# 2023 FHD Workshop - Abstracts

Sept. 19^{th} & 20^{th}, 2023

**List of Abstracts:**

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### Lorenzo Bertini

**Current fluctuations in stochastic lattice gases: Donsker-Varadhan meets Freidlin-Wentzell**

Abstract:

We discuss the large deviations asymptotic of the time-averaged empirical current in stochastic lattice gases in the limit in which both the number of particles and the time window diverges. For some models it has been shown that dynamical phase transitions occur: the optimal density profile to realize such deviations is given by travelling waves rather than by homogeneous profiles. We shall prove a variational representation, proposed by Varadhan, for the corresponding rate function that is obtained by projecting the large deviations at the level of the empirical process.

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### Julian Fischer

**A rigorous approach to the Dean-Kawasaki equation of fluctuating hydrodynamics**

Abstract:

Fluctuating hydrodynamics provides a framework for approximating density fluctuations in interacting particle systems by suitable SPDEs. The Dean-Kawasaki equation – a strongly singular SPDE – is perhaps the most basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of *N* diffusing weakly interacting particles in the regime of large particle numbers *N*. The strongly singular nature of the Dean-Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification: Besides being non-renormalizable by approaches like regularity structures, it has recently been shown to not even admit nontrivial martingale solutions.

In this talk, we give an overview of recent quantitative results for the justification of fluctuating hydrodynamics models. In particular, we give an interpretation of the Dean-Kawasaki equation as a "recipe" for accurate and efficient numerical simulations of the density fluctuations for weakly interacting diffusing particles, allowing for an error that is of arbitarily high order in the inverse particle number.

Based on joint works with Federico Cornalba, Jonas Ingmanns, and Claudia Raithel.

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### Benjamin Gess

**Non-equilibrium large deviations and parabolic-hyperbolic PDE with irregular drift**

Abstract:

Large deviations of conservative interacting particle systems, such as the zero range process, about their hydrodynamic limit lead to the analysis of the skeleton equation; a degenerate parabolic-hyperbolic PDE with irregular drift. In this talk, we present a robust well-posedness theory for such PDEs in energy-critical spaces based on concepts of renormalized solutions and the equation's kinetic form. The relevance of the results toward large deviations in interacting particle systems is demonstrated by applications to the identification of l.s.c. envelopes of restricted rate functions, and to their Gamma-convergence.

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### Rishabh Gvalani

**The thin-film equation with thermal noise**

Abstract:

In this talk, we will study the lubrication approximation of the well-known fluctuating hydrodynamics model introduced by Landau and Lifschitz. The corresponding system is a fourth-order, degenerate, quasilinear singular SPDE commonly referred to as the thin-film equation with thermal noise. We start by presenting an alternative derivation of the equation from thermodynamic principles using as inputs the correct invariant measure for the dynamics (the *1d* Gaussian free field restricted to positive functions) and the correct dissipation mechanism (a weighted version of the *H*^{-1 } inner product). Next, we propose a natural structure-preserving discretisation which preserves the strict positivity of the film height for large enough mobilities. Finally, we study the equation in the framework of the theory of regularity structures: we estimate (uniformly in the regularisation parameter) the appropriately renormalised centered model associated to the equation completing the first step in obtaining a solution theory for the equation.

This talk is based on two separate works with Benjamin Gess (MPI-MiS/Bielefeld), Florian Kunick (MPI-MiS), and Felix Otto (MPI-MiS) and with Markus Tempelmayr (Uni. Muenster), respectively.

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### Carsten Hartmann

**The cross-entropy method as an interacting particle system **

Abstract:

The cross entropy (CE) method has its origin in rare event simulation. It is based on an importance sampling algorithm for estimating rare event probabilities where the idea is to compute an approximation to an optimal change of measure within a parametric family by minimizing the cross entropy between a target distribution that would allow to estimate the rare event probability with minimum variance and the parametric family. The CE method can also be used to solve combinatorial and continuous optimisation problems by iteratively generating a change of measure that concentrates at the minimiser of the objective function. We will look at the CE method from the perspective of interacting particle systems and discuss the relation to consensus- or swarm-based optimisation methods.

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### Daniel Heydecker

**Large Deviations of Landau-Lifschitz-Navier-Stokes & Relationship to the Energy Equality **

Abstract:

The Landau-Lifschitz-Navier-Stokes equations are stochastic partial differential equations, which introduce a stochastic forcing term to describe the macroscopic fluctuations away from the deterministic Navier-Stokes equations. In dimension $*d=3*$, we consider the large deviations of a suitable solution theory in a scaling regime where the noise intensity and correlation length go to zero simultaneously, with a coupled rate. We show that the large deviations reproduce those of the lattice gas studied by Quastel and Yau, which justifies the Landau-Lifschitz-Navier-Stokes as an effective and numerically tractable model to capture deviations from the deterministic limit. We also relate the large deviations to the equality case of the classical energy inequality.

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### Helena Kremp

**A nonlinear SPDE approximation of the Dean-Kawasaki equation **

Abstract:

In this talk we consider a nonlinear SPDE approximation of the Dean-Kawasaki equation for independent particles. Our approximation satisfies the physical constraints of the particle system, i.e. its solution is a probability measure for all times (preservation of positivity and mass conservation). Using a duality argument, we prove that the weak error between particle system and nonlinear SPDE is of the order N^{{-1-1/(d/2+1)} log(N)}. Along the way we show well-posedness, a comparison principle and an entropy estimate for a class of nonlinear regularized Dean-Kawasaki equations with Itô noise.

This is joint work together with Ana Djurdjevac and Nicolas Perkowski.

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### Angeliki Menegaki

**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 (University of Cambridge) and Daniel Marahrens.

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### Fenna Müller

**The Dean-Kawasaki Equation with initial data in the positive tempered distributions**

Abstract:

Motivated from statistical physics, the Dean-Kawasaki Equation aims to describe the den- sity of a system of fluctuating particles. However, it was shown by Konarovskyi, Lehmann and von Renesse that in the space of probability measures the equation only admits solutions given by em- pirical measures. But what happens if we allow solutions and initial conditions with infinite mass, e.g. starting from the Lebesgue measure? In this talk we will show that even allowing infinite mass, the Dean-Kawasaki equation only admits solutions if its initial condition is a suitable multiple of an empirical measure.

Joint work with Vitalii Konarovskyi.

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### Grigorios Pavliotis

**On the Diffusive-Mean Field Limit for Weakly Interacting Diffusions Exhibiting Phase Transitions **

Abstract:

We study the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition. A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature.

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### Milica Tomasevic

**Propagation of chaos for stochastic particle systems with singular mean-field interaction of $***L*^{q }- L^{p}$ type

*L*$ type

^{q }- L^{p}Abstract:

In this work, we prove the well-posedness and propagation of chaos for a stochastic particle system in mean-field interaction under the assumption that the interacting kernel belongs to a suitable $ *L _{t}^{q }- L_{x}^{p }*$ space. Contrary to the large deviation principle approach recently proposed in the litterature (Hoeksama et al, to appeat in Annals IHP), the main ingredient of the proof here are the Partial Girsanov transformations introduced by (Jabir-Talay-Tomašević., ECP 2018) and developed in a general setting here.

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