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19 - 23 DECEMBER, 2011
DYNAMICAL GIBBS-NON-GIBBS TRANSITIONS
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.
Aernout van Enter (Groningen)
Roberto Fernandez (Utrecht)
Frank Den Hollander (Leiden)
Frank Redig (Delft)
Fee is 100 euros. The fee includes lunches, coffee/refreshments
and the conference dinner.
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
Monday December 19
Tuesday December 20
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
Joris Bierkens (SNN Nijmegen)
Relative entropy weighted optimization for Markov chains
Suppose we are given a random variable X with probability
measure P. We are allowed to change the probability measure to a new probability
measure Q in such a way as to minimize the expectation \int X \ d Q. The trivial
solution to this problem puts mass 1 on the event where X is minimal.
Wojciech De Roeck(Universität Heidelberg)
Diffusion in Hamiltonian Models
Irreversible phenomena like diffusion and thermalization obviously occur in our world, which is described by deterministic and time-reversible equations (Newton's equations of motion or the Schrodinger equation in quantum mechanics). However, up to today we seem to lack the tools to describe and derive these phenomena rigorously (apart from a few special models). In physics, one usually models them by stochastic evolution equations and the transition from deterministic equations to stochastic ones remains a leap of faith. Our work treats one of the simplest possible models; a quantum particle interacting with a gas of free bosons. With the help of a time-dependent renormalization group analysis, we prove diffusion.
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.
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
Christof Külske (Ruhr Universität Bochum)
Local discretizations of spin models
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.
Julian Martinez (University of Leiden)
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.
joint work with R. Fernádez and F. den Hollander
Karel Netocny (Academy of Sciences of the Czech Republic)
Steady states and quasistatic transformations out of equilibrium
We discuss some nonequilibrium extensions of standard concepts in equilibrium statistical physics. For a class of (small) stochastic systems undergoing a slow time variation of their parameters, we obtain a well defined quasistatic expansion for the heat exchange with the system's surroundings. Some simplifications and quasi-equilibrium relations are found close to detailed balance where the stationary states become small deformations of equilibrium Gibbs measures. A possible generalization to spatially extended driven systems will also be discussed.
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
Mark Peletier (TU/e)
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.
Michiel Renger (TU Eindhoven)
Extended Wasserstein gradient flows and how they arise
from particle systems
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.
Evgeny Verbitskiy (University of Leiden)
Functions of Gibbs processes: an overview
In this talk I will discuss preservation of the Gibbsianity under renormalization transformations in dimension 1. I will also discuss methods to identify potentials in case renormalization results in Gibbs states.
Mei Yin (University of Texas)
A cluster expansion approach to exponential random graph models
The exponential family of random graphs is among the most widely-studied of network models. A host of analytical and numerical techniques have been developed in the past. We show that any exponential random graph model could be alternatively viewed as an Ising model with a finite Banach space norm, thus making the system treatable by cluster expansion methods in statistical mechanics. In particular, we derive a convergent power series expansion for the limiting free energy in the case of small parameters. This hopefully would help with the application of renormalization group ideas to exponential random graph models.
For all information on how to come to Eindhoven, please check http://www.eurandom.tue.nl/contact.htm