MINI COURSES
Sabine Jansen - LMU München
Duality and intertwining for interacting particles and measure-valued processes
Duality of Markov processes helps in mapping complicated processes to simpler processes. Complicated processes could be birth-death processes, processes with reflecting barriers, systems with infinitely many particles or measure-valued processes; their easier counterparts are pure death processes, absorbing barriers, systems with few or finitely many particles. The tools are well developed in interacting particle systems on lattices a la Liggett and for superprocesses for example in mathematical population genetics. In the minicourse I will give a general introduction to duality and then present some results how to reframe known dualities on the lattice to particles in R^d (joint works with Floreani, Redig, and Wagner). This brings in infinite-dimensional orthogonal polynomials, chaos decompositions, and Fock spaces and factorizable representations of infinite-dimensional Lie groups (current groups).
Sergio Simonella – Università di Roma La Sapienza
Kinetic Limits of Dilute Classical Gases
We review the state of the art in the mathematical justification of fluid equations derived from the fundamental laws of classical mechanics. With current techniques, such problems can be addressed in certain simplified settings, where kinetic theory provides a crucial intermediate step. In particular, we consider deterministic dynamics with random initial data in a low-density scaling. Under suitable assumptions on the initial measure, a well-known strong chaos property is propagated in time, capturing the emergence of irreversibility from reversible microscopic dynamics. Recent work has established the link between microscopic and hydrodynamic scales through an improved ansatz for the propagation of chaos. We illustrate this strategy with the aid of toy models.
[Co-funded by the European Union (ERC CoG KiLiM, PRIN 202277WX43).]
TALKS
José Canizo (Granada)
Long-time diffusive behaviour of some linear kinetic equations
We consider linear kinetic equations of the form $\partial_t f + \frac{1}{\epsilon} v \nabla_x f = \frac{1}{\epsilon^2} L(f)$, for an unknown $f$ which depends on time $t$, position $x$ and velocity $v$, and where $L$ is a linear operator which acts only in the velocity variable, and which typically has a probability equilibrium in $v$. Important examples include the Fokker-Planck operator, nonlocal diffusion operators, linear BGK-type operators, or linear Boltzmann operators. This PDE typically represents a mesoscopic physical model, where we keep track of the probability distribution of the position and velocity of particles. It is well known that when $\epsilon$ tends to $0$, this type of equation has a macroscopic or diffusive limit for the density $\rho(t,x) := \int f(t,x,v) dv$, which is either the standard heat equation, or the fractional heat equation. As a new result, we show that for a fixed epsilon, the behavior of this equation for large times also follows the standard or fractional heat equation, and that the long-time and small-epsilon limits are actually interchangeable in many cases. This is a work in collaboration with Stéphane Mischler (U. Paris-Dauphine) and Niccolò Tassi (U. Granada)
Ana Djurdjevac (Berlin)
Surface Dean-Kawasaki equation
We begin by deriving the classical (flat) Dean–Kawasaki equation and discussing its interpretation, challenges, and issues related to well-posedness and approximation. We then extend the analysis to derive the Dean–Kawasaki equation on surfaces, incorporating both particle interactions and coupling between particles and the underlying geometry. Finally, we outline possible strategies for numerical discretization in this setting. This is joint work with J. Bell (Lawrence Berkley National Lab) and N.Perkowski (FU Berlin).
Clement Erignoux (Lyon)
Hydrostatic and hydrodynamic behavior of SSEP with non-reversible boundary dynamics (JW with Claudio Landim and Tiecheng Xu)
The behavior of the SSEP in contact with reservoirs has been under deep scrutiny in the context of the Macroscopic Fluctuations Theory, as an workable example of non-trivial non equilibrium behavior. Its hydrostatic and hydrodynamic behavior (Law of large numbers), in particular, is easily derived, and one can then look at the associated CLT fluctuations and large deviations associated with it. When the boundary dynamics, however, is not reversible w.r.t. the (product) grand-canonical states for the SSEP, the pictures becomes less clear. In this talk, I will present results on the behavior of the SSEP affected by non-reversible boundary dynamics, some in which a deterministic hydrodynamic limit holds, and some where it does not. I will also briefly present some related w.i.p. with Milton Jara and Beatriz Salvador on the same topic.
Rouven Frassek (Unimore)
Steady states for harmonic processes
We introduce the harmonic process with reservoirs along with its asymmetric and multispecies generalisation. These models are exactly solvable. We discuss representations of the steady state and show how they are related.
Davide Gabrielli (L'Aquila)
TBA
Cristian Giardinà (Unimore)
Large Deviations Around the Fluid Limit
In the hydrodynamic limit, the macroscopic behavior of an interacting particle system on a lattice is described by a deterministic PDE on a continuous domain. This PDE represents the law of large numbers for the empirical density field, and, in a celebrated result [1], Kipnis, Olla, and Varadhan established a large deviation principle (LDP) for the symmetric exclusion process.
In this talk, we turn to a fluid limit setting, where the lattice structure is preserved but both time and the state space are scaled so that the limiting dynamics are governed by a system of ODEs. This scaling can be naturally encoded as a large-spin limit of Markov processes arising from quantum spin chains. After establishing a law of large numbers in this regime, we formulate an LDP around the fluid limit for a broad class of models, encompassing both integrable and non-integrable spin chains. We prove the LDP explicitly for the partial exclusion process (a non-integrable model) on a finite lattice [2]. For the harmonic process (an integrable model) with a step initial condition on the infinite lattice, we compute the cumulant generating function of the current using the inverse scattering method [3].
[1] C. Kipnis, S. Olla, S.R.S. Varadhan, Hydrodynamics and Large Deviations for Simple Exclusion Processes, Comm. Pure Appl. Math. 42 (1989), 115–137.
[2] C. Giardinà, T. Sasamoto, Large spin and large deviations for interacting particle systems [in preparation].
[3] C. Giardinà, K. Mallick, T. Sasamoto, H. Suda, Exact solution of discrete macroscopic fluctuation theory for an integrable spin system [in preparation].
François Golse (Ecole polytechnique)
Numerical Analysis of Quantum Dynamics in the Semiclassical Regime
Milton Jara (IMPA)
Scaling Limit of a population dynamics with age-dependent fitness
Consider the following dynamics. We start with a semi-infinite binary sequence $(...,X_{-1},X_0)$, and we recursively define a process in the following way. At each step, we draw a geometric random variable $Z$ (starting at 1) and we copy what we see $Z$ steps back in the sequence. If we interpret $X_i$ as the trait of an individual that was born at time $i$, the probability of passing ahead this trait (their fitness) decays exponentially with the age of the individual. We show that scaling limits of a generalization of this model are governed by a measure-valued Fleming-Viot process. Joint work with Eldon Barros and Eric Santana (IMPA).
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)
Convergence of dynamical stationary fluctuations
I will present a general black box theorem that ensures convergence of a sequence of stationary Markov processes, provided a few assumptions are satisfied. I will illustrate it on a model of reflected interfaces. This is based on a joint work with Benoît Laslier, Fabio Toninelli and Lorenzo Zambotti.
Bertrand Lods (UniTo)
Non saturation gap for Fokker-Planck-Fermi-Dirac equation
We consider the Fokker-Planck equation for fermions ( Fermi-Dirac-Fokker-Planck equation) which is a correction of the classical Fokker-Planck model that includes the effects of the Planck's exclusion principle. The equation reads
$$\partial_t f(t,v)=\nabla_v \cdot \left(\nabla_v f + v f(1-f)\right)\,.$$
The talk is aimed to present a novel approach yielding the appearance of a non-saturation gap in finite time, i.e. there is $T >0$ and such that solution $f(t,v)$ to the FDFP equation satisfies
$$\inf_v \left(1-f(t,v)\right)=: \kappa_0 >0, \qquad \forall t > T\,.$$
This results in exponential convergence to equilibrium by classical entropy/entropy dissipation method. The approach combines three ingredients: (1) stability of solutions, (2) quantitative convergence of a dense initial data set via a robust entropy theory, and (3) a De Giorgi iteration level argument.
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)
Phase transition in collective dynamics
Certain models of collective dynamics exhibit deceptively simple patterns that are surprisingly difficult to explain. These patterns often arise from phase transitions within the underlying dynamics—transitions that become evident only when one derives continuum equations from the corresponding individual-based models. In this talk, I will explore this subtle yet rich phenomenon and discuss its implications for understanding emergent collective behavior.
Stefano Olla (Paris, GSSI)
Diffusive behaviour in extended completely integrable dynamics
On a diffusive space-time scaling, density fluctuations behave very differently in extended completely integrable systems with respect to chaotic systems. I will expose some recent results concerning the one dimensional hard rods infinite dynamics and the box-ball cellular automata (an ultradiscretization of the KdV equation). Joint works with Pablo Ferrari, Makiko Sasada, Hayate Suda.
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)
Theory of a Multi-component Cosmological Plasma Involving Gravitational and Dark Energy Density Interactions
A multi-component cosmological plasma and its physics are introduced that involve gravitational and dark energy density interactions, following recent and extensive observations on the "Closer Universe". The work is in collaboration with B.Coppi.
Ellen Saada (Université Paris Cité, CNRS)
The multilane exclusion process: hydrodynamics
We consider the simple exclusion process on $\mathbb{Z}\times\{0,1,\cdots,n-1\}$, that is, a ``horizontal ladder'' composed of $n$ lanes. Particles can jump according to a lane-dependent translation-invariant nearest neighbour jump kernel, i.e. ``horizontally'' along each lane, and ``vertically'' along the scales of the ladder. After recalling some results on invariant measures for the dynamics,
we investigate the hydrodynamic behavior and local equilibrium of this model. The dynamics on each lane follows a hyperbolic time scaling, whereas the interlane dynamics has an arbitrary time scaling. We prove that the hydrodynamic behavior of the global density (i.e. summed over all lanes) is governed by a scalar conservation law; the latter, as well as the limit of individual lanes, is the relaxation limit of a weakly coupled hyperbolic system that approximates the particle system. To highlight new phenomena arising beyond the one-dimensional exclusion dynamics, we present the transitions between different possible shapes of the flux function of the two-lane model.
Joint works with Gidi Amir, Christophe Bahadoran, Ofer Busani.
Federico Sau (UniMi)
Versions of Aldous' spectral gap conjecture for stochastic exchange models
In this talk, we present a simple strategy to derive bounds for the spectral gap of a class of continuous-spin stochastic exchange models, which include the Kipnis-Marchioro-Presutti model, the harmonic process, and the immediate exchange model. These bounds are universal, as they hold on arbitrary graphs. Moreover, for some models and in some log-concave regimes, these are sharp, yielding a version of Aldous’ spectral gap conjecture, i.e., the gap is attained on linear functions. Failures and alternatives to this conjecture will also be discussed. Based on the preprint arXiv:2505.02400 (joint with S. Kim and M. Quattropani).