Invited Speakers: Titles and Abstracts
Marzia Bisi
Università di Parma, Italy
Kinetic modelling of knowledge and wealth dynamics in national and global markets
Abstract:
We investigate general kinetic models describing the simultaneous change of multiple microscopic states of the interacting agents. Specifically, we consider a multi-population system where agents are involved in international trades with possible transfers of individuals from a country to another, and trading interaction rules are influenced by the individual knowledge and vice versa. We start from a proper microscopic description of single-agent dynamics, and we build up the corresponding kinetic model. We also derive from the kinetic level the macroscopic equations governing each population density, mean knowledge and national wealth, and we study equilibrium values for suitable choices of microscopic interaction functions. Moreover, we perform a suitable quasi-invariant asymptotic limit of the Boltzmann model, leading to a system of simpler Fokker-Planck-type equations, whose equilibrium distributions are discussed for varying parameters.Andrea Bondesan
Università di Parma, Italy
Perturbation of hypocoercive kinetic operators for elastic and reactive multi-species interactions
Abstract:
In this talk I will present recent results on the spectral properties of linearized Boltzmann operators that are used to model interactions inside inert and reactive gaseous mixtures. In particular, I will show how the hypocoercive nature of the linearized operator for multi-species elastic collisions is preserved under suitable perturbations that include small non-equilibrium dynamics and chemical reactions. Ultimately, the main motivation behind these analyses consists in the rigorous derivation of certain hydrodynamic models for multicomponent flows (Maxwell-Stefan and Fick equations, reaction-diffusion systems).Emeric Bouin
Université Paris-Dauphine & École Normale Supérieure, France
About fractional diffusion limits for linear kinetic equations
Abstract:
In this presentation, I will present results from works with Laura Kanzler and Clément Mouhot on a unified derivation of (potentially) fractional diffusion limits for linear kinetic equations preserving between one and three macroscopic quantities and for which the microscopic equilibrium has an algebraic decay. The result is based on a construction of quasi-eigenmodes, based on the study of spectral projectors or on the extension of an Ellis and Pisnky method to a fractional framework (which is a result of independent interest).François Golse
Ecole Polytechnique, France
Landau Currents anf Kolmogorov-Zakharov Solutions of Boltzmann Type Equations
Abstract:
Abstract: In the 1960s, V.E. Zakharov derived formally a Boltzmann type equation to describe energy transfer in the context of various wave turbulence processes arising in plasma physics or in the theory of surface waves in fluid mechanics. Moreover Zakharov identified special power law solutions of his Boltzmann type equations, expected to describe a universal turbulence spectrum. By analogy with Kolmogorov 1941 theory of fluid turbulence, these solutions are called « Kolmogorov-Zakharov » solutions. Zakharov obtained these solutions by a series of clever, but formal and rather involved substitutions in the collision integral. The purpose of this talk is to offer a simpler, and more rigorous derivation of these solutions.Maria Groppi
Università di Parma, Italy
A mixed Boltzmann-BGK model for gas mixures: mathematical properties and hydrodynamic limits
Abstract:
In this talk we present a hybrid Boltzmann-BGK model for mixtures of monatomic gases, that combines the detailed description of collisions given by the Boltzmann integral operators with the simplicity and the numerical manageability of BGK-type relaxation operators. This kinetic model has the same structure of the full Boltzmann equations, with the collision term of each constituent given by a sum of bi-species operators, that may be chosen of Boltzmann or of BGK type. We prove consistency of the mixed model: conservation properties, positivity of all temperatures, H-theorem, and convergence to a global Maxwellian equilibrium with all species sharing a common mean velocity and a common temperature. The presence of a collision operator for any pair of gaseous components allows for a consistent derivation of evolution equations for the main macroscopic fields in different hydrodynamic regimes, according to the dominant collision process. Specifically, we investigate the classical collision dominated regime, a situation with dominant intra-species collisions, leading to multi-fields macroscopic description, and a mixture with heavy and light particles, leading to a kinetic-fluid description: in all of these frames a Chapman-Enskog procedure allows to obtain an explicit closure of macroscopic equations at Navier-Stokes level, with transport coefficients in agreement with physical expectations.Bertrand Lods
Università di Torino, Italy
Non saturation gap for Fokker-Planck-Fermi-Dirac equation
Abstract:
Alessia Nota
Gran Sasso Science Institute, Italy
Dynamics of Non-Equilibrium Systems: Homoenergetic Flows for the Boltzmann Equation
Abstract:
In this talk, I will consider a particular class of solutions to the Boltzmann equation, known as homoenergetic solutions, which were introduced by Galkin and Truesdell in the 1960s. These are a specific type of non-equilibrium solutions of the Boltzmann equation, useful for modeling the dynamics of Boltzmann gases subjected to mechanical deformations such as shear, expansion, or compression, thereby providing insights into the behavior of open systems. Given that these solutions describe far-from-equilibrium phenomena, their long-time asymptotics cannot always be described by Maxwellian distributions. I will present different possible long-time asymptotics for homoenergetic solutions to the Boltzmann equation, as well as to the Rayleigh-Boltzmann equation in the case of shear deformations. Additionally, I will discuss some conjectures and open problems in this direction.Sergio Simonella
Sapienza Università di Roma, Italy
Cluster Dynamics for the Boltzmann Equation
Abstract:
Cluster dynamics is the property of systems of infinitely many particles of being decomposed onto finite clusters which move independently during a random interval of time. In this talk, I will discuss how to obtain the cluster process for the Boltzmann equation. Beyond its intrinsic theoretical interest, particularly for the quantification of correlations and the analysis of fluctuations, this construction reveals a natural connection with kinetic models arising in the physics of polymers with internal structure.Cinzia Soresina
Università di Trento, Italy
Starvation-driven cell patterning: integrating lab experiments and mathematical modelling
Abstract:
We present a reaction–diffusion model describing the interactions among cells, nutrients, and growth factors, aimed at capturing the emergence of starvation-driven cell pattern formation, a phenomenon recently observed in laboratory experiments under nutrient-limited growth conditions. Experiments and modelling were developed in parallel, enabling progressively more targeted experimental design while enhancing the biological realism of the model. This interdisciplinary feedback loop led to the formulation of new hypotheses and enabled the estimation of several key parameters. Numerical simulations show that the model reproduces pattern formation in both one- and two-dimensional spatial domains. To provide theoretical support for these findings, we performed a Turing instability analysis to investigate the potential for diffusion-driven instability. The analysis indicates that the observed patterns are not driven by chemotaxis; rather, they arise naturally under starvation conditions and display structural similarities to the Klausmeier model for vegetation pattern formation in semi-arid environments, suggesting the robustness of the underlying mechanism across biological scales. Joint work with Anna Bernardi-D'Agostini (UniTn), Giovanni Cappello (UniGrenoble), Simone Pezzuto (UniTn)Mattia Zanella
Università di Pavia, Italy
Large-time behaviour for Lotka-Volterra-type kinetic equations
Abstract:
We study a recently introduced coupled system of kinetic equations describing the time evolution of dynamical densities. By employing energy-type distances, we rigorously establish exponential convergence to equilibrium in appropriate homogeneous Sobolev spaces, with a rate explicitly determined by the dissipative contribution of the interaction term. The analysis reveals the intrinsic energy-dissipation mechanism governing the dynamics and highlights how the evolution of expected quantities drives the emergence of stable equilibrium configurations. This framework provides a unified multiscale connection between classical Lotka-Volterra population models and kinetic equations.