# Stochastic Dynamics Out of Equilibrium

3 April 2017 to 7 July 2017
Europe/Paris timezone

## General seminar

Cristian Giardina (Tuesday, July 4th) [slides]
An introduction to the inclusion process and its scaling limits

Among recently introduced interacting particle systems, the inclusion process is a cognate of the well-know exclusion process in which the exclusion rule is replaced by an attractive interaction among particles. Due to its algebraic properties the inclusion process enjoys duality properties that allow several exact computations. We shall focus on the regime where the inclusion parameter (tuning the spreading of the particles) is scaled to zero and a condensation effect then occurs. We present the results of the work [1] where it is shown that  the dynamics of the inclusion process has up to three relevant time scales. If time permits we also consider the diffusive scaling limit of two inclusion particles in the condensation regime, yielding sticky Brownian motion for the distance between the two particles. This is work in progress with G. Carinci and F. Redig.

[1] A. Bianchi, S. Dommers, C. Giardina', "Metastability in the reversible inclusion process", arXiv:1605.05140

Gioia Carinici (Tuesday, July 4th)
Density fluctuations for the inclusion process

In this talk I will consider the weakly asymmetric inclusion process under diffusive time scaling. I will discuss the behavior of the density fluctuation field and prove the convergence to the unique energy solution of the stochastic Burgers equation. The central point is the derivation of a quantitative version of the non-linear Boltzmann-Gibbs principle, that is proven via duality (joint work with M. Jara, F. Redig).

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Balint Toth (Tuesday, June 27th)
CLT for Lorentz gas beyond Boltzmann-Grad

I will present a coupling argument leading to CLT for the Lorentz trajectory among Poissonian scatterers, partially interpolating between the well understood Boltzmann-Grad limit and the fully open diffusive scaling. The talk  will be based on work in progress joint with Chris Lutsko (Bristol).

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Raffaele Esposito (Tuesday, June 20th)
Stationary solutions to the Boltzmann equation in the hydrodynamic limit

Stationary solutions to the Boltzmann equation are not covered by the Di Perna Lions theory because of the lack of a priori estimates. Very little is known about  in a general geometry. Building on a method we recently developed for the construction of stationary solution with diffuse reflection boundary conditions (thermal walls) in a bounded domain with arbitrary shape, we study the problem in the small Knudsen number limit (hydrodynamic limit). The main technical ingredient in our analysis is a new higher regularity estimate for the macroscopic part of the solution to the steady transport equation. Then we apply it to construct the solution to the steady Boltzmann equation both in a bounded domain and in the complement of a bounded domain (flux around an obstacle). In both cases we prove the convergence to the incompressible Navier-Stokes-Fourier system, as the Knudsen number goes to zero. In the case of bounded domain we also show the exponential dynamical stability of the stationary solution. In the unbounded case extra space decay properties are needed which require a very accurate analysis of the conservation laws.

Davide Gabrielli (Tuesday, June 20th)
Gibbsian stationary non equilibrium states

We study the structure of stationary non equilibrium states for interacting particle systems from a microscopic viewpoint. In particular we discuss two different discrete geometric constructions. We apply both of them to determine non reversible transition rates corresponding to a fixed invariant measure. The first one uses the equivalence of this problem with the construction of divergence free flows on the transition graph. Since divergence free flows are characterized by cyclic decompositions we can generate families of models from elementary cycles on the configuration space. The second construction is a functional discrete Hodge decomposition for translational covariant discrete vector fields. According to this, for example, the instantaneous current of any interacting particle system on a finite torus can be canonically decomposed in a gradient part, a circulation term and an harmonic component. All the three components are associated with functions on the configuration space. This decomposition is unique and constructive. The stationary condition can be interpreted as an orthogonality condition with respect to an harmonic discrete vector field and we use this decomposition to construct models having a fixed invariant measure.

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Stefan Grosskinsky (Thursday, June 1st)
Derivation of mean-field equations for stochastic particle systems

We study the single site dynamics in stochastic particle systems of misanthrope type with bounded rates in a permutation invariant setting on a complete graph. In the limit of diverging system size we establish convergence to a Markovian non-linear birth death chain, described by a mean-field equation known also from the field of exchange-driven growth processes. Conservation of mass in the particle system leads to conservation of the first moment for the limit dynamics, and to non-uniqueness of stationary measures. The proof is based on a coupling to branching processes via the graphical construction, and establishing uniqueness of the solution for the limit dynamics. As particularly interesting examples we discuss the dynamics of two models that exhibit a condensation transition and their connection to exchange-driven growth processes. Joint work with Watthanan Jatuviriyapornchai.

Ines Armendariz (Thursday, June 1st)
Existence of the zero-range process with superlinear growth rates

We use coupling arguments to construct the zero-range dynamics with superlinear, non-decreasing transition rates, and develop some properties of this process. Joint work with E. Andjel and M. Jara.

Christophe Bahadoran (Thursday, June 1st) [slides]

Hydrodynamics with and without local equilibrium

We study the one-dimensional asymmetric nearest-neighbor general zero-range process with deterministic ergodic-like site disorder. Under a slow tail condition near the infimum site speed, a phase transittion occurs. We show that the hydrodynamic limit is giiven by entropy solutions of a scalar conservation law with constant flux function above a critical density. We prove creation and conservation of local equilibrium in strong (pointwise) sense at subcritical hydrodynamic densities, and dynamic escape of mass (loss of local equilibrium) at supercritical hydrodynamic densities. As a byproduct, we obtain convergence of the process from any given configuration with  asymptotic density to the left of the origin. Our approach relates creation of strong local equilibrium to genuine nonlinearity (but not necessarily convexity) of the flux, and yields new results for other (homogenous)  attractive models with product  invariant measures in one dimension. Joint work with T. Mountford, K. Ravishankar and E. Saada.

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Rossana Marra (Thursday, May 11th)
Models for competing interactions

I will discuss kinetic models for systems of particles interacting through potentials which are repulsive and attractive, and the corresponding possible kinds of phase transition.  In particular, I will present a kinetic model for the formation of microemulsion and discuss the microphase separation.

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Nils Berglund (Wednesday, May 3rd) [scanned notes]
Metastability in stochastic PDEs

Stochastic processes subject to weak noise often show a metastable behaviour, meaning that they converge to equilibrium extremely slowly; typically, the convergence time is exponentially large in the inverse of the variance of the noise (Arrhenius law).

In the case of finite-dimensional Ito stochastic differential equations, the large-deviation theory developed in the 1970s by Freidlin and Wentzell allows to prove such Arrhenius laws and compute their exponent. Sharper asymptotics for relaxation times, including the prefactor of the exponential term (Eyring-Kramers laws) are known, for instance, if the stochastic differential equation involves a gradient drift term and homogeneous noise. One approach that has been very successful in proving Eyring-Kramers laws, developed by Bovier, Eckhoff, Gayrard and Klein around 2005, relies on potential theory.

I will describe Eyring-Kramers laws for some parabolic stochastic PDEs such as the Allen-Cahn equation on the torus. In dimension 1, an Arrhenius law was obtained in the 1980s by Faris and Jona-Lasinio, using a large-deviation principle. The potential-theoretic approach allows us to compute the prefactor, which turns out to involve a Fredholm determinant. In dimensions 2 and 3, the equation needs to be renormalized, which turns the Fredholm determinant into a Carleman-Fredholm determinant.

Based on joint works with Barbara Gentz (Bielefeld), and with Giacomo Di Gesù (Vienna) and Hendrik Weber (Warwick).

Generalized arcsine law in fractional Brownian motion

The three arcsine laws for the standard Brownian motion are a cornerstone of extreme value statistics. For a standard Brownian motion evolving in a time window, one can consider the following three observables: (1) the fraction of time it remained positive, (2) the last time it crossed the origin, (3) and the time when it reached its maximum. All three observables have the same cumulative probability distribution expressed as an arcsine function. I shall discuss how these three laws change for a fractional Brownian motion. The fractional Brownian motion is a non-Markovian Gaussian process indexed by Hurst exponent H which generalizes Brownian motion (H=1/2). I shall show that the three observables have different distributions for general H. I shall present a perturbation expansion scheme using which one can derive these probability distributions.

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Baruch Meerson (Wednesday, April 19th) [slides]
Large deviations of surface height in the Kardar-Parisi-Zhang equation

The Kardar-Parisi-Zhang (KPZ) equation describes an important universality class of nonequilibrium stochastic growth models. There has been much recent interest in the complete one-point probability distribution P(H, t) of height H of the evolving interface at time t. I will show how one can use the optimal fluctuation method to evaluate P(H, t) for different initial conditions.

At short times, typical height fluctuations are Gaussian, but the distribution tails are non-Gaussian and highly asymmetric. One tail agrees with the proper asymptotic of the Tracy-Widom distribution (for the flat and curved interface), and of the Baik-Rains distribution (for the stationary interface), previously observed at long times. The other tail behaves differently than the corresponding tail of the TW and BR distributions. This tail also persists at long times.

The case of stationary initial condition is especially interesting. Here at short times the large deviation function of the height exhibits a singularity at a critical value of |H|. The singularity has the character of a second-order phase transition. It results from a symmetry breaking of the most probable history of the interface conditioned on reaching a given height at time t.

Antti Kupiainen (Wednesday, April 19th) [slides]
Renormalizing Rough Stochastic PDEs

I will give an itroduction to a renormalization group approach to construct soutions to nonlinear PDEs driven by a space time white noise.

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Shugo Yasuda (Tuesday, April 11th) [slides]
Monte Carlo method for kinetic chemotaxis model and its applications on traveling pulse and pattern formation

Collective motion of chemotactic bacteria as E. Coli relies, at the individual level, on a continuous reorientation by runs and tumbles, where the length of run is stochastically decided by a stiff response to temporal sensing of chemical cues along the trajectory. This stochastic behavior can be described by a kinetic transport equation, say, kinetic chemotaxis model. In this talk, a Monte Carlo method for run-and-tumble chemotactic bacteria based on the kinetic chemotaxis model and its applications on the traveling pulse and pattern formation in population density of bacteria are presented. Some theoretical results including the asymptotic analysis and instability analysis of the kinetic chemotaxis equation and their comparisons to the numerical results obtained by Monte Carlo simulations are also presented.

[1] S. Yasuda, “Monte Carlo simulation for kinetic chemotaxis model: An application to the traveling population wave”, J. Comput. Phys. 330, 1022–1042 (2017)
[2] B. Perthame and S. Yasuda, “Self-organized pattern formation of run-and-tumble chemotactic bacteria: Instability analysis of a kinetic chemotaxis model”, hal-01494963 (2017)

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