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April 1822, 2016
METASTABILITY in statistical mechanics and stochastic processes
SUMMARY The mathematical study of the phenomenon of
metastability has been a standing issue since the foundation of
Statistical Mechanics. The development of a rigorous mathematical
theory, however, started mostly in the 70’s with the contribution of
researchers of at least three different areas: rigorous statistical
mechanics, reliability theory and dynamical systems. At present, the
subject has achieved a sizeable degree of mathematical maturity, but
researchers face several pressing issues. On the one hand, the diversity
of scenarios leading to the phenomenon has resulted in a multitude of
different approaches with not clearly
ORGANISERS
INVITED SPEAKERS
MONDAY April 18
TUESDAY April 19  programme related to NETWORKS
*************************************************************************************************************************************** Florent Barret Transition times for a class of parabolic SPDEs in one dimension We consider a class of parabolic semilinear stochastic partial differential equations driven by spacetime white noise on a compact space interval. We obtain precise asymptotics of the transition times between metastable states. A version of the socalled EyringKramers formula is proven in an infinite dimensional setting. The proof is based on a spatial finite difference discretization of the stochastic partial differential equation. The expected transition time is computed for the finite dimensional approximation and controlled uniformly in the dimension. Manon Baudel Spectral theory for random Poincare maps We consider an ordinary differential equation with N stable periodic
orbits perturbed by weak noise, for which we want to quantify the rare
transitions between periodic orbits. The behaviour of the stochastic
system is analysed through the spectrum of the kernel of a Markov
process describing successive returns to a Poincare section. Because of
the N stable periodic orbits, N eigenvalues of the Markov process are
expected to be close to 1. Nils Berglund Metastability of stochastic AllenCahn PDEs Consider an AllenCahn SPDE on the ddimensional torus, for d=1 or 2, driven by weak spacetime white noise. We provide EyringKramerstype asymptotics for the mean transition time between stable stationary solutions, going beyond largedeviation (Arrhenius) estimates. In the case d=1, such asymptotics were obtained independently by the speaker and Barbara Gentz (Bielefeld), using a sprectral Galerkin approximation, and by Florent Barret, using finitedifference discretizations. The mean transition time is known for all finite domain sizes, and contains a prefactor expressible in terms of ratios of spectral determinants. In the case d=2, the SPDE is only welldefined after a suitable renormalization is carried out. I will report on joint work with Hendrik Weber (Warwick) and Giacomo di Gesu (CERMICS), yielding an expression for the prefactor when the domain is sufficiently small. Alessandra Bianchi Limiting dynamics of the condensate in the reversible inclusion process on a finite set The inclusion process is a
stochastic lattice gas where particles perform random walks subjected to
mutual attraction, thus providing the natural
bosonic counterpart of the
wellstudied exclusion process. Sander Dommers Metastability of the Ising model on random regular graphs at zero temperature We study the metastability of the ferromagnetic Ising model on a random $r$regular graph in the zero temperature limit. We prove that in the presence of a small positive external field the time that it takes to go from the all minus state to the all plus state behaves like $\exp(\beta n (r/2+O(\sqrt{r})))$ when the inverse temperature $\beta\rightarrow\infty$ and the number of vertices $n$ is large enough but fixed. The proof is based on the socalled pathwise approach and bounds on the isoperimetric number of random regular graphs. Sebastien Dutercq Interface dynamics of a metastable massconserving diffusion We will consider a diffusion process defined by the stochastic
differential equation $dx_t=\nabla V_\gamma(x_t)dt+\sqrt(2\epsilon)dW_t$ Frank den Hollander Metastability on networks This talk presents an
overview on how potential theory for reversible Markov processes can be
used to analyse the metastable behaviour of Metropolis dynamics on
random graphs at low temperature. The analysis starts from a number of
general hypotheses about the energy landscape and leads to three
metastability theorems: (1) the average crossover time from the
metastable state to the stable state satisfies the classical Arrhenius
law, with an exponent and a prefactor that are controlled by the energy
and the entropy of the critical droplet; (2) the critical droplet is the
gate for the metastable crossover; (3) the crossover time divided by its
average is exponentially distributed. Sabine Jansen Lowtemperature LennardJones chains We investigate the lowtemperature behavior of a chain of atoms with LennardJones type interaction. We prove a path large deviations principle for the infinite chain and analyze the principal eigenvector and spectral gap of the transfer operator. There are two interesting regimes: at positive pressure, the spectral gap stays bounded away from zero and correlations decay exponentially fast, uniformly in the temperature. When the pressure goes to zero sufficiently fast, the chain breaks and the spectral gap is exponentially small; this matches known approaches that model fracture as a doublewell problem with one well at infinity. Oliver Jovanovski Metastability for Glauber dynamics on random graphs We give a description of metastability arising in the lowtemperature
Ising model when the setting is a random graph obtained from the
"Configuration model" algorithm. Roman Koteckı Metastability for the WidomRowlinson model The WidomRowlinson model is one of the few models of interacting
particles in the continuum for which a proof of ”liquidgas” coexistence
has been provided. An important feature of this model that facilitates
its investigation is the fact that it can be rewritten as a model of two
species of particles with mutual hardcore interaction. Reviewing first
this feature and its use for the proof of phase coexistence, we will
proceed to a discussion of transition from a metastable supercooled gas
phase to the liquid phase in this model. In particular, the theorem
about Arrhenius law with a nonstandard entropic correction will be
formulated. Claudio Landim Markov chains model reduction We review in this lecture the martingale approach to metastability.
This method has been developed recently to solve the following problem.
Let $E_N$ be a sequence of countable state Christian Maes Drivinginduced stability with longrange effects We give a sufficient condition under which an applied nonequilibrium
driving in a medium always has a stabilizing effect on an attached
quasistatic probe. We show that the resulting Lamb shift in the
symmetric part of the stiffness matrix with respect to equilibrium is
positive and depends strongly on the nonequilibrium density far away
from the probe. For illustration we take the example of diffusive medium
particles with a selfpotential in the shape of a Mexican hat, repulsive
at the origin and attractive near the border of a disk where the
potential energy gets very large. They undergo a rotational force around
the origin. For no or small rotation in the medium, the origin is an
unstable fixed point for the probe and the precise shape of the
selfpotential at large distances from the origin is irrelevant for the
statistical force there. Above a certain threshold of the rotation
amplitude the origin becomes stable for the probe and more details of
the medium start to matter. The stabilization is robust under different
types of rotation, and the temperature dependence gets reversed in the
transition from equilibrium (where lower temperature destabilizes more)
to nonequilibrium (where lower temperature stabilizes more). Patrick Müller Hydrodynamic limits, propagation of chaos and large deviations in local meanfield models with unbounded spins We consider systems of stochastic differential equations that describe the dynamics of unbounded spinsystems where spins interact via longrange 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. Francesca R. Nardi Metastability for General Dynamics with Rare Transitions: Escape
Time and Critical Configurations Elena Pulvirenti Metastability for WidomRowlinson model: II Frank Redig Splitting and exchange models with SU(1,1) symmetry Inspired by a recent model of wealth distribution in econophysics,
the socalled immediate exchange model, we study a general class of
models where at random times a continuous quantity is exchanged among
two agents according to the following mechanism (X,Y) goes to (X(1U)+
YV, XU + Y(1V)) where U,V are independent random variables on [0,1]. We
show the following Bruno Schapira A metastability result for the contact process on finite graphs We show that the contact process on any sequence of growing finite graphs exhibits a metastable behavior, when the infection parameter is larger than the critical rate on $\mathbb Z$. Based on a joint work with Daniel Valesin Metastability for irreversible PCA dynamics In this talk we present a class of parallel
irreversible Markov chains, that are probabilistic cellular automata,
and we show various models in which is possible to describe in details
metastable behaviours. Metastability: old problem and new results Insuk Seo Metastability and EyringKramers Formula for Nonreversible Markov Processes In this talk, we present a metastability result
for a class of nonreversible Markov processes. Our framework is based
on the recent developments of the Dirichlet and the Thomson principles
for the nonreversible Markov chains, and on the martingale approach for
the metastability. In particular, we provide an argument which enables
us to compute the sharp estimates for the mean transition times between
the metastable sets, and as a consequence the EyringKramers formula for
the nonreversible process is obtained. Finally, we discuss the
metastability of the socalled planar Potts model, which is a
generalization of the CurieWeiss model, as an application of our
result. Senya Shlosman Metastable behavior of communication networks with moving nodes I will explain the results of the study of
queuing networks, which consist of servers which can move. Each client
in the network has its destination, but the servers of the network move
on their own. The interplay of the two factors make the overall behavior
of the system quite intricate. Martin Slowik Metastability in stochastic dynamics: Lowlying eigenvalues of the generator via twoscale decomposition Metastability is a phenomenon that occurs in the dynamics of a multistable nonlinear system subject to noise. It is characterized by the existence of multiple, well separated time scales. The talk will be focus on the metastable behavior of reversible Markov chains on countable state spaces. In the talk, I will first review the potential theoretic approach to metastability that has been proven to be a powerful tool to derive sharp estimates on quantities characterizing the metastable behavior, e.g. metastable exit times. In particular, I will discuss systems in which the entropy of microscopic paths plays a crucial role, and precise computations of microscopic point to point capacities are not possible. Second, I will present an approach to derive matching upper and lower bounds on the small eigenvalues of the generator. Cristian Spitoni Sum of exit times in a series of metastable states In this talk we consider the problem of not degenerate in energy metastable states forming a series in the framework of reversible fi?nite state space Markov chains. We assume that starting from the state at higher energy the system necessarily visits the second one before touching the stable state. In this framework we give a sharp estimate of the exit time from the metastable state at higher energy and we prove an addition rule. Furthermore, we discuss the application of this theory to the BlumeCapel model in the zero chemical potential case, and to a reversible Probabilistic Cellular Automaton without selfinteraction. Siamak Taati Metastability of the hardcore process on bipartite graphs The hardcore gas model is widely studied in
statistical mechanics, but also appears in connection with communication
networks. In the dynamic version, particles appear and disappear with
constant rates on the sites of a graph subject to the "hardcore
constraint": two neighbouring sites can hold at most one particle at a
time. When the rate of appearance is large compared to the
disappearance rate, the particles tend to remain close to efficient
maximal packing arrangements. Transition from one locally optimal
arrangement to another requires the formation of a "critical droplet",
which is a configuration solving a particular isoperimetric problem. I
will talk about a description of such transitions in case the underlying
graph is bipartite. We use potentialtheoretic methods and give sharp
asymptotics for the expected transition time for few choices of the
underlying graph. Daniel Valesin The contact process on finite trees and random regular graphs The contact process is a class of interacting
particle systems that models the spread of an infection in a population.
Vertices of a graph are individuals that can either be in state 0
(healthy) or 1 (infected). Infected individuals heal with rate 1 and
transmit the infection to each neighbor with rate $\lambda > 0$. We
study the contact process on trees or locally treelike graphs. Since
seminal works by Pemantle (1992) and Liggett (1996), it has been
established that the contact process on the infinite $d$regular tree ($T_d$)
exhibits two phase transitions. Specifically, depending on the value of
the infection parameter $\lambda$, the process started from finite
occupancy can (1) die out; (2) survive globally but not locally; (3)
survive locally. In this talk, we examine the counterparts of these
distinct regimes in the context of two classes of finite graphs that
locally approximate the tree $T_d$: truncated trees and random regular
graphs. We show that the process exhibits very different behaviors in
these two contexts. Aernout van Enter Metastability, from Ideal to Real to Super To describe metastable phases in infinite volume has been problematical for different reasons. We point out some examples of shortrange models which satisfy the criteria of " Ideal Metastability" due to Sewell, and comment why these criteria still are physically dubious. Sewell's criterion of "Real Metastability" implies exponentially diverging metastability times. I discuss some examples of bootstrap percolation models, in which the metastability times become superexponential, rather than exponential. Alessandro Zocca Hitting time asymptotics for hardcore interaction on grid graphs Motivated by the study of randomaccess networks performance, we consider the hardcore model with Metropolis transition probabilities on finite grid graphs and investigate the asymptotic behavior of the first hitting time between its stable states in the lowtemperature regime. In particular, we develop a novel combinatorial method to show how the orderofmagnitude of this first hitting time depends on the grid sizes and on the boundary conditions for various types of grid graphs. Our analysis also proves the asymptotic exponentiality of the scaled hitting time and yields the mixing time of the process in the lowtemperature limit as sideresult. In order to derive these results, we extended the modelindependent framework for first hitting times known as "pathwise approach" to allow for more general initial and target states.
PRACTICAL INFORMATION ● VenueEurandom, 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).
● RegistrationRegistration is free, but compulsory for speakers and participants. Please follow the link: REGISTRATION
● AccommodationFor 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.
● TravelFor 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 or consult our map with highway connections.
● Conference facilities : Conference room, Metaforum Building MF11&12The meetingroom 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 SecretariatUpon 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 daytoday running of the conference: registration, issuing certificates and receipts, etc.
● CancellationShould you need to cancel your participation, please contact Patty Koorn, the Workshop Officer.
● ContactMrs. Patty Koorn, Workshop Officer, Eurandom/TU Eindhoven, koorn@eurandom.tue.nl SPONSORSThe organisers acknowledge the financial support/sponsorship of:
Last updated
130416,

