**WORKSHOP ORGANIZERS: P. DEGOND, S. KALLIADASIS, G.A. PAVLIOTIS**

**WE ACKNOWLEDGE THE GENEROUS SUPPORT OF THE CNRS-Imperial “Abraham de Moivre” International Research Laboratory, the Quantitative Sciences Research Institute and of the Department of Mathematics at Imperial College.**

**PROGRAMME, TITLES AND ABSTRACTS**

**Tuesday 04 July**

09:15 – 09:30 Welcome Remarks

09:30 – 10:15 **A. GUILLIN **Propagation of chaos for some singular models

10:15 – 11:00 **M. DELGADINO** On the mean field approximation of the Gibbs measure for weakly interacting diffusions.

11:00 – 11:30 Coffee break

11:30 – 12:15 **Ch. SAFFIRIO **Weakly interacting fermions: mean-field and semiclassical regimes.

12:15 – 14:00 Lunch Break

14:00 – 14:45 **J. TUGAUT **Captivity of the solution to the granular media equation

14:45 – 15:30 **A. MENEGAKI **Quantitative framework for hydrodynamic limits

15:30 – 16:00 Coffee Break

16:00 – 16:45 **J. EVANS **Existence and stability of non-equilbrium steady states in a BGK model coupled to thermostats

**Wednesday 05 July**

09:30 – 10:15 **A. EBERLE **Sticky nonlinear SDEs and mean-field systems without confinement

10:15 – 11:00 **R. GVALANI **Logarithmic Sobolev inequalities and equilibrium fluctuations for weakly interacting diffusions

11:00 – 11:30 Coffee break

11:30 – 12:15 **H. DUONG **Asymptotic analysis for the generalized Langevin equation with singular potentials

12:15 – 14:00 Lunch Break

14:00 – 14:45 **K. SPILIOPOULOS **Normalization effects and mean field theory for deep neural networks

14:45 – 15:30** J. REYGNER **** **Dynamical Gibbs principle and associated stochastic control problems

15:30 – 16:00 Coffee Break

16:00 – 16:45 **M. OTTOBRE **McKean Vlasov (S)PDEs

**Thursday 06 July**

09:30 – 10:15 **P. Del MORAL** Some theoretical aspects of Particle Filters and Ensemble Kalman Filters

10:15 – 11:00 **A. SCHLICHTING **Mean-field PDEs on graphs and their continuum limit

11:00 – 11:30 Coffee break

11:30 – 12:15 **N. ZAGLI** Response Theory and Critical Phenomena for Weakly Interacting Diffusions

12:15 – 14:00 Lunch Break

**ABSTRACTS**

**P. Del Moral**

Title: Some theoretical aspects of Particle Filters and Ensemble Kalman Filters.

Abstract: In the last three decades, Particle Filters (PF) and Ensemble Kalman Filters (EnKF) have become one of the main numerical techniques in data assimilation, Bayesian statistical inference and nonlinear filtering. Both particle algorithms can be viewed as mean field type particle interpretation of the filtering equation and the Kalman recursion. In contrast with conventional particle filters, the EnKF is defined by a system of particles evolving as the signal in some state space with an interaction function that depends on the sample covariance matrices of the system. Despite widespread usage, little is known about the mathematical foundations of EnKF. Most of the literature on EnKF amounts to design different classes of useable observer-type particle methods. To design any type of consistent and meaningful filter, it is crucial to understand their mathematical foundations and their learning/tracking capabilities. This talk discusses some theoretical aspects of these numerical techniques. We present some recent advances on the stability properties of these filters. We also initiate a comparison between these particle samplers and discuss some open research questions.

**M. DELGADINO**

Title: On the mean field approximation of the Gibbs measure for weakly interacting diffusions.

Abstract: In this talk we will explore when the invariant Gibbs measure of an N -particle system of weakly interacting diffusion is well approximated by the unique minimiser of the mean field energy. We will compute the exact limit of the associated partition function, by re-interpreting the associated integral as the fluctuation of sampling from the mean field limit measure. With this technique we can obtain sharp estimates all the way up to the phase transition, colloquially defined as the loss of analyticity of the partition function with respect to the inverse temperature.

**H. DUONG**

Title: Asymptotic analysis for the generalized Langevin equation with singular potentials

Abstract: We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori-Zwanzig approach, we represent the system by a class of Markovian dynamics. Under a general set of conditions on the nonlinearities, we study the large-time asymptotics of the multi-particle Markovian GLEs. We show that the system is always exponentially attractive toward the unique invariant Gibbs probability measure. The proof relies on a novel construction of Lyapunov functions. We then establish the validity of the small mass approximation for the solutions by an appropriate equation on any finite-time window. Important examples of singular potentials in our results include the Lennard-Jones and Coulomb functions.

** **This talk is based on a joint work with D. H. Nguyen (UCLA).

**A EBERLE**

TITLE: Sticky nonlinear SDEs and mean-field systems without confinement

ABSTRACT: We consider a class of one-dimensional nonlinear stochastic differential equations with a sticky boundary at zero. It can be shown that there is a phase transition: For small nonlinearities, all mass eventually gets stuck at zero, whereas for larger nonlinearities, there is a nontrivial invariant measure. The solution of the sticky nonlinear SDE provides an upper bound for the distance process between two solutions of a McKean-Vlasov equation without confinement, coupled by a sticky coupling. As a consequence, it can be used to prove exponential convergence to equilibrium for this equation with a sufficiently small nonlinearity. Moreover, uniform in time propagation of chaos for the corresponding mean-field particle system is obtained by an extension of these ideas based on a componentwise sticky coupling (joint work with Alain Durmus, Arnaud Guillin and Katharina Schuh).

**J. EVANS**

Title: Existence and stability of non-equilbrium steady states in a BGK model coupled to thermostats

** **

**A GUILLIN**

Title: Propagation of chaos for some singular models

**R. GVALANI**

Title: Logarithmic Sobolev inequalities and equilibrium fluctuations for weakly interacting diffusions

** **Abstract: We study the mean field limit of interacting diffusions for confining and interaction potentials that are non-convex. The limiting behaviour is described by the nonlocal McKean–Vlasov PDE. We explore the relationship between the limit $N\to\infty$ of the constant in the logarithmic Sobolev inequality (LSI) for the $N$-particle system and the presence or absence phase transitions for the mean field limit, conjecturing a limiting form of the LSI constant. We also explore the consequences of the non-degeneracy of the LSI constant as they relate to uniform-in-time propagation of chaos and equilibrium fluctuations. Our results extend previous results on unbounded spin systems as well as recent results on (uniform-in-time) propagation of chaos using novel coupling arguments. Joint work with Matías Delgadino, Greg Pavliotis, and Scott Smith.

**A. MENEGAKI**

** **Title: Quantitative framework for hydrodynamic limits

** **Abstract: We will present a new quantitative approach to the problem of proving hydrodynamic limits from microscopic stochastic particle systems, namely the zero-range and the Ginzburg-Landau process with Kawasaki dynamics, to macroscopic partial differential equations. Our method combines a modulated Wasserstein distance estimate comparing the law of the stochastic process to the local Gibbs measure, together with stability estimates a la Kruzhkov in weak distance and consistency estimates exploiting the regularity of the limit solution. It is simplified as it avoids the use of the block estimates. This is a joint work with Clément Mouhot and Daniel Marahrens.

** **

**M. OTTOBRE**

** **Title: McKean Vlasov (S)PDEs

Abstract. We consider McKean-Vlasov Stochastic Partial Differential Equations with additive noise; we will focus on well-posedness and long time behaviour, as well as on obtaining these SPDEs from interacting particle systems.

This is based on joint work with L. Angeli, J. Barre’, D. Crisan and M. Kolodziejkzyc.

**J. REYGNER**

** **Dynamical Gibbs principle and associated stochastic control problems

** **In statistical physics, the Gibbs principle describes the asymptotic marginal distribution of a particle when the whole system is conditioned on a large deviation of its empirical measure. We consider a similar question for the case of diffusion processes, and show that the effect of conditioning results in an additional drift which encodes a mean-field like interaction. We also discuss the interpretation of this statement in terms of a stochastic control problem, with a constraint in distribution. This is a joint work with Louis-Pierre Chaintron (École Normale Supérieure) and Giovanni Conforti (École polytechnique).

**CH. SAFFIRIO**

** **TITLE: Weakly interacting fermions: mean-field and semiclassical regimes.

ABSTRACT

The derivation of effective macroscopic theories approximating microscopic systems of interacting particles is a major question in non-equilibrium statistical mechanics. In this talk we will be concerned with the dynamics of systems made of many interacting fermions in the case in which the interaction is singular (e.g. inverse power law). We will focus on the mean-field regime and obtain a reduced description given by the time-dependent Hartree-Fock equation. As a second step we will look at longer time scales where a semiclassical description starts to be relevant and approximate the many-body dynamics with the Vlasov equation, which describes the evolution of the effective probability density of particles on the one particle phase space.

**A SCHLICHTING **

TITLE: Mean-field PDEs on graphs and their continuum limit

Abstract:

This talk reviews some recent results on nonlocal PDEs describing the evolution of a density on graph structures. These equations can arise

from mean-field interacting jump dynamics, but also from applications in the data science field, or they can also be obtained by a numerical

discretization of a continuum problem. We also show how those equations are linked to their continuous counterpart in suitable local limits.

joint work with Antonio Esposito, Georg Heinze, Anastasiia Hraivoronska and Oliver Tse

**K. SPILIOPOULOS**

** **Title: Normalization effects and mean field theory for deep neural networks

** **We study the effect of normalization on the layers of deep neural networks. A given layer $i$ with $N_{i}$ hidden units is allowed to be normalized by $1/N_{i}^{\gamma_{i}}$ with $\gamma_{i}\in[1/2,1]$ and we study the effect of the choice of the $\gamma_{i}$ on the statistical behavior of the neural network’s output (such as variance) as well as on the test accuracy on the MNIST and CIFAR10 data sets. We find that in terms of variance of the neural network’s output and test accuracy the best choice is to choose the $\gamma_{i}$’s to be equal to one, which is the mean-field scaling. We also find that this is particularly true for the outer layer, in that the neural network’s behavior is more sensitive in the scaling of the outer layer as opposed to the scaling of the inner layers. The mechanism for the mathematical analysis is an asymptotic expansion for the neural network’s output and corresponding mean field analysis. An important practical consequence of the analysis is that it provides a systematic and mathematically informed way to choose the learning rate hyperparameters. Such a choice guarantees that the neural network behaves in a statistically robust way as the $N_i$ grow to infinity.

**J. TUGAUT**

TITLE: Captivity of the solution to the granular media equation

ABSTRACT: In this talk, we first recall some facts about the long-time behaviour of the solution to the granular media equation under convexity assumptions. Then, we deal with the non-convex case. In particular, we talk about the phase transition about the number of invariant probabilities. The main part of the talk concerns the characterization of the limiting probability and the whole proof is given in a simple case.

**N. ZAGLI**

Title: Response Theory and Critical Phenomena for Weakly Interacting Diffusions

Abstract:

In this talk, I will present our latest results on the close link between response theory, the existence of collective variables and the development of critical phenomena for noisy systems with mean field interactions.

Such systems are routinely used to model collective emergent behaviours in multiple areas of social and natural sciences as they exhibit, in the thermodynamic limit, continuous and discontinuous phase transitions.

I will show that the perspective given by Response Theory, that aims at establishing a link between natural and forced variability of general physical systems, is particularly useful to understand the physical mechanisms establishing critical phenomena in interacting systems.

Firstly, I will show how to define a set of collective variables for the system starting from the coupling structure among the microscopic agents.

Secondly, I will show that such variables prove to be proper nonequilibrium thermodynamic observables as they carry information on correlation, memory properties and resilience properties of the system.

In particular, the investigation of response properties of such collective variables allows to identify and characterise phase transitions of the system as they manifest as singular values of the susceptibility associated to such thermodynamic variables.

Numerical experiments will be presented to show how the critical pattern of the response arise and can be detected in finite systems.