KCL Disordered Systems Group

Mathematics Department Disordered Systems Group

Academic Staff

Emiriti, Postdocs, Long-Term Visitors

PhD Students

Former group members:

CWH Mace
JM van Mourik (ongoing collaboration)
NS Skantzos (ongoing collaboration)
JAF Heimel
H Chakravorty
A Speranza
JPL Hatchett (ongoing collaboration)
T Nikoletopoulos (ongoing collaboration)
P Mayer
V Del Prete
M Fasolo

Research Activities

The research activities of the group concentrate on the analysis and development of mathematical theories and models with which to describe the statics and dynamics of disordered (or `complex') systems in physics, biology, financial markets, and computer science. Such systems are characterized by microscopic (usually stochastic) dynamic elements with mutual interactions without global regularity but with a significant degree of built-in competition and incompatibility, resulting in the existence of many locally stable states for the system as a whole and a highly non-ergodic `glassy' type of dynamics.

In the field of neural network theory we presently focus on the three areas of (i) the operation of large (symmetric and non-symmetric) recurrent neural networks, (ii) the dynamics of learning in layered neural networks, and (iii) the analysis of neural networks with a coupled dynamics of fast neurons and (adiabatically) slow synapses.

In the area of disordered (or `complex') physical and biological systems we presently study the statics and dynamics of range-free models for spin-glasses (disordered magnetic systems with frustrated exchange interactions), disordered systems of coupled oscillators, models describing cooperative phenomena in populations of trading agents (minority game), kinetically constrained spin systems, and the dynamics of so-called second generation immune system network models. This includes interacting systems with finite connectivity and processes on complex and 'small-world' networks ; see also the Finite Connectivity Workshop at King's College (November, December 2003).

The `glassy' features of disorder and metastability are also essential properties of many `soft materials' (emulsions, foams, dense colloidal suspensions, pastes, slurries). Models for the mechanical behaviour of these `soft glasses' have been developed on this basis, combining a glass transition with the possibility of deformation and flow, and reproducing many important features of the experimental observations.

We are also looking at the effects of polydispersity: In a colloidal suspension, for example, there are normally particles with a continuous range of radii; one thus has an effectively infinite number of distinguishable particle species. This produces interesting kinds of disorder (a dense polydisperse colloid may form a glass, for example, while a monodisperse system would crystallize) and unusual phase behaviour. Dr Sollich is the present coordinator of the Polydispersity Working Group of the UK Soft Condensed Matter Network (see also the Polydispersity Working Group Meeting ) at King's College.

Finally, the group is also involved in the study of protein folding and protein interaction networks. Natural proteins are linear hetero polymers made up out of 20 species of amino acids, and they are involved in virtually any biological function. Random amino acid sequences exhibit many of the essential features of glasses such as disorder, metastability and freezing transitions; natural proteins are specially selected sequences (by nature) which fold into a unique 3-D configuration: the 'native' state. The theoretical study of individial proteins aims at solving the 'protein folding puzzle': predict the native state from the amino acid sequence. At a larger `proteomic' scale, our mathematical analysis concerns the study of interaction processes on protein networks. Similar to networks in areas such as the internet, these were found to have scale-free (i.e. power law distributed) connectivities, as well as the so-called `small world' property.

The mathematical methods used are mainly those of equilibrium and non-equilibrium statistical mechanics and stochastic processes. Within this class a prominent role is played by the tools developed in the disordered systems community, such as replica theory a la Parisi, path integrals and generating functionals (i.e. dynamic mean-field theory) a la De Dominicis, and dynamical replica theory. In the case of neural networks one also finds frequent use for concepts from information theory (e.g. information geometry).

(last update: August 18th 2007 by ACCC)