MINI COURSES
Sabine Jansen - LMU München
TBA
Sergio Simonella – Università di Roma La Sapienza
TBA
TALKS
Mario Ayala Valenzuela (TUMunich)
TBA
José Canizo (Granada)
TBA
Clement Erignoux (Lyon)
TBA
Rouven Frassek (Unimore)
TBA
Davide Gabrielli (L'Aquila)
TBA
Cristian Giardinà (Unimore)
TBA
François Golse (Ecole polytechnique)
TBA
Milton Jara (IMPA)
TBA
Wolfgang Koenig (WIAS)
Spatial particle processes with coagulation: Gibbs-measure approach, gelation, and Smoluchowski equation
We study the spatial Marcus–Lushnikov process: according to a coagulation kernel K, particle pairs merge into a single particle, and their masses are united. We derive an explicit formula for the empirical process of the particle configuration at a given fixed time T in terms of a reference Poisson point process, whose points are trajectories that coagulate into one particle by time T. The non-coagulation between any two of them induces an exponential pair-interaction. Then we first give a large-deviation principle for the joint distribution of the particle histories, in the limit as the number of initial atoms diverges and the kernel scales as K/N. We analyse the minimiser(s) of the rate function and prove a law of large numbers. Furthermore, we give a criterion for the occurrence of a gelation phase transition. This is joint work with Luisa Andreis, Heide Langhammer and Robert Patterson.
Cyril Labbé (Université Paris Cité)
TBA
Bertrand Lods (UniTo)
TBA
Elena Magnanini (WIAS)
Recent results on gelation-type phase transitions in spatial coagulation process
We discuss recent progress on a stochastic coagulation particle system (Marcus-Lushnikov process) in which pairs of particles merge after independent exponential waiting times. This model represents the particle-level counterpart of Smoluchowski’s classical coagulation equation (1917). Specifically, our focus is on a spatial generalization where each particle is characterized by both a mass and a spatial position. After each coagulation event, governed by a positive symmetric kernel, the resulting particle has a mass equal to the sum of the merging masses, and a new spatial location. Depending on the choice of the coagulation kernel, the system may undergo a phase transition known as gelation, corresponding to the emergence, in finite time, of large particles (gels) with macroscopic mass. We establish (1) two sufficient conditions for gelation, one of which settles, as a byproduct, an open conjecture in the non-spatial setting, and (2) a law of large numbers showing convergence of the rescaled process to the solution of a spatial Flory equation. We conclude by outlining some open problems and ongoing work in this framework.
This talk is based on joint works with L. Andreis, and T. Iyer.
Karsten Matthies (Univeristy of Bath)
From a Rayleigh Gas to Boltzmann and Fractional Diffusion Equations
The fractional diffusion equation is rigorously derived as a scaling limit from a deterministic Rayleigh gas, where particles interact via short range potentials with support of size r and the background is distributed in space according to a Poisson process with intensity N and in velocity according to some fat-tailed distribution. As an intermediate step a linear Boltzmann equation is obtained in the Boltzmann-Grad limit as r tends to zero and N tends to infinity with N r^2 =c. The convergence of the empiric particle dynamics to the Boltzmann-type dynamics is shown using semigroup methods to describe probability measures on collision trees associated to physical trajectories in the case of a Rayleigh gas. The fractional diffusion equation is a hydrodynamic limit for times [0,T], where T and inverse mean free path c can both be chosen as some negative rational power r^{-k}. Base on joint work with Theodora Syntaka.
Sara Merino Aceituno (Vienna)
TBA
Stefano Olla (Paris, GSSI)
TBA
Frank Redig (TU Delft)
Gaussian concentration for lattice spin systems
We consider the connection between transportation cost inequalities and Gaussian concentration for lattice spin systems in the thermodynamic limit. More precisely we consider equivalences of the following three statements
μ satisfies a Gaussian concentration inequality for all finite volumes (the so-called ℓq-Gaussian concentration bound).
μ satisfies an inequality between an integral probability metric and the square root of the relative entropy for all finite volumes. We call these inequalities “relative entropy distance inequalities”.
μ satisfies an inequality between a “coupling metric” (a generalized Kantorovich functional) and the square root of the relative entropy for all finite volumes.
In cases where the relative entropy density is well-defined – such as for Gibbs measures associated with absolutely summable, translation-invariant potentials – we show that the thermodynamic limit yields an inequality between the ``d-bar-metric’’ and the square root of the relative entropy density. This gives a uniqueness result for translation invariant Gibbs measures.
Based joint work with Jean Rene Chazottes and Pierre Collet.
Valeria Ricci (UniPa)
TBA
Ellen Saada (Université Paris Cité, CNRS)
TBA
Federico Sau (UniMi)
TBA