• Optimal Transportation
• Workshop: Analytic approaches to scaling limits for random system

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We construct a stochastic model showing the relationship between noise, gradient flows and rate-independent systems. The model consists of a one-dimensional birth-death process on a lattice, with rates derived from Kramers’ law as an approximation of a Brownian motion on a wiggly energy landscape. Taking various limits we show how to obtain a whole family of generalized gradient flows, ranging from quadratic to rate-independent ones, connected via ‘L log L’ gradient flows. This is achieved via Mosco-convergence of the renormalized large-deviations rate functional of the stochastic process.

Filled rubber is an important material for a wide variety of everyday
applications. In the late 1960s, it was noted that filled rubbers
exhibit stress-strain hysteresis, a phenomenon which is now termed the
Mullins effect after one of its discoverers. In this talk, I will
present a possible discrete microscopic model to explain this
phenomenon, and discuss ongoing work with Tony Lelievre and Frederic
Legoll in which we seek to coarse-grain the dynamics in space leading to
a continuum model.

Dawson and Gaertner (1987) showed that the path of the empirical process of n independent identically distributed diffusion processes satisfy a large deviation principle in n with a rate function expressed as the integral of a 'Lagrangian' cost function. In this talk, I will generalise their result to a class of Feller processes on locally compact spaces.

Entropy techniques are powerful methods to study the longtime behaviour of solutions to nonlinear PDEs. They rely on logarithmic Sobolev inequalities. They appear to be particularly well suited for a class of nonlinear PDEs, namely Fokker Planck equations associated to nonlinear stochastic differential equations in the sens of McKean.

I will present a few examples where such techniques prove to be useful: (i) micro-macro models for polymeric fluids, (ii) adaptive biasing force techniques for molecular dynamics and (iii) optimal scaling for high-dimensional Metropolis-Hastings algorithms.

I will present the discrete-to-continuum limit of a non-locally interacting particle system where the unknowns are the positions of the particles on the one dimensional half line. This particle system arises as a model for understanding plasticity of crystals (the particles represent dislocation walls).

Interestingly, the solution to the particle system exhibits a boundary layer which is not recovered by the continuous solution (i.e. the density prole of the particles as predicted by the upscaled problem). I will show how to characterize this boundary layer in a variational framework by using Γ-convergence.

Joint work with C. Le Bris and F. Legoll (École des Ponts, INRIA).

In this work, we introduce a variance reduction approach for the homogenization of a random, linear elliptic second order partial differential equation set on a bounded domain in Rd. The random diffusion coefficient matrix field A(x/ε,ω) is assumed to be uniformly elliptic, bounded and stationary (“periodic in law”). In the limit when ε → ∞, the solution of the equation converges to that of a homogenized problem of the same form, the coefficient field of which is a deterministic and constant matrix A* given by an average involving the so-called corrector function that solves a random auxiliary problem set on the entire space.

In practice, the corrector problem is approximated on a bounded domain Q_N as large as possible. A by-product of this truncation procedure is that the deterministic matrix A* is approximated by a random, apparent homogenized matrix A*_N(ω). We therefore introduce a variance reduction approach to obtain practical approximations of A* with a smaller variance in order to reduce the statistical error. We derive conditions (e.g. exact fraction in a bi-composite) for the selections of finite supercell environments on which we solve the corrector equation.

In this talk I will present some results concerning the dynamics of a system of screw dislocations. Screw dislocations move according to a maximal dissipation criterion, which leads to a differential inclusion. I will show how a suitable theoretical setup will lead to existence and right uniqueness for the equations of motion.

Markov Chain Monte Carlo (MCMC) is a standard methodology for sampling from probability distributions (known up to the normalization constant) in high dimensions.

There are (infinitely) many different Markov chains/diffusion processes that can be used to sample from a given distribution. To reduce the computational complexity, it is necessary to consider Markov chains that converge as quickly as possible to the target distribution and that have a small asymptotic variance. In this talk I will present some recent results on accelerating convergence to equilibrium and on reducing the asymptotic variance for a class of Langevin-based MCMC algorithms.

Due to a series of works, of which many authors will be present at this workshop, the connection between large deviations and gradient flows for the limit equation is well-understood. More particularly, for many systems it is shown that the large deviations of a given system of stochastic processes induce a specific gradient structure on the deterministic limit equation. In this talk I consider the problem in the other direction: given a gradient structure for an evolution, can we construct a system of stochastic processes such that their large deviations induce that gradient structure? And if so, does such construction relate to a particle system? For linear gradient structures, at least formally, the construction is not that involved, but can nevertheless be insightful. For doubly nonlinear structures the matter remains a challenging open problem.

We consider a one-dimensional diagonal hyperbolic system with cumulative distribution functions as initial data. We construct a multitype system of sticky particles and show that the vector of empirical cumulative distribution functions of this particle system provides weak solutions to the hyperbolic system under very weak assumptions.

We then derive uniform stability estimates on the particle system that turn into Wasserstein stability estimates on our solutions, which extends previous results by Bolley, Brenier and Loeper for scalar conservation laws. As a byproduct, we obtain that our solutions are semigroup solutions in the sense of Bianchini and Bressan.

This is a joint work with Benjamin Jourdain (CERMICS, École des Ponts ParisTech).

Coarse-graining is the procedure of approximating a system by a simpler or lower-dimensional one. This is typically achieved by passage to the limit of some parameter in the original system. Variational-evolution structures have been widely used in the case of dissipative (gradient-flow) systems to achieve this. In this talk, which is work with M. Hong Duong, Agnes Lamacz and Mark A. Peletier, I will introduce and apply a variational structure arising from the theory of large deviations to derive the overdamped limit of the Vlasov-Fokker-Planck (VFP) equation. This structure allows us to handle certain evolutionary systems that include non-dissipative effects of which the VFP equation is an example.

Condensing zero range processes are interacting particle systems with zero range interaction exhibiting phase separation at densities above a finite critical density. We prove the hydrodynamic limit of mean zero condensing zero range processes with bounded local jump rate for sub-critical initial profiles, i.e: for initial profiles that are everywhere strictly below the critical density. The proof is based on H.T. Yau's relative entropy method and is made possible by a generalisation of the one block estimate.

A closed hydrodynamic equation for such zero range processes when starting from a general initial profile is not known. We can obtain though a non-closed hydrodynamic description by proving relative compactness results for the processes induced by two important but not conserved quantities at the microscopic level, the empirical diffusion rate σ^N and the empirical current W^N. Denoting by μ^N the empirical density, all limit points of the sequence of laws of the triple (μ^N;σ^N;W^N), N∈ℕ, are concentrated on trajectories (μ;σ;W) satisfying the continuity equation ∂_t μ = Δ σ_t = -div W_t in the sense of distributions. Finally we give some regularity results on the limiting triples (μ;σ;W).

I present a one-dimensional model on a lattice, whose dynamics is governed by coupled differential equations plus random nearest neighbor exchanges. The model has exactly two locally conserved fields. I first derive the system of conservation laws governing the average behavior in the hydrodynamic limit. I then discuss the mode-coupling equations obtained with nonlinear fluctuating hydrodynamics, and illustrate with numerical simulations the predicted peak structure of the steady state space-time correlations (traveling peak with KPZ scaling function and standing peak with a scaling function given by the completely asymmetric Levy distribution with parameter 5/3).

We study effective descriptions for the motion of a particle moving in a bounded periodic potential, as governed by Newton's second law. In particular we seek an effective equation, describing the motion of the particle in a rapidly oscillating potential. The study of such problems was initiated by P. L. Lions, G. Papanicolaou and S. R. S. Varadhan (1987) through the homogenisation of the Hamilton-Jacobi PDE. We propose an alternative approach, that is to use the principle of Maupertuis (1744). In this alternative formulation, which characterises dynamical trajectories as geodesics in a Riemannian manifold, we are able to describe properties of effective Hamiltonians in certain cases; specifically, when the potential energy function takes values in {1,a} for a sufficiently large.

In this talk, we will present quickly some results of the Freidlin-Wentzell theory, then we will give a Kramers' type law satisfied by the McKean-Vlasov diffusion when the confining potential is uniformly strictly convex. We will briefly present two proofs of the result before giving a third one which is simpler, more intuitive and less technical.

Notes

We study the (deterministic) Cahn-Hilliard energy on the torus in a regime in which the system size is large and the mean value is close to -1. We show how the Gamma-limit can be used to establish the existence of a "droplet-shaped" local minimizer and a "droplet-shaped" saddle point in between this minimizer and the uniform state. The sharp isoperimetric inequality gives a quantitative bound on the Fraenkel asymmetry of both critical points. This is joint work with Michael Gelantalis and Alfred Wagner.

We study the totally asymmetric exclusion process on the positive integers with a single particle source at the origin. Liggett (1975) has shown that the long term behaviour of this process has a phase transition: If the particle production rate at the source is below a critical value, the stationary measure is a product measure, otherwise the stationary measure is spatially correlated. Following the approach of Derrida et al. (1993) it was shown by Großkinsky (2004) that these correlations can be described by means of a matrix product representation. In this paper we derive a large deviation principle with explicit rate function for the particle density in a macroscopic box based on this representation. The novel and rigorous technique we develop for this problem combines spectral theoretical and combinatorial ideas and has the potential to be applicable to other models described by matrix products.
This is joint work with Horacio González Duhart and Peter Mörters.