Critical behaviour in spatial particle systems

The workshop is designed as an in-person meeting without additional streaming. There is an upper bound of 30 participants and preference will be given to (the invitees and) members of the SPP2265.

Alessia Nota: 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. Several results have been obtained during the last couple of decades, increasing the mathematical understanding of the Smoluchowski equation, such as well-posedness theory, self-similar behaviour, gelation, as well as some progress on its derivation from particle systems. More recently the analysis of coagulation equations has evolved towards generalizations, including other variables, such as multicomponent equations, or/and other terms, such as additional source or sink terms which naturally appears when modelling open systems. In this mini-course I will report on some recent developments 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 is a universal property of multicomponent systems. I will then discuss the emergence of localization, both in stationary and in time dependent solutions to mass conserving coagulation equations, and some conjectures and open problems in this direction.

Sergio Simonella: Hard-sphere gases: deterministic dynamics with random initial data
We will review the state of the art in the validity problem for a mathematical justification of fluid equations based on fundamental laws of classical mechanics. With the techniques currently available, such problems can be faced in some simple cases, using kinetic theory as an intermediate step. In particular, we will study 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 large deviation estimates and by a theory of small fluctuations, allowing to establish the connection between microscopic and hydrodynamic scales. Many of the open problems left require a deeper understanding of the coupling mechanisms between deterministic and stochastic dynamics.

Luisa Andreis:

Miguel Escobedo: Power laws in ``soft Boltzmann'' equations via Landau currents.
We first present some properties of the Landau mass current, in particular its uniqueness under some gauge condition, and prove that it is zero for the Maxwellians. Then we prove the existence of power laws stationary source solutions of the Boltzmann equation for some soft potentials. We show that on these power laws, the Landau mass current is not zero and there is a mass flux from small to large values of the energies. Finally I will discuss stability properties of the power laws with respect to small radial perturbation in terms of the linerization of the Boltzmann equation around them. (Joint work with F. Golse and L. Saint Raymond)

Marina Ferreira: Smoluchowski coagulation equation with a flux of dust particles
We construct a time-dependent solution to the Smoluchowski coagulation equation with a constant flux of dust particles entering through the boundary at zero. The dust is instantaneously converted into particles and these flux solutions have linearly increasing mass. The construction is made for a general class of non-gelling coagulation kernels for which stationary non-equilibrium solutions, so-called constant flux solutions, exist. The proof relies on several limiting procedures on a family of solutions of equations with sources supported on ever smaller sizes. Flux solutions are expected to converge to a constant flux solution in the large time limit. We show that this is indeed true in the particular case of the constant kernel with zero initial data. (Based on a joint work with Aleksis Vuoksenmaa - U. Helsinki)

Davide Gabrielli: Stationary non-equilibrium states: old and new results
Steady states are the simplest yet interesting situation in which typical non-equilibrium phenomena are observed. I will illustrate a series of results for non-equilibrium stationary states of boundary driven interacting stochastic particle systems. First I will recall some old exact results for specific one-dimensional models such as the simple exclusion and the KMP model, then I will illustrate some recent results on probabilistic and combinatorial representations of stationary states that allow a clearer perspective.

Daniel Heydecker: Bilinear coagulation via random raphs
We consider the interaction clusters of Kac’s model of a dilute gas, where the spatial effects are mimicked by velocity-dependent interaction rates. For a toy model, the coagulation structure can be completely described by random graphs. Using this correspondence, we show that there is a phase transition at a finite gelation time $t_{\rm gel}$, when a single interaction cluster of size comparable to the whole system appears. Our general analysis shows that, except in edge cases, the gelation time is strictly smaller than the mean free time $t_{\rm mf}$.

Wolfgang König: Spatial particle processes with coagulation: Gibbs-measure approach, gelation, and Smoluchowski equation
We study the spatial Marcus–Lushnikov process: according to a coagula- tion 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 par- ticle 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 expo- nential pair-interaction. Then we first give a large-deviation principle for the joint distri- bution of the particle histories, in the limit as the number of initial atoms diverges and the ker- nel 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.

Jani Lukkarinen: Estimation of generation and propagation of chaos via cumulant hierarchies
Propagation and generation of chaos is an important ingredient for rigorous control of applicability of kinetic theory, in general. Chaos is here understood as sufficient statistical independence of random variables related to the "kinetic" observables of the system. Cumulant hierarchy of these random variables thus often gives a way of controlling the evolution and degree of such independence, i.e., the degree of chaos in the system. In this talk, we will consider two, qualitatively different, example cases for which kinetic theory is believed to be applicable: the stochastic Kac model with random velocity exchange and the discrete nonlinear Schrodinger evolution (DNLS) with suitable random, spatially homogeneous initial data. In both cases, we set up suitable random variables and propose methods to control the evolution of their cumulant hierarchies. The talk is based on joint works with Sakari Pirnes and Aleksis Vuoksenmaa, and earlier works with Matteo Marcozzi, Alessia Nota, and Herbert Spohn.

Alexander Zass: Infinite-dimensional diffusions and depletion interaction for a model of colloids
In this talk, we consider a dynamical version of the Asakura--Oosawa model of interacting hard spheres of two different sizes. We study their random diffusion dynamics, modelled with collision local times; describe the reversible measures; and observe the emergence of an attractive short-range depletion interaction between the large spheres. We also study the Gibbs measures associated to this new interaction, exploring connections to percolation and optimal packing. This is joint work with Myriam Fradon.