SUMMARY

The main theme of the workshop is "Variational methods in probability theory and statistical physics", with focus on the application of large deviation theory to interacting particle systems, including gradient flows, polymers and disordered systems.

 

ORGANISERS

INVITED SPEAKERS

PROGRAMME 

Monday March 14

Tuesday March 15

Wednesday March 16

Thursday March 17

Friday March 18

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ABSTRACTS

Stefan Adams

Sample path large deviations and scaling limits for weakly pinned integrated random walks

We study scaling limits and the corresponding large deviation principle of the integrated random walk perturbed by an attractive force towards the origin. In particular we analyze the critical situation that the rate function admits more than one minimiser leading to concentration of measure problems. The integrated random represents interface models with Laplacian interaction.

Erwin Bolthausen

Localization properties of random surfaces with pinning

We present two recent results on random surfaces with local pinning The first one (joint work with T. Chyonobu and T. Funaki) on a classical gradient interface model at a critical parameter and the second one (jointworkd with A. Cipriani, N. Kurt) on the decay of correlations in a membrane model.
 

Anton Bovier/Patrick Müller

Hydrodynamic limits, propagation of chaos and large deviations in local mean-field models with unbounded spins

We consider systems of stochastic differential equations that describe the dynamics of unbounded spin-systems where spins interact via long-range spatially variable interactions. We prove the convergence
of certain empirical mesures under proper rescaling to solution of a pde and we show that propagation of chaos holds.
Finally, we derive a large deviation principle for on the space of paths of measures.

Paul Chleboun

Large deviations of the empirical current in zero-range processes on a ring

We examine atypical current fluctuations in totally asymmetric zero-range processes in one dimension with periodic boundary conditions. For large systems, by calculating the Jensen-Varadhan action functional, we are able to predict the time dependent optimal profiles which realise currents below the typical value. Under certain conditions on the jump rates, we demonstrate that these systems can exhibit a dynamical phase transition. Above a critical non-typically current the optimal macroscopic density profile is given by a traveling wave with a shock and anti-shock pair, while rare events below the critical current are realised by condensed configurations, in which a positive fraction of all the particles accumulate on a single site in the thermodynamic limit. Heuristics are supported by simulations using 's-ensemble' cloning methods.
(work in progress with Andrea Pizzoferrato and Stefan Grosskinsky)

Jin Feng

On the metric nature of some Hamilton-Jacobi equations for infinite particles

In this talk,  a class of Hamilton-Jacobi PDEs in the space of probability measures will be formulated as equations in length metric spaces. These equations arise from both statistical mechanics as well as continuum mechanics applications. We develop an abstract well-posedness theory by introducing a new notion of viscosity solution.
The "metric nature" in the title reflects an observation that the space of probability measures can be viewed as an infinite dimensional polyhedron, a singular space where a good choice of metric can capture the singularities. In such context, a metric formulation of differential calculus using local Lipschitz constant gives more precise subtle information than the usual smooth calculus. Two situations illustration the above claim better. The first one is based on a joint work with L Ambrosio. We use the geometric tangent cone concept, instead of the usual locally linear tangent space structure, in the Wasserstein space setting to handle mass condensation property of some Hamilton-Jacobi equations.  In second situation, we make the observation that the local Lipschitz constant is metric dependent. With a change of base metric, the notion of derivative may change, hence re-normalizing the Hamilton-Jacobi PDEs. Also relying upon a few other techniques from the weak KAM theory in Lagrangian dynamics, we develop an abstract well-posedness theory for a Hamilton-Jacobi PDE modeling infinite particles with singular attractive pair-wise potentials. 

Davide Gabrielli

Discrete and continuous variational problems for current fluctuations

I will first discuss current fluctuations for interacting particle systems in the hydrodynamic scaling limit and the related variational principles. In particular I will discuss the additivity principle, the dynamic approach and the existence of dynamic  phase transitions.
Then I will discuss a joint large deviations principle for the empirical measure and flow for a continuous time Markov chain. I will discuss how to use it to study current fluctuations for a system of particles and the corresponding discrete variational problems.
I will discuss  solvable cases, in particular the boundary driven zero range model for which both the continuous and the discrete variational principles can be solved.

Giambattista Giacomin

Pinning and disorder relevance for the lattice Gaussian free field in dimension three or more

I will present recent results on the localization transition for a Gaussian free field on Z_d , d≥3, interacting with a quenched disordered substrate that acts on the interface when the interface height is close to zero. The substrate has the tendency to localize or repel the interface at different sites and one can show that a localization-delocalization transition takes place when varying the average pinning potential
h: the free energy density is zero in the delocalized regime, that is for h smaller than a threshold hc, and it is positive for h>hc. We compute hc and we show that the transition happens at the same value as for the annealed model. However we can show that the critical behavior of the quenched model differs from the one of the annealed one. While the phase transition of the annealed model is of first order, we show that the quenched free energy is bounded above by (h−hc)2+ times a positive constant and that, for Gaussian disorder, the quadratic behavior is sharp. Therefore this provides an example in which a "relevant disorder critical exponent" can be made explicit.

Stefan Grosskinsky

Metastability in condensing zero-range processes

We consider a zero-range process with jump rates decreasing with the occupation number, which is known to exhibit a condensation phenomenon where a finite fraction of all particles concentrates on a single lattice site. We derive a scaling limit for the asymptotic stationary dynamics of the condensate location in the thermodynamic limit on a one-dimensional torus. Our proof follows previously developed methods using potential theory and a martingale approach, which have been applied to zero-range processes on finite lattices. The main challenge and novelty of our paper arises from the absence of attractor states which complicates the proof of equilibration within metastable wells and requires a coupling argument to get uniform bounds on exit rates from wells.
(joint work with Ines Armendariz and Michalis Loulakis)

Sabine Jansen

Surface energy and transfer operators for a chain of atoms at low temperature

We analyze the surface Gibbs free energy and the transfer operator for a chain at atoms in the limit where the temperature goes to zero. The interaction potential is of Lennard-Jones type. Our main results are:
(1) At zero temperature, the surface energy is the minimizer of a well-behaved functional on a space of square summable sequences and it is characterized by a Bellman equations, a fixed point equation from discrete optimal control theory.
(2) At small positive temperature, the spectral gap of the transfer operator stays bounded away from zero.
(3) To leading order, the asymptotics of the principal eigenfunction is described by the unique solution of the Bellman equation. As an application we prove path large deviations principles for the Gibbs measures for the infinite and semi-infinite chains and prove exponential decay of correlations, uniformly in the temperature.
In additions we establish a relation between the decay of correlations at small positive temperature and the decay of boundary layers at zero temperature.
(joint work in progress with Wolfgang König, Bernd Schmidt and Florian Theil)

Wolfgang König

A variational formula for the free energy of an interacting many-body system

We consider N bosons in a box in the d-dimensional space in a large box under the influence of a mutually repellent pair potential. The particle density is kept fixed and positive. Our main result is the identification of the limiting free energy at positive, sufficiently high temperature in terms of an explicit variational formula. The thermodynamic equilibrium is described by the symmetrised trace of the negative exponential of the corresponding Hamilton operator. The well-known Feynman-Kac formula reformulates this trace in terms of N interacting Brownian bridges. Due to the symmetrisation, the bridges are organised in an ensemble of cycles of various lengths. The novelty of our approach is a description in terms of a marked Poisson point process whose marks are the cycles. This allows for an asymptotic analysis of the system via a large-deviation analysis of the stationary empirical field. The resulting variational formula ranges over random shift-invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process. Our formula is not able to express the unboundedly long cycles; as a result we derive only lower and upper bounds. Our results and their shortcomes are at the heart of Bose-Einstein condensation.

Tom Kurtz

The semigroup approach to large deviation theorems for Markov processes

The semigroup approach to proving large deviation results for Markov processes developed in Feng and Kurtz (2006) will be outlined and illustrated by application to models arising in systems biology.

Carlangelo Liverani

Large deviations and metastability in deterministic systems

Lots of results are available for large deviations and metastability in random particle systems. On the contrary next to nothing is known when the dynamics is deterministic and the only randomness lies in the initial condition. I will describe a simple (but highly non trivial) system in which precise results can be obtained. this can be considered as the zero step in the direction of studying some Hamiltonian interacting particles systems.

Christian Maes

The physics in large deviation functionals

We give some physics interpretation to static and dynamic large deviation functionals that have appeared in statistical mechanics. For equilibrium, the macroscopic fluctuations are directly related to heat and work. For nonequilibrium, there are relations with entropy production rates, with dynamical activity and statistical forces in general.

Angela Stevens

Directed cell motion and a hydrodynamic limit for chemotaxis

In this talk the first equation of a chemotaxis system is derived as a hydrodynamic limit from a stochastic interacting many particle system on the lattice. The cells do interact with attractive chemical molecules and among themselves on the same lattice site. The chemical environment is assumed to be stationary with a slowly varying mean. This results in a non-trivial macroscopic chemotaxis equation for the cells.

Directed cell motion - like chemotaxis - is induced by polymerization and depolymerization of actin filaments within the cellular cytoskeleton. If time permits, a one-dimensional hyperbolic-parabolic model for the dynamics of the actin cytoskeleton is derived and the emergence of Dirac measures for the respective filament tips is discussed.
These can be interpreted as sharp polymerization fronts which can be observed experimentally.
(joint works with S. Grosskinsky, J. Fuhrmannn, D. Marahrens)

Florian Theil

Orientational order in two dimensions

A classic phenomenon is Statistical Mechanics is the emergence of crystalline phases at low temperature. Until recently not much was known about this problem in the case of atomistic systems with unbounded degrees of freedom. I will explain the link between crystal formation and orientational order. Then I will demonstrate that orientational order emerges in many realistic two-dimensional systems.

Balint Toth

Super-diffusive bounds for self-repelling motions in low dimension

I will give a survey of the resolvent method for obtaining bounds on diffusivity of random motions. This powerful method was initiated by Landim-Quastel-Salmhofer-Yau (2004) and Yau (2004) and leads to variational poblems. So it fits well to the central theme of this workshop. I will show some applications to diffusion in random drift field and self-repelling Brownian polymer.
(joint work with B. Valko)

Ofer Zeitouni

Large deviations for the two-dimensional two-component plasma

We derive a large deviations principle for the two-dimensional two-component plasma in a box. As a consequence, we obtain a variational representation for the free energy, and also show that the macroscopic empirical measure of either positive or negative charges converges to the uniform measure.

Johannes Zimmer

Nonlinear diffusion: from particle models to gradient flows

This talk will study scale-bridging from a thermodynamic perspective, focusing on gradient flows. We discuss the interplay between particle models and their thermodynamic description at hand of a class of nonlinear diffusion equations. It will first be shown how an underlying particle model can reveal an underlying (geo-)metric structure of the governing PDE, notably a gradient flow setting for a class of nonlinear diffusion equations. Large deviation arguments will be discussed, and it will be sketched how this can link mesoscopic fluctuations and stochastic PDEs in a way that can allow to derive stochastic "corrections" for deterministic PDEs.
 

PRACTICAL INFORMATION

●      Venue

Eurandom, Mathematics and Computer Science Dept, TU Eindhoven,

Den Dolech 2, 5612 AZ  EINDHOVEN,  The Netherlands

Eurandom is located on the campus of Eindhoven University of Technology, in the Metaforum building (4th floor) (about the building). The university is located at 10 minutes walking distance from Eindhoven main railway station (take the exit north side and walk towards the tall building on the right with the sign TU/e).
Accessibility TU/e campus and map.

 

 

●      Registration

Registration is free, but compulsory for speakers and participants. Please follow the link: REGISTRATION PAGE

 

 

●      Accommodation

For invited participants, we will take care of accommodation. Other attendees will have to make their own arrangements.

We have a preferred hotel, which can be booked at special rates. Please email Patty Koorn for instructions on how to make use of this special offer.

For other hotels around the university, please see: Hotels (please note: prices listed are "best available"). 

More hotel options can be found on the webpages of the Tourist Information Eindhoven, Postbus 7, 5600 AA Eindhoven.

 

●      Travel

For those arriving by plane, there is a convenient direct train connection between Amsterdam Schiphol airport and Eindhoven. This trip will take about one and a half hour. For more detailed information, please consult the NS travel information pages or see Eurandom web page location.

Many low cost carriers also fly to Eindhoven Airport. There is a bus connection to the Eindhoven central railway station from the airport. (Bus route number 401) For details on departure times consult http://www.9292ov.nl

The University  can be reached easily by car from the highways leading to Eindhoven (for details, see our route descriptions).

 

●      Conference facilities : Conference room, Metaforum Building  MF11&12

The meeting-room is equipped with a data projector, an overhead projector, a projection screen and a blackboard. Please note that speakers and participants making an oral presentation are kindly requested to bring their own laptop or their presentation on a memory stick.

 

●      Conference Secretariat

Upon arrival, participants should register with the workshop officer, and collect their name badges. The workshop officer will be present for the duration of the conference, taking care of the administrative aspects and the day-to-day running of the conference: registration, issuing certificates and receipts, etc.

 

●      Cancellation

Should you need to cancel your participation, please contact Patty Koorn, the Workshop Officer.

There is no registration fee, but should you need to cancel your participation after January 2, 2014, we will be obliged to charge a no-show fee of 30 euro.

 

●      Contact

Mrs. Patty Koorn, Workshop Officer, Eurandom/TU Eindhoven, koorn@eurandom.tue.nl

 

SPONSORS

The organisers acknowledge the financial support/sponsorship of:

 

 

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Last updated 10-03-16,
by PK

 P.O. Box 513, 5600 MB Eindhoven, The Netherlands
tel. +31 40 2478100  
  e-mail: info@eurandom.tue.nl