Ginzburg-Landau (GL) dynamics are popular interacting particle systems. Macroscopic fluctuations theory (MFT) is now considered as the cornerstone of non-equilibrium statistical mechanics for diffusive systems. In this talk I will consider GL dynamics with long range interactions so that the system is superdiffusive and hydrodynamic limits are given by (non-linear) fractional diffusion equations. I will discuss issues concerning the establishment of a MFT for these GL dynamics. Joint work with R. Chetrite, P. Gonçalves and M. Jara.

Imagine three gamblers with respectively $A$, $B$, $C$ at the start. Each time, a pair of gamblers are chosen (uniformly at random) and a fair coin is flipped. Of course, eventually, one of the gamblers is eliminated and the game continues with the remaining two until one winds up with all $A+B+C$. In poker tournaments (really) it is of interest to know the chances of the six possible elimination orders (e.g. $3,1,2$ means gambler $3$ is eliminated first, then gambler 1, leaving 2 with all the cash). In particular, how do these depend on $A,B,C$? For small $A,B,C$, exact computation is possible, but for $A,B,C$ of practical interest, asymptotics are needed. The math involved is surprising; Whitney and John domains, Carlesson estimates. To test your intuition, recall that if there are two gamblers with $1$ and $N-1$ the chance that the first winds up with all $N$ is $1/N$. With three gamblers with $1,1$ and $N-2$ the chance that the third is eliminated first is $\frac{\operatorname{Const}}{N^3}$. We don't know the answer for four gamblers. This is a report of joint work with Stew Ethier, Kelsey Huston-Edwards and Laurent Saloff-Coste.

We present results concerning the qualitative and quantitative description of interacting systems with particular emphasis on those possessing a phase transition under the change of relevant system parameters. For this, we first discuss and identify different kinds of phase transitions (continuous and discontinuous) for mean-field limits of interacting particle systems on continuous and discrete state spaces. Since phase transitions are intimately related to the metastability of the stochastic particle system, we show how a suitable mountain pass theorem in the space of probability measures can describe the metastable behaviour of the underlying finite particle system. We also show that the particle system close to a discontinuous phase transition shows coarsening where smaller clusters, which grow through coagulation events. We provide numerical experiments that those phenomena can be also observed by a SPDE of Dean-Kawasaki type consisting of the McKean-Vlasov equation with some suitable conservative noise. Furthermore, we propose a simplified description of the growth process, which might give insights into the relevant time-scales. joint works with José Carrillo, Rishabh Gvalani and Greg Pavliotis.

In this talk, we study mixing times and the so-called cutoff phenomenon, which was introduced by Diaconis and Aldous in the study of a quantitative convergence to equilibrium for card-shuffling Markov models. We will estimate the mixing times and show cutoff phenomenon for the ergodic Langevin dynamics with a strongly coercive potential and driven by an additive noise with small amplitude. More precisely, when the driven noise is the Brownian motion, the total variation distance between the current state and its equilibrium decays around the mixing time from one to zero abruptly. When the noise is the $\alpha$-stable with index $\alpha \gt 3/2$, cutoff phenomenon still holds while for $\alpha\leq 3/2$ our coupling techniques do not apply, and therefore we cannot conclude if the cutoff phenomenon still holds. In the case of degenerate potential, cutoff phenomenon does not hold, however precise estimates for the mixing times can be obtained. The talk is based on series of papers with Milton Jara (IMPA, Brazil), Michael Högele (Universidad de los Andes, Colombia), Juan Carlos Pardo (CIMAT, Mexico) and Conrado da Costa (Durham University, UK).

The Dyson Brownian motion (DMB) is a system of infinitely many interacting Brownian motions with logarithmic interaction potential, which was introduced by Freeman Dyson '62 in relation to the random matrix theory. In this talk, we reveal that an infinite-dimensional differential structure induced by the DBM has a Bakry-Émery lower Ricci curvature bound. As an application, we show that the DBM can be realised as the unique Wasserstein-type gradient flow of the Boltzmann-Shannon entropy associated with $\operatorname{Sine}_\beta$ ensemble.

We obtain the hydrodynamic limit of symmetric long-jumps exclusion in $\mathbb{Z}^{d}$ (for $d≥1$), where the jump rate is inversely proportional to a power of the jump's length with exponent $\gamma+1$, where $\gamma≥2$. Moreover, movements between $\mathbb{Z}^{d-1} \times \mathbb{Z}_{-}^{*}$ and $\mathbb{Z}^{d-1} \times \mathbb{N}$ are slowed down by a factor $\alpha n^{-\beta}$ (with $α>0$ and $β≥0$). In the hydrodynamic limit we obtain the heat equation in $\mathbb{R}^{d}$ without boundary conditions or with Neumann boundary conditions, depending on the values of $β$ and $γ$. The (rather restrictive) condition in previous works (for $d=1$) about the initial distribution satisfying an entropy bound with respect to a Bernoulli product measure with constant parameter is weakened or completely dropped.

We present a method for simulating the stochastic relativistic advection-diffusion equation using the Metropolis algorithm. This approach simulates dissipative dynamics by randomly transferring charge between fluid cells, combined with ideal hydrodynamic time steps. Charge transfers are accepted or rejected based on entropy as a statistical weight in a Metropolis step. This reproduces expected dissipative strains in relativistic hydrodynamics within a specific hydrodynamic frame known as the density-frame. Numerical results, with and without noise, are compared to relativistic kinetics and analytical expectations. Notably, unlike other numerical approaches, this method is strictly first order in gradients and lacks non-hydrodynamic modes. The simplicity and convergence properties of the Metropolis algorithm make it promising for simulating stochastic relativistic fluids in heavy ion collisions and critical phenomena.

Reference: https://arxiv.org/abs/2403.04185

Severals SPDEs arise from SDE dynamics under partial conditioning of the noise. My talk will circulate on three concrete examples, the Zakai equation from non-linear filtering, the pathwise control problem suggested by Lions-Sougandis, and last not least a rough PDE approach to pricing in non-Markovian stochastic volatility models. Underlying all these examples is the notion of rough stochastic differential equations, recently introduced by K. Lê, A. Hocquet and the speaker.

Many natural systems can be maintained in a stationary state through the exchange of matter, energy or information with their surroundings. These various currents break time-reversal invariance, generating a continuous increase of entropy in the universe. Such systems are out of equilibrium and can not be described by the Laws of Thermodynamics, or by using the classical principles of statistical physics, à la Gibbs-Boltzmann. In the last decades, however, important advances in our understanding of non-equilibrium processes have been achieved. Concepts of rares events, large deviations, fluctuations relations and macroscopic fluctuations provide a unified framework with the emergence of some universal features. The objective of this talk is to review these new ideas in non-equilibrium statistical physics and to illustrate them by examples inspired from soft-condensed matter.

The asymmetric simple exclusion process is a model of interacting particles that appears in many realistic descriptions of low-dimensional transport with constraints and plays the role of a paradigm to understand the behaviour of non-equilibrium systems. The aim of this talk is to review some representative exact results about this model, to describe the methods involved and present some recent developments. In particular, by using the mathematical arsenal of integrable probabilities developed to solve the one-dimensional Kardar-Parisi-Zhang equation, we shall derive the exact finite-time distribution of a tagged particle in the symmetric simple exclusion process.

Gianni Jona-Lasinio and his collaborators have proposed in the early 2000’s a non-linear action functional that encodes the macroscopic fluctuations and the large deviations for a wide class of diffusive systems out of equilibrium, by generalizing a variational principle due to Kipnis, Olla and Varadhan. This theory, called the Macroscopic Fluctuation Theory (MFT) shows that large deviations far from equilibrium can be found by solving two coupled non-linear hydrodynamic equations. In this talk, we shall show that the MFT equations for the symmetric exclusion process are classically integrable and can be solved with the help of the inverse scattering method, originally developed to study solitons in dispersive non-linear wave equations (such as KdV or NLS).

We will discuss a weak universality phenomenon in the context of two-dimensional fractional nonlinear wave equations. For a sequence of Hamiltonians of high-degree potentials scaling to the fractional $Φ_2^4$, we will present a sufficient and almost necessary criteria for the convergence of invariant measures to the fractional $Φ_2^4$. Then we will discuss the convergence result for the sequence of associated wave dynamics to the (renormalized) cubic wave equation. This extends a result of Gubinelli-Koch-Oh to a situation where we do not have any local Cauchy theory with highly supercritical nonlinearities. This is a joint work with Chenmin Sun and Weijun Xu.

In this talk we consider the polymer measure with selfinteractions, which is called the Edward model. The two-dimensional case had been studied around 2000, and the polymer measure and the associated Dirichlet form were constructed. Here, we consider the three-dimensional case, which requires harder calculation than the two-dimensional case. It is known that, to construct the measure we need the renormalization, because of the singularity of the interactions. Moreover, the renormalization constant coincides with that of the $\Phi ^4$-quantum field measure. In this talk, I will explain the strategies to construct the measure and the Dirichlet form.

This talk is based on jointworks with Sergio Albeverio, Makoto Nakashima and Song Liang.

We derive the general structure for returning to the steady macroscopic nonequilibrium condition, assuming a first-order relaxation equation obtained as zero-cost flow for the Lagrangian governing the dynamical fluctuations. The main ingredient is local detailed balance from which a canonical form of the time-symmetric fluctuation contribution (aka frenesy) can be obtained. That determines the macroscopic evolution. As a consequence, the linear response around stationary nonequilibrium gets connected with the small dynamical fluctuations, leading to fluctuation-response relations. Those results may be viewed as nonequilibrium extension of the well-known structure where the relaxation to equilibrium is characterized by a (dissipative) gradient flow on top of a Hamiltonian motion.

In this talk, I am going to present a generalization of the Porous Media Model (PMM) analogue of the Bernstein polynomial basis, in the context of gradient models. The PMM is a symmetric nearest-neighbour process associated with the Porous Media Equation, and the corresponding dynamics are kinetically constrained, in the sense that particles diffuse in the lattice under a set of conditions on local configurations. While in the PMM, the occupation values of two neighbouring sites are exchanged only if there are "enough" groups of particles around them, our generalized model describes a system where, at very low densities, there is no interaction, while at high densities, there is no space for movement. I am going to present the construction of the model and its main properties, and, if time permits, discuss its extension in a long-range interaction context.

We present a random interface model on the one-dimensional torus of size $N$ with a weak perturbation, i.e. an asymmetry $\sim N^{-\gamma}$ of the direction of growth that switches from up to down based on the sign of the area underneath. The evolution of the interface can be studied in terms of the density field of an underlying, non-Markovian exclusion process. We compute the order of the correlation functions of this process for the invariant measure of the interface model, and investigate the stationary fluctuations of the density field: we establish the convergence to an Ornstein-Uhlenbeck equation for $\gamma>\frac{8}{9}$, and discuss the limit for $\frac{1}{2}\leq \gamma<\frac{8}{9}$. Based on joint work with Martin Hairer and Patrícia Gonçalves.

Reaction-diffusion equations arise naturally when modelling multi-component systems of interacting populations. These equations are widely employed to describe pattern formation phenomena across various biological, chemical and physical processes. The kinetic theory of statical mechanics provides a powerful framework to describe different types of interactions at multiple spatial or temporal scales. Through appropriate hydrodynamic limits of the kinetic systems, macroscopic equations can be derived, describing observable quantities and explaining how macroscopic phenomena emerge from the underlying microscopic dynamics. In this talk, I will apply these tools to study the evolution and interactions of competing bacterial populations on a leaf surface. Specifically, I will consider self and cross diffusion effects and investigate Turing instability properties leading to the formation and persistence of stationary spatial patterns.

This work is a collaboration with D. Cusseddu (University of Minho), M. Bisi and R. Travaglini (University of Parma, Italy).

In this talk we consider a system of repelling small and large (hard) spheres in a continuous medium. This could describe colloidal particles (large spheres) within a substrate (small spheres). Alternatively, it could provide an idealized picture of what may happen in a phase transition when the new phase is getting formed (large spheres) but we still have small isolated particles (small spheres). One interesting phenomenon is that despite the repulsive forces between all particles, when we look at the effective system of only big spheres, an attractive force emerges between them which is usually referred to as "depletion attraction". The question we address in this talk is how to compute the free energy of the system, in particular for the renormalized one, i.e., when we first integrate over the small spheres. We will discuss a sufficient condition for the convergence of the related cluster expansion, which involves the surface of the large spheres rather than their volume (as it would have been the case in a direct application of existing methods to the binary system). This is based on joint works with Sabine Jansen, Giuseppe Scola and Xuan Nguyen.