The following topics are targeted:

1. Gibbs-non-Gibbs transitions in mean-field and lattice systems,  possibly also quantum-spin systems. 

2. Non-equilibrium large deviations of interacting particle systems. 

3. Stochastic control theory for interacting diffusions.

The workshop is intended, in part, to review progress made since  the workshop, organized in December 2003 at EURANDOM, on Gibbs 

versus non-Gibbs in Statistical Mechanics and Related Fields. Proceedings of this workshop appeared as a special volume of Markov 

Processes and Related Fields in 2004.

Organizers

Aernout van Enter (Groningen)

Roberto Fernandez (Utrecht)

Frank Den Hollander (Leiden)

Frank Redig (Delft)

Fee

Fee is 100 euros. The fee includes lunches, coffee/refreshments and the conference dinner.

Exceptions:
- Speakers and Organizers
- Participants from TU/e only pay 35 euro, if they want to join the dinner

Bank details are given on the registration form.
 

Registration

Registration is obligatory for all participants (organizers and speakers too!).

Please indicate on the registration form your attendance, participation in the lunches, dinner.

Link to Registration form

Participants have to make their own hotelbooking. However, they can get a reduced rate if they book our preferred hotel.
Please send an email to Patty Koorn for instructions on how to obtain this special price.

For other bookings we suggest to consult the web pages of the Tourist Information Eindhoven, Postbus 7, 5600 AA Eindhoven.
 

Preliminary Programme

Monday December 19

Tuesday December 20

Wednesday December 21

Thursday December 22

Friday December 23

Abstracts

Stefan Adams (Warwick University)

Large deviations for stochastic processes

Large deviation theory deals with the decay of the probability of increasingly unlikely events. It is one of the key techniques of modern probability, a role which is emphasised by the award of the 2007 Abel prize to S.R.S. Varadhan, one of the pioneers of the subject. Large deviation theory is a part of probability theory that deals with the description of events where a sum of random variables deviates from its mean by more than a “normal" amount, i.e., beyond what is described by the central limit theorem. The mini-course will give an overview and glimpse of new techniques for large deviations for a class of stochastic processes. The course will rely on the recent book by Feng & Kurtz on Large Deviations for Stochas- tic Processes, and one of our aims is to elaborate on some of the key ideas and to provide an overview. In the first part we will review basic large deviations techniques adapted for stochastic processes. Beginning with the work of Cramér and including the fundamental work on large deviations for stochastic processes by Freidlin and Wentzell and Donsker and Varadhan, much of the analysis has been based on change of measure tech- niques. However, recently another methodology for large deviations analogous to the Prohorov compactness approach to weak convergence of probability measures has been developed. The main theme of the course and the book by Feng & Kurtz is the development of this approach to large deviation theory as it applies to sequences of cadlag stochastic processes. This approach involves verification of exponential tightness and unique characterisation of the possible rate function. We conclude henceforth our first part with results on ex- ponential tightness and give Puhalskii’s analogue of the Prohorov
compactness theorem. In the second part, we focus on Markov processes and give large deviation results based on the convergence of corresponding semigroups. Again this has an analogy with the use of convergence of linear semigroups in proofs of weak convergence of Markov processes. The success of this approach depends heavily on the idea of a viscosity solution of a nonlinear equation. In the last part we shall demonstrate the effectiveness of these methods with examples of large deviatio results for Markov processes including the Donsker-Varadhan theory for occupation measures and weakly interacting stochastic particles.

Joris Bierkens (SNN Nijmegen)

Davide Gabrielli (Universita dell'Aquila)

Interacting particle systems out of equilibrium

I will discuss the statistical properties of the stationary non equilibrium states (SNS) for models of stochastic interacting particles evolving on lattices. A typical example is a boundary driven stochastic lattice gas. I will consider two different approaches.

1) A microscopic approach that describes a SNS computing exactly the stationary measure of the Markov process. Due to the irreversibility this is in general hard, nevertheless in some cases some beautiful combinatorial representations are available.

2) A macroscopic approach that associates to a configuration of particles an empirical measure and then studies its asymptotic behavior in the limit of large number of degrees of freedom. In particular I will show how to establish a large deviation principle for the empirical measure of the SNS using a dynamic variational approach. The starting point is a dynamic large deviation principle from the hydrodynamic scaling limit of the model. Extending the classic Freidlin and Wentzell theory to an infinite dimensional framework, we can then identify the rate functional of the SNS with the quasi-potential associated to the dynamic rate functional. A central role in this approach is played by an infinite dimensional Hamilton-Jacobi equation.

I will illustrate all the results showing explicit computations for specific models like exclusion processes, zero range models and the KMP model.

Bert Kappen (Radboud University Nijmegen and University College London)

To compute a course of actions such as is required in intelligent biological systems or robots in the presence of uncertainty is the topic of
stochastic optimal control theory. Such computations require the solution of complex partial differential equations and these computations become intractable for most problems. I will introduce a class of control problems for which the optimal cost-to-go is expressed as a path integral, which plays the role of a partition function. The optimal control is computed as an expectation value with respect to the associated Gibbs measure. Noise plays the role of temperature and it is shown that there exist qualitatively different control solutions for high and low noise, separated by a phase transition. Surprisingly, the path integral formulation allows one to compute the optimal control solution without knowledge of the plant. We demonstrate this for some simple examples.

Christof Külske (Ruhr Universität Bochum)

Local discretizations of spin models
(Joint work with Aernout van Enter and Alex Opoku)

We describe what may happen to the Gibbs property of spin models under local discretizations, where we put a particular emphasis on massless models. We also discuss continuous symmetry breaking and Kosterlitz-Thouless behavior in the discretized rotor-model in this context.

Arnaud Le Ny (Université de Paris-Sud XI)

Dobrushin program for the 2d-Ising model : results and perspectives

In this talk, we shall use the seminal example of the decimation of the Ising model to review the state of achievement of the Dobrushin program of restoration of Gibbsianness and try to indicate a few possible applications and perspectives.

Christian Maes (KU Leuven)

We consider 3D active plane rotators, where the interaction between the spins is of XY-type and where each spin is driven to rotate. For the clock-model, when the spins take N\gg1 possible values, we conjecture that there are two low-temperature regimes. At very low temperatures and for small enough drift the phase diagram is a small perturbation of the equilibrium case. At larger temperatures the massless modes appear and the spins start to rotate synchronously for arbitrary small drift. For the driven XY-model we prove that there is essentially a unique translation-invariant and stationary distribution despite the fact that the dynamics is not ergodic.

Variational description of Gibbs-non-Gibbs dynamical transitions for the Curie-Weiss Model

We consider a Curie-Weiss model, with external magnetic field h subject to an independent spin-flip dynamic.
For this model, Gibbs-non-Gibbs transitions are shown to be equivalent to the occurrence of bifurcations in the set of global minima of the large deviation rate function for the trajectories of the magnetization conditioned on their endpoint.

Alex Opoku (University of Leiden)

Good transforms and their goodness

The images of Gibbs measures under transformations such as time evolution, discretization of spin space, renormalization-group map, etc., need not be Gibbs measures.
The case of transformations that preserves the Gibbs property will be discussed in this talk. We will also exhibit the continuity
properties ("goodness") of the conditional distributions of the resulting Gibbsian image measures.

Mark Peletier (TU/e)
(joint work with Stefan Adams, Nicolas Dirr, and Johannes Zimmer)

Understanding the origins of the Wasserstein gradient flows

Many evolutionary systems described by parabolic partial differential equations can be written as a gradient flow of some energy with respect to some metric. When present, this gradient-flow structure provides both high-level insight into the behaviour of the system, and low-level, practical tools for the analysis of the system and its solutions.
Since the pioneering work of Jordan, Kinderlehrer, and Otto (1998) an impressive collection of evolutionary PDEs has been formulated as a gradient flow of some energy with respect to the Wasserstein metric.
In this talk I will briefly introduce the Wasserstein metric and the concept of a Wasserstein gradient flow, and then turn to the main topic of the talk. This is the question: how can we understand _why_ the Wasserstein metric appears in so many systems?
I will focus on the simplest of all examples, the linear diffusion equation, which is the Wasserstein gradient flow of the entropy. I will show how the gradient-flow structure is intimately connected to the large-deviation behaviour of a system of independent Brownian particles. This connection gives us an explanation for the prevalence of Wasserstein gradient flows, and suggests directions for the derviation of a large number of related systems.

Michiel Renger (TU Eindhoven)

Extended Wasserstein gradient flows and how they arise from particle systems

The statistical mechanics programme has provided us a deep understanding of the connection between stochastic particle systems at the microscopic level and thermodynamics on the macro level. The discovery that diffusion is a Wasserstein gradient flow of entropy can potentially extend this knowledge to the non-equilibrium case. In 2010, Adams, Dirr, Peletier and Zimmer revealed a connection between stochastic particle systems and the Wasserstein gradient flow formulation of the diffusion equation.
I will discuss how this connection can be used as a general methodology for deriving natural gradient flow structures for more complex diffusion processses. More specifically, I will discuss the case of diffusion in a force field, diffusion with internal decay and diffusion with decay at a boundary.

Wioletta Ruszel (University of Groningen)

Evolving continuous models - what can go wrong?

I will review some recent result about propagation, loss and recovery of continuous (compact and unbounded) spin models subjected to time-evolution.

Practical information

Conference Location
The workshop location is EURANDOM,  Den Dolech 2, 5612 AZ Eindhoven, Laplace Building, 1st floor, LG 1.105.

EURANDOM is located on the campus of Eindhoven University of Technology, in the 'Laplace Building' (LG on the map). The university is located at 10 minutes walking distance from Eindhoven railway station (take the exit north side and walk towards the tall building on the right with the sign TU/e).

For all information on how to come to Eindhoven, please check http://www.eurandom.tue.nl/contact.htm

Contact
For more information please contact Mrs. Patty Koorn
Workshop officer of  Eurandom

Sponsored by:

STAR

Last updated 23-12-11,
by PK