Describe the project’s objectives and research goals.
Define the problems being addressed by positioning them in relation to the current state of knowledge.
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 non-intersecting 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.
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.
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.
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.
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 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.
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.