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Interuniversity Attraction Poles (IAP)
Phase VI
2007 – 2011
ANNEX I
TO CONTRACT P6/02
TECHNICAL SPECIFICATIONS : SECTION I
Information on the network
Title of the project : Nonlinear systems, stochastic processes, and statistical
mechanics
Name of the coordinator : Pierre VAN MOERBEKE
Institution : UCL
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I. 1. NETWORK COMPOSITION
BELGIAN PARTNERS

Coordinator : Partner 1 (P1)
Name : Pierre VAN MOERBEKE
Institution : Universite Catholique de Louvain
Institution’s abbreviation : UCLouvain
Partner 2 (P2)
Name : Pierre GASPARD
Institution : Universite Libre de Bruxelles
Institution’s abbreviation : ULB
Partner 3 (P3)
Name : Arnoldus KUIJLAARS
Institution : Katholieke Universiteit Leuven
Institution’s abbreviation : KULeuven
Partner 4 (P4)
Name : Joris VAN DER JEUGT
Institution : Universiteit Gent
Institution’s abbreviation : UGent

Mention only one name per partner. The person listed here should be the one in charge of the operational
aspects of the project. Indicate the full name (family name + first name) of the partner.
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EUROPEAN PARTNERS

(if applicable)
EU-Partner 1 (EU1)
Name : Thomas KRIECHERBAUER
Institution : Ruhr Universitat Bochum
Institution’s abbreviation : RUB
Country : Germany
EU-Partner 3 (EU3)
Name : Boris DUBROVIN
Institution : SISSA (Trieste)
Institution’s abbreviation : SISSA
Country : Italy
EU-Partner 2 (EU2)
Name : Antti KUPIAINEN
Institution : University of Helsinki
Institution’s abbreviation : HU
Country : Finland

Mention only one name per partner. The person listed here should be the one in charge of the operational
aspects of the project. Indicate the full name (family name + first name) of the partner.
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I. 2. TITLE AND SUMMARY OF THE PROJECT
Indicate clearly and briefly the project’s major objectives and provide a concise description of the
project.
A. Title and summary in English (2 pages maximum)
Today a major trend in fundamental sciences is the study of nonlinear systems and the appearance of
complex behavior. Such systems are characterized by many interacting entities as in macroscopic
physics where systems composed of many microscopic particles present collective and nonlinear
phenomena such as phase transitions, or transport properties under nonequilibrium conditions. As a
starting point such a study requires the microscopic description in terms of interacting particles and the
next step is to derive the collective and transport properties by mathematical methods.
New challenges have recently appeared in this field concerning the understanding of the dynamics and
the fluctuations in non-equilibrium conditions. In order to make progress in this field, a wide range of
knowledge will be required, from the theory of chaotic and integrable systems to the theory of
stochastic processes and statistical mechanics. Indeed, the different systems are governed by a
Hamiltonian microscopic dynamics from which we need to infer the properties of their time evolution.
These systems differ by the behavior of their trajectories; they may be periodic, quasiperiodic as it is
often the case in celestial mechanics, or appear in the form of fluctuations as in statistical mechanics
and random matrix theory. These fluctuations can be described by invariant probability measures. The
fact is that such invariant measures are well-known at equilibrium but largely unknown in nonequilibrium steady states. One famous problem in this context is the derivation of collective transport
properties such as heat conduction and the Fourier law and the understanding of the corresponding
fluctuations. In this regard, large-deviation relationships are currently studied under the names of
‘fluctuation theorem’ and ‘non-equilibrium work theorem’. These large-deviation relationships play a
fundamental role in characterizing the invariant measure of non-equilibrium steady states. The
understanding of these invariant measures has much to gain from dynamical systems theory and a
precise knowledge of the trajectories followed by the particles composing the system.
Along similar lines, the study of matrix models and random matrices is an extremely lively and
important domain of research, which bridges several areas of theoretical physics, mathematics and
statistics and which has strikingly deep connections with a variety of problems, e.g., with
combinatorics, combinatorial probability related to statistical mechanics, number theory, random
growth and random tilings, and questions of communication technology.
The main question is to investigate the mean density of the spectrum for large size random matrices
(with certain symmetry conditions) and its fluctuations about this equilibrium distribution. The
probability distributions for the spectrum of random matrices are conveniently described by Fredholm
determinants of kernels; the limiting kernels depend on the different regimes (scalings), leading to
different statistical behaviors near the edge of the spectrum, near a gap in the spectrum or in the bulk of
the spectrum. Interesting and novel statistical behaviors have come up and are related to non-linear
equations (Painleve equations) and new non-linear partial differential equations; they, moreover, all
seem to be “universal”, in that the limit only depends on the coarse features of the matrices, like
symmetry conditions. Some of the random matrix models are closely related to the free energy and the
Yang-Baxter equation for statistical mechanical models (six-vertex models, impenetrable Bose gas,
etc…).
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Dyson has introduced dynamics in the random matrix models in order to account for slowly varying
physical parameters; the spectrum then behaves as large systems of Brownian motions repelling one
another by a Coulomb force. The behavior of these infinite-dimensional diffusions near critical points
(edge, gap, etc…) display striking phase transitions, which we expect to study from the point of view
of asymptotics, transition probabilities and large deviations. The methodology consists of a formulation
in terms of the Riemann-Hilbert problem, which enables one to use the powerful steepest descent
methods, which have been developed over the last ten years. The use of integrable equations
(Korteweg-de Vries equations,..) and of the Virasoro algebra related to the underlying matrix models is
a very efficient method to find differential equations for the transition probabilities.
Related to the previous work, quantum theories with non-commutative geometry have received much
attention. Many simple approaches have been considered, usually based on deformations of canonical
commutation relations of position and momentum operators. In the context of Wigner Quantum
Systems (WQS), the approach is more fundamental. It is essentially based on the requirement that
Hamilton’s equations and the Heisenberg equations should be identical as operator equations in the
state space (Hilbert space), giving rise to certain compatibility conditions. This approach leads, for
example for the oscillator model, to relations with Lie superalgebras, since the compatibility conditions
(usually triple operator identities involving anti-commutators) have a natural solution in terms of Lie
superalgebras.
The main themes are the following:
The spectrum of random matrices, critical behavior and phase transitions
Transport and fluctuation theory
Quantum dynamical systems, dynamical entropy and semiclassical methods
Statistical mechanics of complex dissipative systems, transport properties and relaxation in
Hamiltonian dynamical systems and self organized criticality.
Integrability and non-integrability of Hamiltonian systems, master-symmetries, zero-dispersion KdV
equation and extensions of the Tracy-Widom distribution.
These are wide open fields, still at their cradle and which have acquired international visibility and
recognition. Two large European projects are devoted to these matters: an EU-project (ENIGMAUCL/KUL) and two “European Science Foundation”-projects (MISGAM-UCL/KUL and
STOCHDYN-ULB). The purpose of this proposal is to set up a network pooling the strong theoretical
expertise available in Belgium in this field in order to create a force to make significant progress on
these challenging issues.
There is a great enthusiasm for creating such a network (the first of this kind in Belgium!), aiming at
stimulating the exchange of ideas, ranging from theory to applications, and at improving crossfertilization of researchers and students across boundaries. The topics involved are highly related, and at
the same time have an impact on many different areas of geometry, combinatorics, probability, statistics
and physics. Many interesting questions - fundamental and applied - are ready to be tackled. With this in
mind, we propose a very broad network to provide ample opportunities of interaction and growth. A PAIIUAP grant will provide the necessary impetus to improve communication between researchers. Yet the
project is sufficiently focused in order to serve as a fruitful training ground for young researchers. The
main objective is to attract young graduate students from Belgium and from other countries, who will be
trained in the institutions of the network. The network will also attract young PhD’s from all over the
world to do post-doctoral research in these areas.
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This team combines researchers from the departments of mathematics and physics form the four
universities, Jean Bricmont, Pierre Bieliavsky, Luc Haine, Philippe Ruelle, Pierre Van Moerbeke
(UCLouvain), Pierre Gaspard (Université Libre de Bruxelles), Mark Fannes, Arnoldus Kuijlaars,
Christian Maes and Walter Van Assche (KULeuven) and Joris Van der Jeugt (Universiteit Gent).
These teams provide complementary skills in an interdisciplinary environment, and yet with common
objectives.
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B. Title and summary in Dutch (2 pages maximum)
Titel van het voorstel (Nederlands) : Niet-lineaire systemen, stochastische processen en
statistische mechanica
Samenvatting van het voorstel (Nederlands) :
Een belangrijke trend in het hedendaagse fundamenteel onderzoek is de studie van niet-lineaire systemen en
het optreden van complex gedrag. Zulke systemen worden gekarakteriseerd door een groot aantal
interagerende deeltjes zoals in macroscopische fysica waar systemen die opgebouwd zijn uit veel
microscopische deeltjes collectieve en nonlineaire fenomenen ondergaan zoals fase-overgangen en transport
eigenschappen onder niet-evenwichts voorwaarden. Als vertrekpunt vereist zulk een studie een
microscopische beschrijving in de vorm van interagerende deeltjes. De volgende stap is om de collectieve en
transport eigenschappen af te leiden met wiskundige methoden.
In dit gebied zijn recent nieuwe uitdagingen verschenen betreffende het begrip van de dynamica en de
fluctuaties in niet-evenwichts voorwaarden. Om vooruitgang te kunnen maken is een brede waaier van
kennis vereist van de theorie van chaotische en integreerbare systemen tot de theorie van stochastische
processen en statistische mechanica. Immers, de verschillende systemen worden beschreven door een
microscopische Hamiltoniaanse dynamica waarvan we de eigenschappen en tijdsevolutie willen afleiden.
Deze systemen verschillen in het gedrag van hun trajectorieen: deze kunnen periodiek zijn, quasi-periodiek
zoals vaak het geval is in de hemelmechanica, of verschijnen in de vorm van fluctuaties zoals in statistische
mechanica en random matrix theorie. Deze fluctuaties worden beschreven door middel van invariante
kansmaten. Het is het geval dat zulke invariante maten in evenwicht goed bekend zijn maar grotendeels
onbekend in niet-evenwichts toestanden. Een beroemd probleem in deze context is het afleiden van
collectieve transport eigenschappen zoals warmtetransport en de Fourierwet en het begrip van de
overeenkomstige fluctuaties. In dit verband worden tegenwoordig grote afwijkingen bestudeerd onder de
namen “fluctuation theorem” en “non-equilibrium work theorem”. Deze grote afwijkingen spelen een
fundamentele rol in de karakterisering van de invariante maat van niet-evenwichts toestanden. Het begrip
van deze invariante maten komt van de theorie van dynamische systemen en een preciese kennis van de
trajectorieen die gevolgd worden door de deeltjes in het systeem.
Op een vergelijkbare wijze is de studie van matrixmodellen en random matrices een uiterst levendig en
belangrijk onderzoeksdomein dat verschillende gebieden uit de theoretische fysica, wiskunde en statistiek
verbindt. Het heeft verrassend diepe verbanden met een veelheid van problemen zoals met combinatoriek,
combinatorische kansrekening in statistische mechanica, getaltheorie, random groei en random
vlakvullingen, en vraagstukken uit de communicatietheorie.
Het belangrijkste probleem is het onderzoek naar de verwachte dichtheid van het spectrum van grote random
matrices (met zekere symmetrie voorwaarden) en de fluctuaties rond deze evenwichtsverdeling. De
kansverdelingen voor het spectrum van random matrices worden mooi beschreven door Fredholm
determinanten van kernen. De limieten van de kernen hangen af van de verschillende toestanden (schalingen)
en dit leidt tot verschillend statistisch gedrag rond de rand van het spectrum, rond een gat in het spectrum of
in het inwendige van het spectrum. Interessant en nieuw statistisch gedrag verschijnt hierbij die verbant
houden met niet-lineaire vergelijkingen (Painleve vergelijkingen) en nieuwe niet-lineaire partiele
differentiaalvergelijkingen. Bovendien lijken ze allemaal “universeel” te zijn in de zin dat de limiet alleen
afhangt van de grove karakteristieken van de matrices, zoals de symmetrie eigenschappen. Sommige random
matrix modellen zijn nauw verwant met de vrije energie en de Yang-Baxter vergelijking van modellen uit de
statistische mechanica (six vertex model, ondoordringbaar Bose gas, enz..)
Dyson introduceerde dynamica in de random matrix modellen om rekening te houden met langzaam
varierende fysische parameters. Het spectrum gedraagt zich dan als een groot systeem van Brownse
bewegingen die elkaar afstoten volgens een Coulomb kracht. Het gedrag van dit soort oneindig-dimensionale
diffusies rond kritieke punten (rand, gat, enz.) vertoont opmerkelijke faseovergangen die we wensen te
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bestuderen vanuit het gezichtspunt van asymptotiek, overgangswaarschijnlijkheden en grote afwijkingen. De
werkwijze bestaat uit een formulering van een Riemann-Hilbert probleem, hetgeen ons in staat stelt om de
krachtige steilste afdalingsmethode die de laatste tien jaar is ontwikkeld, te gebruiken. Het toepassen van
integreerbare vergelijkingen (Korteweg-de Vries vergelijkingen,…) en van de Virasoro algebra die
samenhangt met het onderliggende matrix model is een zeer effectief middel om de
differentiaalvergelijkingen voor de overgangswaarschijnlijkheden te vinden.
Samenhangend met het bovenstaande hebben kwantum theorieen met niet-commutatieve meetkunde veel
aandacht gekregen. Vele eenvoudige benaderingen zijn beschouwd, doorgaans gebaseerd op deformaties van
kanonieke commutatieregels voor positie en momentum operatoren. In het kader van Wigner Quantum
Systems (WQS) is de aanpak meer fundamenteel. Het is essentieel gebaseerd op het vereiste dat Hamilton’s
vergelijkingen en de Heisenberg vergelijkingen identiek zouden moeten zijn als operatorvergelijkingen in de
toestandsruimte (Hilbert ruimte), hetgeen aanleiding geeft tot zekere compatibiliteitsvoorwaarden. Deze
aanpak leidt, bijvoorbeeld for het oscillator model, tot verbanden met Lie superalgebras, omdat de
compatibiliteitsvoorwaarden (doorgaans drievoudige operator-identiteiten met anti-commutatoren) een
natuurlijke oplossing in termen van Lie superalgebras hebben.
De belangrijkste themas zijn de volgende:
1 Het spectrum van random matrices, kritiek gedrag en faseovergangen
2 Transport en fluctuatietheorie
3 Kwantum dynamische systemen, dynamische entropieen en semiklassieke methoden
4 Statistische mechanica van complexe dynamische systemen, transporteigenschappen en relaxatie in
Hamiltoniaanse dynamische systemen en zelfgeorganiseerde kritiekheid.
5. Integreerbaarheid en niet-integreerbaarheid van Hamiltoniaanse systemen, meester-symmetrieen, kleine
dispersie KdV vergelijking en uitbreidingen van de Tracy-Widom verdeling.
Het betreft grote open gebieden die in volle ontwikkeling zijn en die internationaal actief beoefend en
gewaardeerd worden. Twee grote Europese projecten zijn gewijd aan deze gebieden: een EU-project
(ENIGMA-UCL/KUL) en twee ‘European Science Foundation’ projecten (MISGAM-UCL/KUL en
STOCHDYN-ULB). Het doel van dit voorstel is om een netwerk te vormen dat de sterk aanwezige
theoretische kennis in Belgie kan bundelen om zo belangrijke vooruitgang te kunnen boeken.
Het team bestaat uit onderzoekers uit departementen wiskunde en natuurkunde van vier universiteiten: Jean
Bricmont, Pierre Bielavsky, Luc Haine, Philippe Ruelle, Pierre Van Moerbeke (UCL), Pierre Gaspard
(ULB), Mark Fannes Arnoldus Kuijlaars, Christian Maes, Walter Van Assche (KULeuven) en Joris Van der
Jeugt (UGent). Deze partners zorgen voor complentaire vaardigheden in een interdisciplinaire omgeving,
met tevens gemeenschappelijke doelstellingen.
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C. Title and summary in French (2 pages maximum)
Titre de la poposition (Français) : Systèmes non-linéaires, processus stochastiques et
mécanique statistique
Résumé de la proposition (Français) :
Aujourd’hui une voie importante en science fondamentale est l’étude des systèmes non linéaires et la
présence de comportements complexes. De tels systèmes sont caractérisés par plusieurs entités
interagissantes comme en physique macroscopique ou des systèmes composés de plusieurs particules
microscopiques présentent des comportements collectifs et non linéaires comme des transitions de phase ou
des propriétés de transport sous des conditions de non-équilibre. Comme point de départ une telle étude
requiert une description microscopique en termes de particules interagissantes et l’étape suivante est de
déduire des propriétés collectives et de transport par des méthodes mathématiques.
De nouveaux défis sont apparus récemment dans ce domaine, en ce qui concerne la compréhension de la
dynamique et des fluctuations dans des conditions de non-équilibre. Pour progresser dans ce domaine un
large éventail de connaissances sera requis, d’une part dans la théorie du chaos et des systèmes intégrables et
d’autre part dans les processus stochastiques et la mécanique statistique.
En effet, ces différents systèmes sont régis par une dynamique microscopique hamiltonienne, ce qui devrait
nous permettre de déduire des propriétés d’évolution dans le temps. Ces systèmes diffèrent les uns des
autres par le comportement de leurs trajectoires, celles-ci peuvent être périodiques ou quasi-périodiques
comme c’est souvent le cas en mécanique céleste, ou apparaître sous forme de fluctuations comme en
mécanique statistique et en théorie des matrices aléatoires. Ces fluctuations peuvent être décrites comme des
mesures de probabilités invariantes. Le fait est que ces mesures invariantes sont bien connues à l’équilibre
mais essentiellement inconnues dans des états de non-équilibre. Un problème célèbre dans ce contexte est la
dérivation de propriétés de transports collectifs tels que la conduction de chaleur et la loi de Fourier et la
compréhension des fluctuations correspondantes. Dans ce contexte, des relations de grandes déviations sont
à l’étude actuellement sous les noms de « théorème de fluctuations » et de « théorème du travail en régime
de non-équilibre ». Ces relations de grandes déviations jouent un rôle fondamental dans la caractérisation
des mesures invariantes des états stationnaires de non-équilibre. La théorie des systèmes dynamiques et la
compréhension des trajectoires des particules du système devraient contribuer à la découverte de ces
méthodes invariantes.
Dans la même ligne, l’étude des modèles matriciels et des matrices aléatoires est un domaine très vivant et
important de la recherche qui établit des ponts entre divers domaines de la physique théorique, des
mathématiques et de la statistique et qui établit des liens très profonds avec plusieurs problèmes, par exemple
avec la combinatoire, la probabilité combinatoire liée à la mécanique statistique, la théorie des nombres, les
modèles de croissances et de pavages aléatoires et les questions de technologie de la communication.
La question majeure est d’examiner la densité moyenne du spectre pour des matrices aléatoires de grande
taille (satisfaisant à certaines conditions de symétrie) et leurs fluctuations autour de cette distribution
d’équilibre. Ces distributions de probabilités pour le spectre de matrices aléatoires sont décrites par des
déterminants de Fredholm des noyaux. Ces noyaux limites dépendent de régimes différents (changement
d’échelles) menant à des comportements statistiques différents près du bord du spectre, près d’un vide dans
le spectre ou dans le « gros » du spectre. Des comportements statistiques intéressants et nouveaux sont
apparus et sont liés à des équations non-linéaires (équations de Painlevé) et des équations aux dérivées
partielles non-linéaires nouvelles ; de plus ces distributions revêtent toutes un caractère « universel » au sens
que la limite ne dépend que des propriétés grossières de la matrice comme des conditions de symétrie.
Certains de ces modèles matriciels aléatoires sont étroitement liés à l’énergie libre et l’équation de YangBaxter pour des modèles de mécanique statistique (Six-Vertex Models Impenetrable Bose gas,etc…)
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Dyson a introduit de la dynamique dans les modèles de matrices aléatoires qui reflètent des paramètres
physiques évoluant lentement ; le spectre se comporte alors comme des grands systèmes de mouvements
browniens séparés les uns des autres par une force de Coulomb. Le comportement de ces diffusions de
dimension infinie près du point critique (bord, vide, etc…) donnent lieu à des transitions de phase
intéressantes que nous comptons étudier du point de vue de l’asymptotique, des probabilités de transition et
des grandes déviations. La méthodologie consiste à formuler la question en termes du problème de
Riemann-Hilbert, ce qui nous permet l’utilisation de méthodes du col qui ont été développées les dix
dernières années. L’utilisation des équations intégrables (équations de Korteweg-deVries,…) et de l’algèbre
de Virasoro (liées aux modèles matriciels sous-jacents) conduit à une méthode très efficace pour trouver les
équations différentielles des probabilités de transition.
En relation avec les travaux mentionnés ci-dessus, les théories quantiques à géométrie non commutative ont
retenu toute l’attention. Plusieurs approches simples ont été considérées basées sur les déformations de
relation de commutation des opérateurs de position et du moment. Dans le contexte des systèmes quantiques
de Wigner (WQS), cette approche est plus fondamentale. Celle-ci est basée essentiellement sur le fait que
les équations d’Hamilton et les équations d’Heisenberg sont identiques comme équations d’opérateurs dans
les espaces des états (espaces d’Hilbert), donnant lieu à certaines conditions de compatibilité. Cette
approche conduit, par exemple pour l’oscillateur, à des relations avec les super-algèbres de Lie ; en effet les
conditions de compatibilité (souvent des identités triples entre anti-commutateurs) ont une solution naturelle
en termes de super-algèbres de Lie.
Les thèmes de recherche principaux sont les suivants :
1°) le spectre de matrices aléatoires, comportement critique et transition de phase
2°) théorie du transport et des fluctuations
3°) système dynamique quantique, entropie dynamique et méthodes semi-classiques
4°) mécanique statistique de systèmes complexes dissipatifs, théorie du transport et relaxation dans les
systèmes dynamiques hamiltoniens
5°) intégrabilité et non-intégrabilité des systèmes hamiltoniens, symétries maîtresses, équations de KdV à
dispersion nulle et extension de la distribution de Tracy-Widom.
Ce sont des domaines de recherche très vierges, ayant acquis visibilité et reconnaissance internationales.
Trois grands projets européens visent cette problématique, un projet de l’Union européenne (ENIGMA –
UCL/KUL) et deux projets de la « European Science Foundation » (MISGAM – UCL/KUL et STOCHDYN
– ULB). L’objectif de ce projet est de constituer un réseau profitant de la grande expertise dans ce domaine
en Belgique en vue de créer une force permettant de faire des progrès significatifs dans ces domaines
d’avenir.
Cette équipe contient des chercheurs des départements de mathématique et physique de quatre universités :
Jean Bricmont, Pierre Bieliavsky, Luc Haine, Philippe Ruelle, Pierre Van Moerbeke (UCLouvain), Pierre
Gaspard (Université Libre de Bruxelles), Mark Fannes, Arnoldus Kuijlaars, Christian Maes et Walter Van
Assche (KULeuven) et Joris Van der Jeugt (Universiteit Gent). Ces équipes fournissent des talents
complémentaires dans un environnement inter-disciplinaire mais ayant néanmoins des objectifs communs.
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I. 3. OBJECTIVES, MOTIVATION AND STATE OF THE ART (5 pages maximum)
Describe the project’s objectives and research goals.
Define the problems being addressed by positioning them in relation to the current state of
knowledge.
Workpackage 1: Random matrices
We study critical phenomena and phase transitions related to random matrix models. Important
questions in random matrix theory deal with the mean eigenvalue density of large size random
matrices, its fluctuations around this equilibrium distribution and the local statistical behavior of
eigenvalues. The mathematical tools to study these problems for large classes of random matrices have
been developed only in the last decade. By now the local behavior near generic points in unitary
random matrix models is well-understood. The study near critical points presents many new challenges
and open problems.
The random matrix model with external source presents an extension where new critical phenomena
have been observed. This model may be understood in terms of non-intersecting Brownian motions. A
phase transition occurs when two groups of non-intersecting paths come together and merge to
continue as one group. This critical behavior is modelled by Pearcey integrals and in the limit by the
Pearcey process which is an example of a new type of infinite dimensional diffusion for which usual
descriptions break down. A new type of critical behavior is expected when the two groups of nonintersecting paths meet in one point and then immediately split again. This case should be related to an
as yet unknown special function and a likewise unknown infinite dimensional diffusion. We intend to
continue our study of such “Markov clouds” and the PDEs that govern their transition probabilities.
We also intend to explore the universality of the critical behavior in random matrix models with
external source as well as possible higher order critical behavior.
Another line of research concerns the PDEs that are satisfied by statistical distributions that arise from
random matrix theory. The PDEs for the probability that no eigenvalues belong to an interval or
several intervals extend the ones that are described by Painleve functions. It is of substantial interest to
extend the methods of Virasoro constraints to other situations and to discover the integrable model that
is underlying all these cases.
Workpackage 2 : Transport and fluctuation theory
We plan to explore the nonperturbative generalizations of the fluctuation-dissipation theorem, which is
one of most exciting new that has appeared concerning fluctuations in nonequilibrium systems. One
considers the physical entropy production and one gives general relations concerning its transient and
its steady state fluctuations. The result can be presented in various ways, e.g. by relating the
irreversible work with a change in equilibrium free energy as in the Jarzynski relation. One of our
contributions there was the realization that these nonequilibrium fluctuations were all, without
exception, the consequence of one general fact; that the variable and fluctuating entropy production as
conceived by Boltzmann is really equal to the source of time-reversal breaking in the action governing
the space-time histories.
On the other hand, different versions of the fluctuation theorem have been recently proposed for
quantities such as a fluctuating entropy or the fluctuating currents in either nonequilibrium steady
states or transient situations. The study of these fluctuation theorems is one of the goal of the present
project because they represent the latest advances in nonequilibrium statistical mechanics and a
breakthrough in the development of this field.
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Important questions are open concerning the consequences of the fluctuation theorem, in particular, on
the nonlinear response coefficients. These consequences may concern different applications such as
the characterization of the fluctuations of electric currents in mesoscopic conductors, chemical
reactions at the nanoscale, nonequilibrium Brownian motion, as well as molecular motors.
There is today also major effort, to which we will contribute, towards the derivation of hydrodynamic
and kinetic equations out of equilibrium. Finally, let us mention another open problem, on which we
plan to work, the one of phase transitions for systems out of equilibrium.
Workpackage 3: Quantum systems
The increasing relevance of quantum technology offers an important challenge to better understand
quantum dynamics both from a theoretical and practical point of view. The lack of a configuration
space, due to the basic non-commutativity, calls for a more abstract treatment that often leads to
counterintuitive phenomena. A solid theoretical basis is therefore, more than in a classical context,
essential also for the actual development of quantum devices.
A broad range of problems and techniques within this context is proposed in this package, from the
study of dynamical systems arising from representations of Lie superalgebras to essentially random
systems. The proposed techniques range from the construction of explicit models satisfying strong
geometrical constraints to a statistical approach of randomizing systems.
There is apparently no unique natural extension of the concept of dynamical entropy to the quantum
world. It is nevertheless of fundamental importance to quantify different aspects of the complexity of a
quantum dynamics and to relate such quantities to structures of spectra, to fluctuations, to transport
phenomena, ... Moreover, techniques devised for quantum systems should also prove useful in studying
dissipative classical systems as they exhibit some quantum features (a well-known example of this
connection is the Feynmann-Kac formula).
On the opposite side is the search and construction of particular quantum systems arising, not from the
usual canonical quantisation, but rather from the requirement that the equations of motion coincide
with their classical analogues. Group theoretical techniques will be used for this purpose. A detailed
analysis of such models could give a good handle on understanding dynamical aspects of
entanglement.
Workpackage 4 : Nonlinear dynamics
We plan to work within the domain of dynamical systems and of their applications to statistical
mechanics. Ever since the beginning of statistical mechanics, there has been a debate about the relative
role of statistical versus dynamical arguments in deriving the macroscopic behavior of matter from its
microscopic behaviour. The contemporary version of that debate is to know to what extent the
nonlinearities and the apparently chaotic behavior of the microscopic components affect the values of
transport coefficients, the existence of autonomous transport equations, and the exact nature of
dissipative effects.
Many natural complex systems can be modeled as coupled nonlinear elements as it is the case in
neuronal networks, or coupled map lattices. These networks may present transitions in their dynamical
behavior which can be either stationary, oscillatory, or chaotic. The question of synchronization is also
an important issue in many applications. Since these systems have many variables and are spatially
distributed, the tools of statistical mechanics such as the theory of critical phenomena are very
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appropriate although the lack of conserved quantities in dissipative systems constitutes a significant
change with respect to the usual situations.
Transport properties such as viscosity or heat conduction manifest themselves in spatially extended
Hamiltonian dynamical systems. The ergodic properties of these systems determine their frequency
spectrum. These ergodic properties rely on specific aspects of their dynamics such as dynamical
instability which may guarantee a continuous spectrum and therefore a monotonous decay of the time
correlation functions. Recent work has shown that these resonances can be used in the transport theory
in order to give a framework to such physical concepts as the dispersion relations of diffusion. In the
present project, we intend to extend previous results on diffusion and reaction-diffusion systems to
viscosity and heat conductivity.
Another aspect where nonlinearities enter the present project has to do with what is commonly called
self-organized criticality. Organization is in a way an anti-theme to dissipation. In the preceding
decades, various toy-models have been proposed, ranging from specific systems of differential
equations to cellular automata.
Workpackage 5 : Integrable systems
We intend to study certain aspects of integrable systems, that partly complement and extend the study
of random matrix theory from workpackage 1. Indeed, many of the distribution functions of random
matrix theory are expressible in terms of the integrable Painleve equations, while random matrix
partition functions satisfy integrable hierarchies.
The Tracy-Widom distribution is an important distribution function that describes fluctuations in a
wide variety of processes. This distribution is expressed in terms of a solution of the Painleve II
equation with parameter 0. We are interested in a new generalization of the Tracy-Widom distribution
that involves the Painleve II equation with a general parameter.
The partition function of 2D quantum gravity is a solution of the Korteweg-de Vries hierarchy, which
in addition is a fixed point of its master symmetries. We further explore the master symmetries of
discrete integrable hierarchies and their relation with bi-Hamiltonian structures and the bispectral
problem. Some random matrix partition functions are expressed as determinants of the moment
matrices of orthogonal polynomials or of some of its generalizations. After perturbation of the
orthogonality weights the determinants were shown to satisfy integrable equations. In the study of the
generalizations (bi-orthogonal polynomials, multiple orthogonal polynomials) the Riemann-Hilbert
matrix plays an important role.
Certain aspects of the limiting behavior of eigenvalues of random matrix theory have an analog in
singular limits of integrable systems. Based on our experience with critical phenomena in random
matrices we plan to study the zero dispersion limit of the Korteweg-de Vries equation at the time of
shock formation. Our goal is to show that the the onset of oscillations is described by the second
member of the Painleve I hierarchy, thereby proving part of a recent conjecture of Dubrovin.
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I. 4. DETAILED DESCRIPTION OF THE PROJECT (15 pages minimum, 25 pages
maximum)
- Submit a general description of the project as well as a description detailing each workpackage and
indicate the partners involved in each workpackage.
- Illustrate by means of a table or scheme the interaction between the partners within a workpackage
and the interaction between the workpackages.
- Describe and justify the methods and proposed approaches in relation to the state of the art.
- Describe and justify how the contribution of the different partners will be integrated.
Workpackage 1: Random matrices
Pierre Van Moerbeke (UCL), Arno Kuijlaars (KULeuven), Pierre Gaspard (ULB)
1. Introduction and background
Random matrix theory (RMT) is an extremely lively, exciting and vibrant domain of research, which
bridges several areas of theoretical physics, mathematics and statistics and which has strikingly deep
connections with a variety of problems, e.g., with combinatorics, combinatorial probability related to
statistical mechanics, number theory, random growth and random tilings, and questions of
communication technology.
RMT has its origins in the 1920s in the works of Wishart in mathematical statistics and in the 1950s in
the works of Wigner, Dyson and Mehta on the spectra of heavy nuclei. During the last 15 years, the
subject has developed fast and has found applications in many branches of mathematics and physics,
ranging from quantum field theory to statistical mechanics, integrable systems, from number theory to
communication technology, statistics, and probability. The relationship between RMT and integrable
systems is especially deep. It is interesting to point out that the US National Science Foundation has
identified ``random matrix theory and its ties to classical analysis, number theory, quantum mechanics,
and coding theory'' as one of its emerging areas.
The main question is to investigate the mean density of the spectrum for large size random matrices
(with certain symmetry conditions), its fluctuations about this equilibrium distribution and the
statistical behavior of the spectrum near critical places, like near an edge of the spectrum, near a gap in
the spectrum, in the bulk of the spectrum, etc.
Dyson has introduced dynamics in the random matrix models in order to account for slowly varying
physical parameters; the spectrum then behaves as large systems of Brownian particles on the line
repelling one another by a Coulomb force, thus leading to non-intersecting Brownian motions. In the
large size limit, the Dyson non-intersecting Brownian motions lead to critical infinite-dimensional
diffusions, opening up an entirely new world of stochastic processes.
2. Project description
A main aim of this part of the project is to study critical phenomena in random matrix models.
Critical behavior in unitary random matrix models
In a ground-breaking paper Deift et al. [1] applied the Riemann-Hilbert steepest descent technique to
orthogonal polynomials and they used it to prove universality results for unitary random matrices in the
regular case. They also classified three types of possible singular behavior and it is conjectured that all
types have universality properties that can be described by Painlevé functions. Progress has been made
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in the understanding of some of the singular cases in the work of Kuijlaars and students. Singular
behavior in the bulk is described with the Painlevé II equation [2,3], and singular behavior at the edge
is described with the Painlevé I equation and its hierarchy [4,5]. A complete description of all singular
cases is expected to be associated to the full Painlevé I and Painlevé II hierarchies, and to some as yet
unknown hierarchy that would describe singular behavior outside of the spectrum.
[1] P. Deift, T. Kriecherbauer, K.T-R McLaughlin, S. Venakides, X. Zhou, Uniform asymptotics for
polynomials orthogonal with respect to varying exponential weights and applications to universality
questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), no. 11, 1335-1425.
[2] P. Bleher and A.Its, Double scaling limit in the random matrix model: the Riemann-Hilbert
approach. Comm. Pure Appl. Math. 56 (2003), no. 4, 433-516.
[3] T. Claeys and A.B.J. Kuijlaars, Universality of the double scaling limit in random matrix models,
arxiv:math-ph/0501074, to appear in Comm. Pure Appl. Math.
[4] M. Duits and A.B.J. Kuijlaars, Painlevé I asymptotics for orthogonal polynomials with respect to a
varying quartic weight, arxiv:math.CA/0605201.
[5] T. Claeys and M. Vanlessen, The existence of a real pole-free solution of the fourth order analogue
of the Painleve I equation, arxiv:math-ph/0604046.
Random matrices with external source
Bleher and Kuijlaars initiated the study of unitary random matrices with external source with the aid of
the Riemann-Hilbert problem for multiple orthogonal polynomials [1]. Already the simplest Gaussian
model exhibits new critical behavior that is described by Pearcey integrals [2,3,4]. The steepest descent
analysis of the Riemann-Hilbert problem [4] turned out to be more complicated than initially expected.
Since the Riemann-Hilbert problem is 3x3 matrix valued (or higher), we encountered new features that
are not present in the 2x2 case Along the way we had to develop new technical details in the
asymptotic analysis. It is expected that the critical behavior is again universal and extends to more
general potentials. A good understanding of an underlying equilibrium problem for logarithmic
potentials will be crucial. Complications arise since the equilibrium problem may not be posed on the
real line but instead on curves in the complex plane. Higher order critical phenomena are also possible
for unitary random matrix models with external source. The Pearcey integrals are but the first member
of a complete hierarchy that is not yet understood.
[1] P.M. Bleher and A.B.J. Kuijlaars, Random matrices with external source and multiple orthogonal
polynomials, Int. Math. Res. Not. 2004, no. 3, 109-129.
[2] E. Brézin and S. Hikami, Level spacing of random matrices in an external source, Phys. Rev. E 58
(1998), 7176-7185.
[3] C. Tracy and H. Widom, The Pearcey process, Comm. Math. Phys. 263 (2006), no. 2, 381-400.
[4] P.M. Bleher and A.B.J. Kuijlaars, Large n limit of Gaussian random matrices with external source,
Part III: Double scaling limit, arxiv:math-ph/0602064.
Non-intersecting Brownian motion
The Gaussian unitary random matrix model with external source has an equivalent formulation in
terms of non-intersecting one-dimensional Brownian motions [1]. The Pearcey integrals appear when
two groups of Brownian paths come together and merge to become one group [2]. From this point of
view one expects different singular behavior when two groups come together and immediately split
again. This situation can be modelled by a 4x4 matrix valued RH problem [3]. It will be extremely
interesting to identify the special functions connected to this singular case. They should satisfy a fourth
order ODE which is as yet unknown. Having the correct special functions we can apply the steepest
descent method to this case, where the main new contribution will be the construction of a local
parametrix with the new special functions.
[1] A.I. Aptekarev, P.M. Bleher, and A.B.J. Kuijlaars, Large n limit of Gaussian random matrices with
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external source. II. Comm. Math. Phys. 259 (2005), no. 2, 367-389.
[2] C. Tracy and H. Widom, The Pearcey process, Comm. Math. Phys. 263 (2006), no. 2, 381-400.
[3] E. Daems and A.B.J. Kuijlaars, Multiple orthogonal polynomials of mixed type and nonintersecting Brownian motions, arxiv:math.CA/0511470.
Dynamics of eigenvalues
Classically chaotic quantum systems typically have irregular spectra with the properties of random
matrices. If the system depends on contral parameters such as external fields, the eigenvalues move
with the parameters in a motion described by generalized Calogero-Moser completely integrable
systems [1]. A statistical mechanics can be set up for these Hamiltonian systems and a connection can
be established in this way with random matrix theory. As in random matrix theory, there exist different
kinds of systems depending on the potential for the motion of the eigenvalues. We intend to study these
dynamics of eigenvalues which are complementary to the random-matrix theory.
[1] P. Gaspard, S.A. Rice, H.J. Mikeska, and K. Nakamura, Parametric motion of energy levels:
Curvature distribution, Physical Review A 42 (1990), 4015-4027.
PDE’s for statistical distributions in random matrix theory
It has been known by physicists since the 80’s that many matrix models are closely related to
integrable systems, that they satisfy Virasoro constraints and that, in many cases, their expansion are
related to topological features in complex geometry. This idea enabled Adler-Van Moerbeke to
establish a precise correspondence between Hermitian matrix integrals, associated orthogonal
polynomials, some solutions to the Toda lattice and the Virasoro constraints. The key to the “string
equation” associated to these matrix integrals is a formula of Adler-Shiota-Van Moerbeke between
master symmetries of the Toda lattice and the Virasoro constraints. This circle of ideas applied to
random matrix theory gave rise to a single PDE for the probability that all spectral points belong to an
interval or several intervals, extending the case of one single semi-infinite interval for the Gaussian,
Laguerre and Jacobi ensemble (Painleve IV, V, VI). In the large size limit, this also gave rise to a
single PDE for the multi-interval case, an extension of the Painleve II equation appearing in the
celebrated Tracy-Widom distribution. The Adler-van Moerbeke method does not require a precise
knowledge of the kernels, but provides immediately the differential equation for the distribution. These
methods have been applied successfully to random matrix ensembles of symmetric and symplectic
matrices, to chains of coupled Gaussian Hermitian random matrices, with or without external potential,
to integrals over the Grassmannian space of p-planes in n-dimensional complex spaces, to various
unitary matrix ensembles appearing in the context of the distribution of the length of the longest
increasing sequences in random permutations, words and in percolation problems. Many nonintersecting Brownian motions (Dyson Brownian motion) can be transformed, via a change of
variables, to matrix models of the type described above. In many other situations, one leaves the realm
of matrix models. It is a challenging question how to proceed in those cases. What is the underlying
integrable model and what are the Virasoro constraints?
M. Adler, T. Shiota and P. Van Moerbeke: A Lax pair representation for the vertex operator and the
central extension, Comm. Math. Phys., 171, 547-588 (1995)
M. Adler, T. Shiota and P. Van Moerbeke: Matrix integrals, Toda symmetries, Virasoro constraints
and orthogonal polynomials, Duke Math. J., 80, 863-911 (1995)
M. Adler and P. Van Moerbeke: The spectrum of coupled random matrices, Annals of Math., 149, 921-976 (1999).
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M. Adler and P. Van Moerbeke: Hermitian, symmetric and symplectic random ensembles: PDE's for
the distribution of the spectrum, Annals of Math., 153, 149—189 (2001).
Infinite dimensional diffusions
Dyson considered Hermitian matrix models whose entries evolve according to independent OrnsteinUhlenbeck motions. Their eigenvalues behave like non-intersecting Brownian motions, held together
by a drift term. Putting dynamics in other types of Hermitian matrix models and taking large size
limits, leads to striking phase transitions for these finite-dimensional diffusions near critical points
(edge, gap, etc…), in the large n limit.
The transition probabilities for n-dimensional diffusions are solutions of second order parabolic PDE's
(heat-like equations) in n variables. The Pearcey process, appearing near a gap in the equilibrium
distribution and governed by the Pearcey kernel, is an infinite-dimensional diffusion and could be
thought of as a “continuous Markov cloud”, for which an effective description by heat-like equations
breaks down! Nevertheless we propose for such infinite-dimensional diffusions, like the Pearcey
process, non-linear equations for the transition probability p(t,x,y). It is of interest to study such
``Markov clouds" and the PDE's governing their transition probabilities.
F.J. Dyson: A Brownian-Motion Model for the Eigenvalues of a Random Matrix, Journal of Math.
Phys., 3, 1191--1198 (1962)
E. Brezin and S. Hikami: Universal singularity at the closure of a gap in a random matrix theory,
Phys. Rev., E 57, 4140--4149 (1998).
C. Tracy and H. Widom: The Pearcey Process, Comm. Math. Phys. 263 (2006), no. 2, 381-400.
M. Adler, T. Shiota and P. Van Moerbeke: A PDE for the Gaussian ensemble with external source and
the Pearcey distribution, Comm. Pure and Appl. Math, 2006 (to appear).
.
A Pearcey matrix integral?
On the one hand, the Witten-Kontsevich integral is a matrix Airy function and is a solution of the KdV
equation, singled out by its Virasoro constraints. On the other hand, the distribution of the largest
eigenvalue in Hermitian random matrix models is governed by the Fredholm determinant of the Airy
kernel. Close inspection shows a very intimate relationship between the two problems: the Fredholm
determinant of the Airy kernel is a ratio of two functions, one of which is the Witten-Kontsevich
integral. What is the analogue of the Witten-Kontsevich integral for the Pearcey distribution and what
would be their geometrical significance in terms of moduli space of curves, for instance?
M. Adler, T. Shiota and P. Van Moerbeke: Random matrices, vertex operators and the Virasoro
algebra, Phys. Lett. A208, 67-78 (1995) and Random matrices, Virasoro algebras and “noncummutative” KP, Duke Math. J., 94, 379-431 (1998).
A stochastic differential equation for the Airy process.
The Airy process, introduced by Praehofer and Spohn in the context of polynuclear growth models and
investigated by Johansson, describes the motion of the largest eigenvalue of a Hermitian matrix model
whose entries evolve according to independent Ornstein-Uhlenbeck motions for large size matrices,
after some rescaling. The Airy process is a stationary, non-Markovian process, which locally behaves
as Brownian motion; having been observed in several percolation and growth model, it will certainly
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sooner or later acquire the status of “universal process”. The transition probability for this process is
thus time-independent and is given by the Tracy-Widom distribution, whereas the transition
probabilities for several times are given by a non-linear third-order PDE, with quadratic non-linearity.
It is a challenging problem to discover the stochastic differential equation for this process in terms of
Brownian motion. This requires a delicate limit, involving a linear statistics (depending on the largest
eigenvalue) over all the remaining eigenvalues. An appropriate Riemann-Hilbert method should
resolve this problem.
M. Adler and P. van Moerbeke: PDE's for the joint distributions of the Dyson, Airy and Sine processes,
The Annals of Probability, 33, 1326-1361 (2005)
P. Deift, T. Kriecherbauer, K.T-R McLaughlin, S. Venakides, X. Zhou, Uniform asymptotics for
polynomials orthogonal with respect to varying exponential weights and applications to universality
questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), no. 11, 1335-1425.
K. Johansson: The Arctic circle boundary and the Airy process, Annals of Probability, 33, 1-30 (2005).
A. Okounkov and N. Reshitikhin: Random skew plane partitions and the Pearcey process,
math.CO/0503508 (2005).
M. Praehofer and H. Spohn: Scale Invariance of the PNG Droplet and the Airy Process, J. Stat. Phys.,
108, 1071-1106 (2002).
C.A. Tracy and H. Widom : Differential equations for Dyson processes, Comm. Math. Phys. 252, 7-41
(2004).
Workpackage 2 : Transport and fluctuation theory
Jean Bricmont (UCL), Pierre Gaspard (ULB), Chris Maes (KUL)
The study of macroscopic phenomena can be divided into two sets of problems : one is to write down
approximate phenomenological equations describing the phenomena under consideration and to study
the behaviour of the solutions of those equations ; the other is to relate these equations, if possible, to
the known microscopic properties of matter. Usually, this connection relies on probabilistic
considerations – it is one form or another of the law of large numbers that connects the microscopic
properties to the macrocopic ones. Starting in the 19th century, and continuing through the 1960-70’s,
this program has been accomplished for systems in equilibrium, i.e. systems that have been isolated for
a sufficiently long time, including phase transitions and critical phenomena. Concerning the latter, one
of the main tools that have contributed to their analysis is the renormalization group, a remarkable idea
that has also greatly contributed to our understanding of quantum field theory and that has been applied
to apparently unrelated fields, such as dynamical systems.
However, much less is understood about systems that are not isolated or that are evolving between
equilibrium states. The very rich phenomenology of systems out of equilibrium stretches from some of
the last great unsolved problems of classical physics, such as turbulence, over areas of biophysics,
nanophysics and astrophysics. Multidisciplinary components are linked with the study of complex
systems in general, as we will also describe in workpackage 4. Some qualititative understanding of the
microscopic basis of the second law of thermodynamics exists, as well as many phenomenological
equations, such as Fick’s law or Fourier’s law, but their microscopic derivation remains rather obscure
and there is nothing similar to a general framework such as the one of Gibbs states that has been so
useful in the study of equilibrium phenomena. Our goal in this workpackage, is to explore and to
develop some of the new directions that have emerged recently in the study of nonequilibrium systems.
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1. Fluctuation theorems and their consequences.
Nonequilibrium statistical mechanics means foremost to develop a fluctuation theory for systems that
are driven away from equilibrium. In equilibrium, there is a close connection between the variational
principle by which an equilibrium is characterized and the structure of fluctuations at equilibrium.
Depending on the situation, the functional that gets varied is the entropy or some other related
thermodynamic potential. That functional governs the equilibrium fluctuations. Today we have a well
developed theory, called the theory of large deviations, that connects these principles with probability
theory. The basic insights are however coming from physics and the foundation for it were laid by
pioneers like Laplace, Boltzmann and Einstein. These principles form the real basis of equilibrium
statistical mechanics, and hence are also founding the relations of equilibrium statistical
thermodynamics.
There is very little of that in nonequilibrium physics, or in irreversible thermodynamics far from
equilibrium. Yet, some time ago, starting in the mid 1990’s, some exciting new and general results
have appeared concerning fluctuations in nonequilibrium systems.
In a way these are for a good part generalizations of the fluctuation-dissipation theorem, see e.g. [5], to
a nonperturbative context. One considers the physical entropy production and one gives general
relations concerning its transient and its steady state fluctuations. The result can be presented in
various ways, e.g. by relating the irreversible work with a change in equilibrium free energy as in the
Jarzynski relation [9]. Most often however it concerns a symmetry in the fluctuations or large
deviations of some dissipation function. We refer to [6, 7, 9, 10, 4] for more information. One of our
contributions there was the realization that these nonequilibrium fluctuations were all, without
exception, the consequence of one general fact; that the variable and fluctuating entropy production as
conceived by Boltzmann is really equal to the source of time-reversal breaking in the action governing
the space-time histories.
The natural continuation of the work on fluctuation symmetries goes in various directions. It is true
that the fluctuation theorems as we know them today yield Green-Kubo relations, but these are only
first order (linear) relations around equilibrium. No systematic perturbation theory seems to follow,
one other very useful tool in the equilibrium counterpart. In fact, to go beyond linear order, we need to
understand also the modifications by the nonequilibrium driving to the fluctuations of time-symmetric
observables (like the current squared, or like kinetic energy etc). Similarly, while there is an
approximate variational principle close to equilibrium, the so called minimum entropy production
principle, we need to pick up other quantities than the variable time-antisymmetric entropy production
to vary over, when interested in nonlinear contributions to the steady state.
Perhaps surprisingly, these questions are often easier to handle in the quantum context. The reason is
the (sometimes very) discrete nature of the states, and the related possibility to work with spins and
energy levels on the lattice. It gives, among other things, physical motivations to some stochastic
processes, modeling a particular nonequilibrium phenomenon. There are other things that are much
harder here, having to do with the nature of entropy and with the notion of path-dependence and
history. Nevertheless, some of the most exciting applications are also to be found in a quantum
context, e.g. in questions of nanoscale engineering and ratchet devices.
On the other hand, different versions of the fluctuation theorem have been recently proposed for
quantities such as a fluctuating entropy or the fluctuating currents in either nonequilibrium steady
states or transient situations. The study of these fluctuation theorems is one of the goal of the present
project because they represent the latest advances in nonequilibrium statistical mechanics and a
breakthrough in the development of this field.
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In this context, the fluctuation theorem for the currents can be investigated in the theory of stochastic
processes on the basis of graph theory [1]. Important questions are open concerning the consequences
of the fluctuation theorem, in particular, on the nonlinear response coefficients. These consequences
may concern different applications such as the characterization of the fluctuations of electric currents
in mesoscopic conductors, chemical reactions at the nanoscale, nonequilibrium Brownian motion, as
well as molecular motors. Our aim is here to investigate the fluctuation theorem and its applications.
Furthermore, interesting questions concern the range of validity of the fluctuation theorem. Indeed, the
fluctuation theorem is a relation among the large-deviation functions in nonequilibrium systems. The
large-deviation functions may present singularities which are often interpreted as dynamical phase
transitions and which are important to study in model systems because they may limit the range of
validity of the fluctuation theorem which should thus be generalized.
Another important question is to establish the fluctuation theorem in the framework of the Hamiltonian
systems. Till now, the theorem has been proved for thermostated nonHamiltonian dynamical systems
as well as for stochastic processes. In principle, the microscopic motion of atoms is ruled by
Hamiltonian classical or quantum mechanics and it is therefore a fundamental question to have direct
proof of the theorem starting from the Hamiltonian microscpic dynamics.
Furthermore, we intend to develop the formalism which expresses the thermodynamic entropy
production as the difference between a time-reversed entropy per unit time and the usual (KolmogorovSinai) entropy per unit time [8]. The interest of this approach is that it relates the entropy production to
a time asymmetry in the dynamical randomness of the nonequilibrium fluctuations. Certainly, this type
of relationships are interesting because they connection a macroscopic quantity such as the entropy
production to more microscopic quantities which characterize the fluctuations and it is at this
mesoscopic level that we are facing new results.
2. Transport equations
Nonequilibrium conditions are created by some driving mechanism. These can be surface driving, like
in heat conduction, or bulk driving as with a bulk external field. As a result, some quantity like mass
or energy is being transported. With the transport comes dissipation. Even for situations that have
been with us for more than a century, we have often no rigorous control over the derivation of the
transport law. There is today a major effort to derive hydrodynamic and kinetic equations also out of
equilibrium. A famous example concerns the derivation of Fourier’s law, [2]. In order to derive
microscopically that law, one usually considers a Hamiltonian system coupled to reservoirs at différent
temperatures. If the reservoirs are modelled by stochastic processes, then one can hope to study the
properties of that processes and to establish Fourier’s law for its stationary state. Recent progress in
that direction has been accomplished recently in [3], who use a closure of the equations describing the
stationary state, similar to the one leading to Boltzmann’s equation, and derive Fourier’s law within
that approximation.
Another issue of much current interest is the rectification problem in transport. Its solution would teach
us again important information about transport coefficients, but its theoretical interest goes far beyond
that. The point has again to do with going beyond linear response theory. In linear order, gradient or
field reversal is equivalent with time-reversal. That is not necessarily so starting from second order
around equilibrium. It is therefore expected that the problem of rectification, and the associated ratchet
effects, see e.g. [11], will be important studies in the construction of nonequilibrium statistical
mechanics.
Finally, let us mention another open problem, the one of phase transitions for systems out of
equilibrium, but which we will discuss in workpackage 4.
References:
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[1] D. Andrieux and P. Gaspard, Fluctuation theorem for transport in mesoscopic systems, Journal of
Statistical Mechanics: Theory and Experiment (2006) P01011.
[2] F. Bonetto, J.L. Lebowitz and L. Rey-Bellet: Fourier’s Law: A Challenge for Theorists, in :
Mathematical Physics 2000, Imp. Coll. Press, London 2000, pp. 128--150.
[3] J. Bricmont, A. Kupiainen, On the derivation of Fourier's law for coupled anharmonic oscillators,
preprint.
[4] C. Bustamante, J. Liphardt and F. Ritort, The nonequilibrium thermodynamics of small systems,
Physics Today 58, (2005), 43-48.
[5] S.R. de Groot and P. Mazur, Non-equilibrium Thermodynamics, North Holland Publishing
Company (1962).
[6] D.J. Evans, E.G.D. Cohen and G.P. Morriss, Probability of second law violations in steady flows,
Physical Review Letters, 71, (1993), 2401.
[7] G. Gallavotti and E.G.D. Cohen, Dynamical ensembles in nonequilibrium Statistical Mechanics,
Physical Review Letters, 74, (1995), 2694. -, Dynamical ensembles in stationary states, Journal of
Statistical Physics, 80, (1995) 931.
[8] P. Gaspard, Time-reversed dynamical entropy and irreversibility in Markovian random processes,
Journal of Statistical Physics 117 (2004), 599-615.
[9] C. Jarzynski, Nonequilibrium Equality for Free Energy Differences, Physical Review Letters, 78,
(1997), 2690; Microscopic analysis of Clausius Duhem processes, Journal of Statistical Physics, 96
(1999), 415.
[10] C. Maes, On the origin and the Use of Fluctuation Relations for the Entropy, in : Poincaré
Seminar 2003, Eds. J. Dalibard, B. Duplantier and V. Rivasseau, Birkhauser (Basel), 145 (2004).
[11] P. Reimann, Brownian motors: Noisy transport far from equilibrium. Phys. Rep. 361, (2002), 57.
Workpackage 3: Quantum systems
Mark Fannes (KUL), Pierre Gaspard (ULB), Joris Van der Jeugt (Ugent), Pierre Bieliavsky
(UCL)
Quantum dynamical systems
In general, the behaviour of complex quantum dynamical systems is still rather poorly understood.
With ever decreasing time-scales in experiments and applications, a good understanding of quantum
phenomena, before some Markovian regime sets in, is an important issue. Even the simplest instances,
such as quantum spin chains or lattices, present serious challenges. This is due to the basic noncommutativity in quantum systems that immediately delocalizes observables.
Sufficiently randomizing classical dynamical systems can efficiently be analyzed in terms of shifts on
symbolic sequences, i.e. analysis of time series. Tools like the Kolmogorov-Sinai invariant allow both
to quantify the chaoticity and to model the system. Finer results can be obtained such as understanding
the return statistics.
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The situation with quantum dynamical systems is very different. Developing measures, analogous to
the Kolmogorov-Sinai invariant, has proven to be a difficult and long-lasting endeavour which,
moreover, led to quite different approaches [E. Stormer, in “Classification of Nuclear C*-Algebras.
Entropy in Operator Algebras”, eds. M. Rordam and E. Stormer, Springer Verlag, Berlin (2002), 148198; R. Alicki and M. Fannes, “Quantum Dynamical Systems.” Clarendon Press, Oxford (2001)]. The
fundamental reason for this state of affairs is that, due to non-commutativity, simple models of a
system in terms either of classical shifts or of shifts on spin chains never return isomorphisms.
Within this context, we propose to investigate whether more complex, but still essentially finite
dimensional systems, can be used to model a quantum dynamical system. We also want, for
quantizations of classical chaotic systems, to better understand the regime beyond the initially classical
evolution of coherent states (and before saturation effects set in).
Furthermore, the analysis of return statistics, which has been made for a random dynamics, should be
extended to a more realistic dynamics [M. Hirota, B. Sausol and S. Varienti, Commun. Math. Phys.
206 (1999) ,33 ; M. De Cock, M. Fannes and P. Spincemaille, J. Phys. A 32 (1999), 6547]. Another
goal in this area is to gain a better understanding of the chaotic properties of dissipative maps. This is
an important issue as such maps describe the influence of an uncontrollable environment on an almost
isolated system.
Quantum generalization of dynamical entropies
The generalizations to quantum mechanics of the Kolmogorov-Sinai entropy per unit time and the
time-reversed entropy per unit time [P. Gaspard, J. Statistical Phys. 117 (2004) 599] are fundamental
questions in the context of this collaboration. Indeed, it has long been expected that such
generalizations could be possible and useful in particular to characterize the fluctuations in quantum
systems. In view of the importance of these concepts in classical systems, it is important to pursue the
effort and bring together the different aspects. One issue is that we have a perspective given by the
fluctuation theorem that the two aformentioned quantities should be closely related to entropy
production. The investigation of these problems are among the most fundamental in this field.
Furthermore, it is also important to generalize the fluctuation theorem to quantum systems.
Semiclassical methods
Semiclassical methods will be developed for Schrödinger operators of quantum mechanics as well as
for Fokker-Planck operators of the theory of stochastic processes. The most advanced semiclassical
methods are based on Gutzwiller trace formula which reduces the trace of the resolvent operator to a
sum over periodic orbits and their properties. This method can be systematically expanded in powers
of the small parameter of the theory as shown previously [P. Gaspard, Progr. Theor. Phys. Suppl. 116
(1994) 59]. The method can also be applied to Fokker-Planck operators in noisy classical systems [P.
Gaspard, J. Statistical Phys. 106 (2002) 57] where it can be used to obtain excellent approximations for
the leading eigenvalues. We intend to further develop the method and its various applications.
Transport properties in quantum systems
At low temperature, the quantum effects manifest themselves in the transport properties. The transport
by diffusion on surfaces is an important case with many applications which we want to study [M.
Esposito and P. Gaspard, J. Statistical Phys. 121 (2005) 463]. The quantization of cross transport
properties as it is the case for instance in the quantum Hall effects will also be developed. A better
understanding of the electronic Hamiltonian in many-electron systems is also important to develop for
applications in this context.
Geometrical and informational aspects of states and maps
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The recent rather explosive expansion of quantum information theory has revived the interest in
probabilistic and geometric aspects of state spaces and quantum maps. A central feature, which breaks
the unitary invariance that was very prominent in traditional quantum probability, is the presence of
distinguished parties and the study of non-classical correlations between these parties, i.e.
entanglement [I. Bengtsson and K. Zyczkowski, “Geometry of Quantum States: An Introduction to
Quantum Entanglement.” Cambridge University Press, Cambridge (2006)]. While an isolated quantum
system is ideally described by a pure state, a wave function, decoherence effects or measurements by
some parties, immediately require mixed states (density matrices). It is therefore reasonable to assume
that relevant geometrical or informational properties should be extendible from pure to mixed states.
It is well-known, for pure states, that there are non-classical invariants such as the Bargman invariants.
How do these extend to the general case? Which is the minimal dimension in which we can realize
multiplets of states with given invariants? Gaining insight in such matters will improve our
understanding of the geometry of quantum state spaces and allow a better selection of extension of
quantities known for pure states to general ones.
Another set of questions is related to the ill-understood properties of quantum maps. Such maps
possess a rich structure, combining usual positive definiteness with properties that remind somewhat of
the Schur product. A clear sign of this lack of understanding is the difficulty to resolve the additivity
problem for quantum channels [C.K. King and M.B. Ruskai, in “Quantum Information, Statistics and
Probability”, ed. O. Hirota, Rinton Press (2004); R. Alicki and M. Fannes, Open Systems and
Information dynamics 11 (2004), 1]. We plan to approach these questions, not only from an algebraic
point of view but also using global topological properties and relations with mean-field models in
quantum statistical mechanics.
Wigner quantum systems and geometrical properties
During recent years quantum theories with non-commutative geometry have received much attention.
The literature on the subject is vast, and it is no longer of purely theoretical interest. Many simple
approaches have been considered, usually based on deformations of canonical commutation relations
of position and momentum operators. In the context of Wigner Quantum Systems (WQS’s), the
approach is more fundamental. In this project, we propose to study properties of Wigner Quantum
Systems, in particular (spectral) properties of the Hamiltonian and of position and momentum
operators. This topic has connections with the previous one, in the sense that geometric aspects of state
spaces of simple quantum systems – of relevance in quantum information theory – will be investigated
from the point of view of WQS’s.
In [E.P. Wigner, Phys. Rev. 77 (1950), 711], Wigner observed – on the simple example of a onedimensional harmonic oscillator – that this quantum system also allows solutions for which both
Hamilton’s equations and the Heisenberg equations are satisfied, but not the canonical commutation
relations. In other words, the canonical commutation relations are sufficient but not necessary
conditions for Hamilton’s equations and the Heisenberg equations to be compatible. The non-canonical
solutions of quantum systems where both Hamilton’s equations and the Heisenberg equations hold are
known as WQS-solutions.
Wigner’s work led to the theory of parabosons and parafermions in quantum field theory, and because
of this attention its impact for ordinary quantum systems was somewhat overlooked. Another reason
why WQS’s did not receive immediate attention was because no general solutions could be constructed
for the compatibility conditions of simple WQS’s (apart from the one-dimensional harmonic
oscillator). It was only much later – after the theory of Lie superalgebra’s was completed – that Palev
[T.D. Palev, J. Math. Phys. 23 (1982), 1778] observed that classes of WQS-solutions for the ndimensional harmonic oscillator are described by means of representations of the Lie superalgebras
osp(1|2n) and sl(1|n). This algebraic or representation theoretic approach to quantum systems has
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revived the interest in WQS’s [A. Degasperis and S.N.M. Ruijsenaars, Ann. Phys. 293 (2001), 92;
N.M. Atakishiyev, G.S. Pogosyan and K.B. Wolf, Phys. Part. Nuclei 36 (2005), 247; M.A. Lohe, Rep.
Math. Phys. 57 (2006), 131].
In a WQS, suppose that the system with n position operators qi and n momentum operators pi, i=1,…,n,
is described by the Hamiltionian H, usually of the form H  12
pi2  V (q1 ,..., q n ) . Hamilton’s

i
equations for such a system are of the form q k 
i

H
H
and p k  
; the Heisenberg equations
q k
p k
i

read p k   [ p k , H ] and q k   [ q k , H ] . The requirement that these equations should be
identical as operator equations in the state space (Hilbert space) give rise to certain compatibility
conditions, extending the usual canonical commutation relations. The study of operator solutions for
these compatibility conditions is difficult, however. Since the Hamiltonian can usually be written in
symmetrized form involving anti-commutators, the compatibility conditions become triple operator
identities involving commutators and anti-commutators. Such relations have a natural solution in terms
of Lie superalgebras.
The relation to Lie superalgebras implies that solutions of the WQS are described by means of unitary
representations of the corresponding superalgebra. So far, mainly simple systems such as the N-particle
D-dimensional oscillator have been studied. For the oscillator WQS, both general linear and
orthosymplectic Lie superalgebras play a role [N.I. Stoilova and J. Van der Jeugt, J. Phys. A: Math.
Gen. 38 (2005), 9681]. The physical properties of the system are then deduced from the operator
actions in the representation space. This had already led to interesting models with a non-commutative
and discrete spatial structure [R.C. King, T.D. Palev, N.I. Stoilova and J. Van der Jeugt, J. Phys. A:
Math. Gen. 36 (2003), 4337 and 11999]. Some of the physical properties are not always easy to
interpret properly or correctly.
So far, only rather simple unitary representations of Lie superalgebras (namely Fock type
representations) have been studied in this context. In the frame of this project, we plan to construct
other classes of unitary representations, and study the physical properties of WQS’s in such
representations. Among these properties are: energy spectra, angular momentum contents, spectrum of
position and momentum operators, classical limits of solutions, etc.
Furthermore, the study will be extended to more advanced quantum systems, such as a onedimensional string of n harmonic oscillators coupled by “springs”, and described by the Hamiltonian
H 
k
pk2 qk2

 c(qk  qk 1 ) 2 , with some coupling constant c. Herein, k=1,…,n and periodic
2
2
boundary conditions are assumed. Such systems are of relevance in entanglement dynamics [e.g. M.B.
Plenio, J. Hartley and J. Eisert, New J. Phys. 6 (2004), 36]. For this last quantum system, recent work
shows that there is still an sl(1|n) solution of the compatibility conditions, provided the coupling is
weak (c smaller than 1/n). In other words, in such a case one can construct finite-dimensional operator
solutions of the given quantum system. This implies, in particular, that also the position operators of
this system have a discrete spectrum.
Our purpose is, amongst others, to establish all Lie superalgebraic solutions of such Wigner Quantum
Systems. For each type of solution, the physical properties of the system will be determined. Since this
involves spectra of operators from a Lie superalgebra, the methods used are mainly algebraic or group
theoretical: representation theory, branchings of representations with respect to subalgebras and Young
diagram techniques.
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Noncommutative Geometry, Deformation Quantization and Number theory
Non-commutative geometry as introduced by Alain Connes in the 80's [1] generalizes correspondences
between spaces and commutative algebras such as Gelfand's correspondence between compact spaces
and unital commutative C*-algebras.
Several recent works have indicated deep relations between non-commutative geometry and analytical
number
theory [see e.g. 2,3]. In [2], the key object is a universal deformation formula for the actions of a
particular Hopf algebra. A fundamental feature of the associated calculus is its equivariance under the
Lie algebra sl(2). However, these deformations belong to the formal category; no topological operator
representation has been found as yet.
An important challenge consists therefore in finding such representations. Another question of interest
consists in finding a framework were the above-mentioned calculus would be equivariant under the
actions of the modular or Fuchsian groups. This would define a notion of non-commutative Riemann
surfaces. So far, such a notion has not been satisfactorily defined except in genus zero or one. A
definition in higher genus would lead in particular to a notion of non-commutative Anosov flows.
[1] Connes, Alain, Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math. No.
62 (1985), 257--360.
[2] Connes, Alain; Moscovici, Henri, Rankin-Cohen brackets and the Hopf algebra of transverse
geometry. Mosc. Math. J. 4 (2004), no. 1, 111--130, 311.
[3] Connes, Alain; Moscovici, Henri, Modular Hecke algebras and their Hopf symmetry. Mosc. Math.
J. 4 (2004), no. 1, 67--109, 310.
Workpackage 4 : NONLINEAR DYNAMICS
Jean Bricmont (UCL), Pierre Gaspard (ULB), Chris Maes (KULeuven)
The domain of ‘complex systems’ started in the 1970’s and was quite flourishing by the 1980’s. That
came at the same time of other developments. There was, on the one hand, the ever growing power,
availability and efficiency of fast computers. An essential element however was the new interest in
modern mechanics. Today we speak about the theory of dynamical systems or about nonlinear
dynamics. Since the 1970’s important mathematical results such as chaos theory, contributed much to
the dynamical side of various problems. Nonlinear phenomena and complex systems were studied in a
great variety of contexts, and a link was established between dynamical systems and statistical physics.
Since at least the time of Boltzmann, pioneer of statistical mechanics, the question was often repeated
about the relative importance of statistical versus dynamical arguments in deriving macroscopic
behavior. After all, heat is molecular motion and studies of dynamics, even for few particles, benefit
much from a probabilistic treatment. This issue is very much open when one speaks about
nonequilibrium statistical mechanics, see also workpackage 2. To what extent and how exactly do the
nonlinearities and the apparently chaotic behavior of the microscopic components contribute to the
values of transport coefficients, to the existence of autonomous transport equations, and to the exact
nature of dissipative effects? In this workpackage, we plan to try to answer some aspects of this general
question.
1. Statistical mechanics of complex dissipative systems.
Many natural complex systems can be modeled as coupled nonlinear elements as it is the case in
neuronal networks. These networks may present transitions in their dynamical behavior which can be
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either stationary, oscillatory, or chaotic. The question of synchronization is also an important issue in
many applications. Since these systems have many variables and are spatially distributed, the tools of
statistical mechanics such as the theory of critical phenomena are very appropriate although the lack of
conserved quantities in dissipative systems constitutes a significant change with respect to the usual
situations. On the other hand, the sensitivity to initial conditions in the chaotic regimes justifies the use
of probabilistic considerations and the generalized Liouville equation which provides a framework to
set up descriptive equations. In this part of the project, we intend to apply the methods of statistical
mechanics to study these transitions and the critical behavior at the transitions. Networks with scaling
geometries will also be studied.
A related class of models is provided by coupled map lattices. While dynamical systems with few
degrees of freedom are often modelled by iterated maps, partial differential equations or extended
dynamical systems may be similarly modelled by coupling an infinite lattice of such maps. While those
systems are well understood when they are weakly coupled, see [1, 7], the problem of phase transitions
in those systems, i.e. of non equilibrium phase transitions is quite open. Such systems possess also
stationary non equilibrium measures, the SRB measures, that are quite well understood and this can be
used to analyze some of the new ideas introduced in non equilibrium statistical mechanics, in particular
those discussed in workpackage 2.
2. Study of transport properties and relaxation in Hamiltonian dynamical systems.
Transport properties such as viscosity or heat conduction manifest themselves in spatially extended
Hamiltonian dynamical systems. The ergodic properties of these systems determine their frequency
spectrum. These ergodic properties rely on specific aspects of their dynamics such as dynamical
instability which may guarantee a continuous spectrum and therefore a monotonous decay of the time
correlation functions. Such decay can be understood in terms of Pollicott-Ruelle resonances. Recent
work has shown that these resonances can be used in the transport theory in order to give a framework
to such physical concepts as the dispersion relations of diffusion for instance [2, 5]. The modes
associated to these decay are known as the hydrodynamic modes which here find a more rigorous
foundation. In the present project, we intend to extend previous results on diffusion and reactiondiffusion systems to viscosity and heat conductivity. Such extensions would be based on the concept
of Helfand moment.
On the other hand, we intend to investigate the relationships between the transport properties in chaotic
dynamical systems in the escape-rate formalism where formulas have been proposed connection the
transport coefficients to the Lyapunov exponents, the fractal dimensions, and the Kolmogorov-Sinai
entropy per unit time [3]. These chaotic properties and the transport coefficients can be studied in
detail in model systems such as hard-ball systems [2,3,4].
3. Self organized criticality.
Another aspect where nonlinearities enter the present project has to do with what is commonly called
self-organized criticality. Organization is in a way an anti-theme to dissipation. In the preceding
decades, various toy-models have been proposed, ranging from specific systems of differential
equations to cellular automata. One issue that we plan to study is the emergence of power laws in the
statistics of waiting time distributions of various signals [6]. The standard sandpile models,
paradigmatic models of self-organized criticality, are not able to produce these power laws --- they
have Poissonian signals and exponentially distributed waiting time distributions. The signal here is
composed of large avalanches. We will study ways of driving the system, like in the deposition of
sand on the lattice, which could produce more interesting time sequences, without putting it in by hand.
References:
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[1] J. Bricmont, A. Kupiainen, High-temperature expansions and dynamical systems, Communications
in Mathematical Physics, 178, (1996), 703-732.
[2] P. Gaspard, I. Claus, T. Gilbert, and J. R. Dorfman, The Fractality of the Hydrodynamic Modes of
Diffusion, Physical Review Letters, 86 (2001), 1506-1509.
[3] P. Gaspard and G. Nicolis, Transport Properties, Lyapunov Exponents, and Entropy per Unit Time,
Physical Review Letters, 65 (1990), 1693-1696.
[4] P. Gaspard and H. van Beijeren, When do tracer particles dominate the Lyapunov spectrum?
Journal of Statistical Physics, 109 (2002), 671-704.
[5] C. Maes and E. Verbitskiy: Large deviations and a fluctuation symmetry for chaotic
homeomorphisms, Communications in Mathematical Physics, 233, (2003), 137-151.
[6] C. Maes, F. Redig, F. Takens, A. Van Moffaert and E. Verbitsky: Intermittency and weak Gibbs
states, Nonlinearity, 13, (2000), 1681-1698.
[7] C. Maes and A. Van Moffaert : Stochastic Stability of Weakly Coupled Lattice Maps , Nonlinearity,
10, (1997), 715-7303.
Workpackage 5: Integrable systems
Luc Haine and Pierre Van Moerbeke (UCL), Arno Kuijlaars and Walter Van Assche
(KULeuven), Joris Van der Jeugt (UGent)
1. Introduction and background
The theory of integrable systems originates from the study of dynamical systems where one can
identify certain nonlinear partial differential equations with special properties which allow for explicit
solution techniques. The prototype example is the Korteweg-de Vries equation which can be solved
using the inverse scattering transform. This discovery in 1967 was rapidly followed by many more
examples and a systematic theory of integrable (or soliton) equations emerged. This also includes
integrable difference equations such as the Toda lattice and integrable ordinary differential equations
which are known as Painlevé equations. The theory of integrable systems now has many connections
to many branches of mathematics and mathematical physics. Techniques from integrable systems
could be applied to problems outside of dynamical systems as well and a new sort of "integrable
mathematics" has emerged. The subject has strong ties to orthogonal polynomials and classical special
functions as well as the new special functions of Painlevé type.
2. Project description
The theory of integrable systems is a vast one. We will only deal with a number of aspects of it in this
project.
Zero dispersion KdV equation at shock time
If in the Korteweg-de Vries (KdV) equation we let the dispersion term tend to zero, we obtain the Hopf
equation which develops shocks in finite time. In the KdV equation with small dispersion the shocks
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are resolved by means of rapid oscillations [1]. We intend to study the zero dispersion limit at the time
of shock formation and show that the onset of oscillations is described by the second member of the
Painlevé I hierarchy. This would resolve a special case of a recent conjecture of Dubrovin [2] about
Hamiltonian perturbations of hyperbolic systems. Since we had experience with a similar situation in
random matrix models [3,4] we expect that similar techniques could be applied, at least for certain
initial conditions.
[1] P.D. Lax and C.D. Levermore, The small dispersion limit of the Korteweg-de Vries equation. I, II
and III, Comm. Pure Appl. Math. 36 (1983), 253-290, 571-593, 809-829.
[2] B. Dubrovin, On Hamiltonian perturbations of hyperbolic systems of conservation laws, II:
universality of critical behaviour, arxiv:math-ph/0510032.
[3] T. Claeys and A.B.J. Kuijlaars, Universality of the double scaling limit in random matrix models,
arxiv:math-ph/0501074, to appear in Comm. Pure Appl. Math.
[3] T. Claeys and M. Vanlessen, The existence of a real pole-free solution of the fourth order analogue
of the Painlevé I equation, arxiv:math-ph/0604046.
An extension of the Tracy-Widom distribution
The Tracy-Widom distributions [1] are central distribution functions in the modern theory of integrable
systems. They first arose as the distribution function for the largest eigenvalue of random matrices, but
later were found to be new universal limit laws for a wide variety of processes in combinatorial and
statistical mechanics problems [2,3]. The GUE Tracy-Widom distribution is related to the Painlevé II
equation with parameter 0. We are interested in a generalization of the Tracy-Widom distribution that
is related to the Painlevé II equation with general parameter. We want to relate this distribution
function to the largest eigenvalue of certain random matrices and possibly to other stochastic models.
[1] C.A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys.
159 (1994), no. 1, 151-174.
[2] J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing
subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), no. 4, 1119-1178.
[3] K. Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), no. 2,
437-476.
Master symmetries of discrete integrable equations and the bispectral problem
Lax-integrable equations are known to possess non-isospectral additional (master) symmetries, which
are intimately related with the bi-Hamiltonian nature of these equations. Master symmetries came back
to the forefront, with E. Witten’s 1991 discovery that the partition function of 2D-quantum gravity is a
solution of the Korteweg-de Vries (KdV) equation, which in addition is a fixed point of its master
symmetries. In its original form, the bispectral problem of J.J. Duistermaat and F.A. Grünbaum [1] is a
purely continuous and vast generalization of the notion of a family of classical orthogonal polynomials.
The problem is deeply connected with the KdV-equation [1] and its master symmetries [2]. We
propose to further explore the master symmetries of discrete integrable hierarchies, their relations with
bi-Hamiltonian structures and discrete bispectral problems as studied in [3,4]. We expect that this study
will reveal new relations with the representation theory of the Virasoro and W-algebras as already
shown in [5], their q-deformations, discrete versions of the string equations [3,6] and special solutions
of the Painlevé equations connected with random matrix theory [7,8].
[1] J.J. Duistermaat and F.A. Grünbaum, Differential equations in the spectral parameter, Comm.
Math. Phys. 103, 177-240 (1986).
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[2] F. Magri and J.P. Zubelli, Differential equations in the spectral parameter, Darboux
transformations and a hierarchy of master symmetries for KdV, Comm. Math. Phys. 141, 329-351
(1991).
[3] F.A. Grünbaum and L. Haine, Some functions that generalize the Askey-Wilson polynomials,
Comm. Math. Phys. 184, 173-202 (1997).
[4] L. Haine and P. Iliev, Askey-Wilson type functions, with bound states, Ramanujan J. 11, 285-329
(2006).
[5] L. Haine and E. Horozov, Toda orbits of Laguerre polynomials and representations of the Virasoro
algebra, Bull. Sc. Math., 2e série, 117, 485-518 (1993).
[6] T.H. Koornwinder, The structure relation for Askey-Wilson polynomials, arXiv:math.CA/0601303
v2 (2006).
[7] M. Adler, T. Shiota and P. van Moerbeke, Random matrices, Virasoro algebras, and
noncommutative KP, Duke Math. J. 94, no 2, 379-431(1998).
[8] L. Haine and J.P. Semengue, The Jacobi polynomial ensemble and the Painlevé VI equation, J.
Math. Phys. 40, no 4, 2117-2134 (1999).
The galoisian approach to the non-integrability of Hamiltonian systems.
S. Kowalevski’s famous method for obtaining a new case of integrability of the rigid body system with
a fixed point, imposing that the general solution is a meromorphic function of complex time, is still
very interesting and modern. The relation of her method with the complex Liouville integrability of
Hamiltonian systems is a difficult open problem. In [1], it was shown that if a complex Hamiltonian
system satisfies the Kowalevski-Painlevé property, then the monodromy around the poles of the
variational equations along any solution is trivial. An interesting question is to analyze the relation
between this result and more generally the notion of algebraic complete integrability [2] and the results
of J.J. Morales-Ruiz and J.P. Ramis [3], showing that complex Liouville integrability forces the
differential Galois group of the variational equation along any solution to be virtually abelian.
References:
[1] L. Haine, The algebraic complete integrability of geodesic flow on SO(N), Commun. Math. Phys.
94, 271-287 (1984).
[2] M. Adler, P. van Moerbeke and P. Vanhaecke, Algebraic integrability, Painlevé geometry and Lie
algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 47, Springer-Verlag Berlin
Heidelberg (2004).
[3] J.J. Morales-Ruiz and J.P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems,
Methods and Applications of Analysis 8, no 1, 33-96 (2001).
Orthogonal polynomials and integrable systems
It has been known classically that orthogonal polynomials, bi-orthogonal polynomials and multiple
orthogonal polynomials all can be expressed as a determinant of a moment matrix. Upon perturbing the
weight(s) by multiplication with an exponential of Σ t(i)z^i, the determinant of the moment matrix is
shown to satisfy hierarchies of integrable equations, like the KP equation, multi-component KP
equation, Toda lattices, etc. The Riemann-Hilbert matrix for the corresponding polynomials plays a
crucial role in this matter; see [1], [2], [3] and [4].
[1] M. Adler, P. Van Moerbeke and P. Vanhaecke: Multiple orthogonal polynomials, Multi-component
KP and non-intersecting Brownian motions, to appear.
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[2] A.S. Fokas, A.R. Its and A.V. Kitaev: The isomonodromy approach to matrix models and 2D
quantum gravity, Comm. Math. Phys. 147 (1992), no 2, 395-430.
[3] P. Deift, T. Kriecherbauer, K.T-R McLaughlin, S. Venakides, and X. Zhou: Uniform asymptotics
for polynomials orthogonal with respect to varying exponential weights and applications to
universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), no. 11, 13351425.
[4] E. Daems and A.B.J. Kuijlaars, Multiple orthogonal polynomials of mixed type and nonintersecting Brownian motions, arxiv:math.CA/0511470.
IAP – Phase VI
Annex I : Technical specifications – Section I
Page 30 of 37
Project P6/02
I. 5. PARTICIPATION OF THE PARTNERS IN THE DIFFERENT WORKPACKAGES
Tick off in the table the participation of the different partners in the different workpackages (delete
not used rows and columns in the table). Mention for each partner his/her name and the institution’s
abbreviation.
PARTNER
P1
P2
P3
P4
EU1
EU2
EU3
Name : P. VAN MOERBEKE
Institution : UCL
Name : P. GASPARD
Institution : ULB
Name : A. KUIJLAARS
Institution : KULeuven
Name : J. VAN DER JEUGT
Institution : UGent
Name : T. KRIECHERBAUER
Institution : RUB
Name : A. KUPIAINEN
Institution : HU
Name: B. Dubrovin
Institution: SISSA (Trieste)
IAP – Phase VI
WP1
WP2
WP3
WP4
WP5
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Annex I : Technical specifications – Section I
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Project P6/02
I. 6. MAIN SKILLS OF THE PARTNERS
Describe the main skills of each of the partners in relation to the project (15 lines maximum per
partner).
Delete not used lines.
P1
- Name : Pierre Van Moerbeke
Institution : UCL
Main Skills :This team consists of mathematicians and physicists. The expertise is as
follows
On the mathematics side (Pierre Bieliavsky, Luc Haine and Pierre Van Moerbeke) are:
- The study of integrable systems, discrete and continuous,
- master symmetries for integrable systems and the Virasoro algebra,
- matrix integrals, underlying integrable systems and orthogonal polynomials
- statistical distributions for Random matrices and critical infinite-dimensional diffusions
- master symmetries and bispectral problems
- solvable symmetric spaces and star-products
On the physics side (Jean Bricmont and Philippe Ruelle):
-the study of non equilibium statistical mechanics
-extended dynamical systems
-dynamical phase transitions
-general study of critical phenomena and their description by field theories
-conformal field theory
P2
-
- Name : Pierre GASPARD
Institution : ULB
Main Skills :The group is composed of physicists with expertise on:
Nonequilibrium statistical mechanics,
Nonlinear dynamics,
Theory of stochastic processes,
Quantum mechanics.
P3
- Name : Arnoldus KUIJLAARS
Institution : KULeuven
Main Skills : This team consists of mathematicians and physicists. The expertise is as
follows
Mathematics (Arno Kuijlaars and Walter Van Assche):
Random matrices
Riemann-Hilbert problems, asymptotics
Orthogonal polynomials and special functions
Painleve equations
Mathematical physics (Mark Fannes and Christian Maes):
Statistical physics
Nonequilibrium systems and complex phenomena
Quantum spin systems, quantum dynamical systems,
Quantum information theory
P4
- Name : Joris VAN DER JEUGT
Institution : UGent
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Annex I : Technical specifications – Section I
Page 32 of 37
Project P6/02
-
-
-
Main Skills :
Group theoretical methods in physics (use of Lie groups and Lie algebras in physical models).
Representation theory of Lie algebras, Lie superalgebras and quantum groups with applications in
mathematical physics (branching rule techniques; supersymmetric Schur functions and characters of Lie
superalgebra representations).
Special functions, orthogonal polynomials and applications (special functions of hypergeometric type
appearing in certain Lie group representations; relations and symmetry properties for ordinary and basic
hypergeometric series).
Binary coupling trees and 3nj-coefficients in coupling-recoupling theory (angular momentum algebra).
Numerical and computational methods, computer physics. - Wigner quantum systems (non-canonical
solutions of quantum systems), and generalizations of quantum statistics (generalizing para-Bose and paraFermi statistics).
EU1 - Name : Thomas KRIECHERBAUER
Institution : RUB
Main Skills :
- Random matrix theory, integrable systems, Riemann-Hilbert analysis,
- Operator theory, Szego limit theorems,
- Combinatorics and large deviations
EU2 - Name : Antti KUPIAINEN
Institution : HU
Main Skills :
- Turbulence,
- Stochastic Loewner equation,
- KAM theory
EU3 - Name : Boris DUBROVIN
Institution : SISSA
Main Skills :
- Integrable systems (classical and quantum)
- Topologocal field theory,
- Frobenius manifolds
IAP – Phase VI
Annex I : Technical specifications – Section I
Page 33 of 37
Project P6/02
I. 7. NETWORK ORGANISATION AND MANAGEMENT (4 pages maximum)
Describe the network’s organisation and the practical terms governing collaboration and interaction
between the partners (meetings, newsletters, doctoral school, ...).
The members of the network have a notable experience in scientific activities; they have organized numerous
international conferences on the subject, are members of steering committees of existing European networks
and have been in the scientific committees of international conferences.
The organisation of the network will include the following items:
A steering Committee will be responsible for the scientific strategy of the Network, for the organisation of
conferences in the area and for taking the important decisions concerning the network.
Large meetings: The plan is to have 2 meetings per year (fall and spring), where the researchers of the
network explain their work in a seminar; the lectures must be accessible to all the partners of the network,
including graduate students.
Weekly seminars: The partners of the network have already a number of existing seminars, which will be
further enlarged to include all partners, For instance;
-
random matrices (KUL-UCL)
Orthogonal polynomials and special functions (KUL-UCL)
seminar of the Interdisciplinary Centre for Nonlinear Phenomena and Complex Systems (ULB)
Non-linear seminar (UCL, together with physics, applied mathematics and CORE)
Graduate courses: We intend to have graduate courses and mini-courses, given by the faculty of the various
partner-universities and by visitors from abroad. They will aim at training graduate students and young
researchers in the field and will be integrated in the existing doctoral schools. When distinguished visitors will
lecture in our universities, we plan on organizing an informal lunch, encouraging direct contact between the
lecturer and graduate students/postdocs.
Thematic semesters: We plan on organising semesters with a definite subtheme, in conjunction with the
existing European networks on the subject (ENIGMA, MISGAM, STOCHDYN).
Website: A website with a weekly newsletter will inform the partners of seminars, courses, doctoral thesis
defenses and other activities. Also a site will be made, establishing a link with all the publications written within
the project.
International conference on random matrices and stochastic dynamics: P. Van Moerbeke plans on
organizing an international conference at the “Royal Academy of Belgium”, co-sponsored by ENIGMA (EU)
and MISGAM (ESF) in June 2007. The purpose is to put our graduate students and postdocs in close contact
with the international community in the field, among other things, by organizing poster sessions.
International conference on random matrices (Luminy): A. Kuijlaars is in the process of organizing an
international conference in November 2006 at Luminy, cosponsored by MISGAM (ESF), with a broad
participation of international experts.
Graduate students and postdocs: depending on the available budgets, we plan on hiring two graduate
students and one postdoc per university and per year for the 5 years. The priority will be on graduate
students.
IAP – Phase VI
Annex I : Technical specifications – Section I
Page 34 of 37
Project P6/02
I. 8. RE-ORGANISATION OF THE PROJECT (maximum 3 pages)
To be completed only if the initial proposal has to be adapted as a result of the selection outcome. If
this implies changes in the composition of the network and/or the budget, it may be that it is not
longer possible to pursue (achieve) the originally proposed objectives.
In this case, describe and clarify the re-organisation of the project compared to the initial proposal.
IAP – Phase VI
Annex I : Technical specifications – Section I
Page 35 of 37
Project P6/02
I. 9. BUDGET (global distribution per partner for the 5 years)
(in EURO, without decimals)
The detailed distribution per partner is given in Section II
Name Partner
Institution
Budget
P1
Pierre VAN MOERBEKE
UCL
800 000,00 EUR
P2
Pierre GASPARD
ULB
600.000,00 EUR
P3
Arnoldus KUIJLAARS
KULeuven
600.000,00 EUR
P4
Joris VAN DER JEUGT
UGent
500.000,00 EUR

Thomas KRIECHERBAUER
RUB
34.000,00 EUR

Antti KUPIAINEN
HU
33.000,00 EUR

Boris DUBROVIN
SISSA
33.000,00 EUR
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
P16
EU1
EU2
EU3
EU4

TOTAL BUDGET

2.600.000,00 EUR
The budget for the EU-partner is the budget attributed by the IAP-programme only (without the 50%
contribution of the EU-partner)
IAP – Phase VI
Annex I : Technical specifications – Section I
Page 36 of 37
BELGIAN
SCIENCE
POLICY
Project P6/02
Wetenschapsst
raat 8 rue de la
Science
B-1000
BRUSSELS
I. 10. PREVIOUS IAP-PHASES
Tel. +32 2 238
34 11  Fax
+32 2 230 59
To be completed only if the present network
12
programme.
www.belspo.be
was funded during earlier phases of the IAP
Mention the earlier phases of the IAP programme (I, II, III, IV, or V) and the titles of projects in which
the partners of the present network has participated.
IAP – Phase VI
Annex I : Technical specifications – Section I
Page 37 of 37