this talk I will present some recent results on the problem of existence or non-existence of stationary solutions, both for single and multi-component systems, under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. The most striking feature of these stationary solutions is that, whenever they exist, the solutions to multi-component systems exhibit an unusual “spontaneous localization” phenomenon: they localize along a line in the composition space as the total size of the particles increase. This localization is a universal property of multicomponent systems and it has also been recently proved to occur in time dependent solutions to mass conserving coagulation equations.
Matteo Quattropani (University of Tor Vergata, Rome)
Cutoff at the entropic time for Repeated Block Averages
We consider a discrete-time randomized dynamics over the n-simplex, where at each step a random set, or block, of coordinates is evenly averaged. It is not hard to check that in the long run the process will eventually converge to the uniform distribution. In the talk I will discuss the speed of such a convergence as a function of the distribution of the block size, providing sharp conditions for the emergence of the cutoff phenomenon, i.e., a dynamical phase transition in which the equilibrium state is reached abruptly on a given time scale. In particular, I will show how this process frames within the realm of process exhibiting cutoff at the so-called “entropic time”, i.e., the location of the cutoff time can be interpreted as the first time at which the entropy of the system becomes of order log(n). The talk is based on a joint work with P. Caputo (Roma) and Federico Sau (Trieste).
Marielle Simon (GSSI, University of Lyon)
A few scaling limits results for the facilitated exclusion process
I will present some recent results which have been obtained for the facilitated exclusion process, in one dimension. This stochastic lattice gas is subject to strong kinetic constraints which create a continuous phase transition to an absorbing state at a critical value of the particle density. If the microscopic dynamics is symmetric, its macroscopic behavior, under periodic boundary conditions and diffusive time scaling, is ruled by a non-linear PDE belonging to free boundary problems (or Stefan problems). One of the ingredients is to show that the system typically reaches an ergodic component in subdiffusive time. When the particle system is put in contact with reservoirs of particles (which can either destroy or inject particles at both boundaries), we observe an usual impact on the boundary values of the empirical density. Based on joint works with O. Blondel, H. Da Cunha, C. Erignoux, M. Sasada and L. Zhao.
Sergio Simonella (Sapienza University of Rome)
Foundations of kinetic theory: recent progress and open directions.
We consider deterministic, time-reversible dynamics with random initial data, in a low-density scaling. Under suitable assumptions on the initial measure, a strong chaos property is propagated in time, which also encodes the transition to irreversibility. This result is complemented by a theory of fluctuations, allowing to establish the connection between microscopic and hydrodynamic scales, for perturbations of a global equilibrium. Many of the open problems left require a deeper understanding of the coupling mechanisms between deterministic and stochastic dynamics.
Funded by the European Union.
Gunter Schütz (IST Lisbon, Jülich)
Exclusion processes with long range correlations
We discuss some exclusion processes with nearest-neighbor jumps that occur naturally in a variety of contexts, ranging from random matrix theory to molecular motors in biological systems, and prove that they exhibit long-range correlations in their stationary distribution. This is in contrast to the conventional expectation that one-dimensional interacting particle systems exhibit short-range correlations even when there are long-range interactions in the sense that particle jumps are not limited to a finite range. In two examples we find phase separation and using reverse duality on microscopic level demonstrates a breakdown of the usual large-scale hydrodynamic description based on the law of large numbers and local equilibrium. In a third example computations using free-fermion methods suggest that the particle current is a non-local functional given by the Hilbert transform of the local density and correlations in the invariant measure have a large-scale description in terms of conformal field theory. Joint work with V. Belitsky, J. Dubail, N.P.N. Ngoc and A. Zahra.
Romina Travaglini (INdAM & University of Parma)
A mathematical model for bacterial strains on a leaf surface
We present a mathematical model applied to bacterial aggregation on leaf surfaces, a phenomenon influenced by spatial heterogeneity in water and nutrient availability, and interactions among bacterial populations. These interactions, which may be cooperative or competitive, can result in diverse spatial patterns such as co-aggregation or segrega- tion. By leveraging insights from the kinetic theory of active particles and classical macroscopic models, we derive a reaction-diffusion system for two interacting bacterial populations on a leaf surface. We perform numerical simulations that contribute to the understanding of microbial spatial organization and pattern formation.
The slides of the talks will be uploaded here