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Nonlinear systems, stochastic processes, and statistical mechanics

Research project P6/02 (Research action P6)

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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 non-equilibrium 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…).

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 (ENIGMA-UCL/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 cross-fertilization 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 PAI-IUAP 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.

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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