Interacting Random Systems

The group contributes to the following mathematical research topics of WIAS:

In the mathematical modeling of many processes and phenomena in the Sciences and Technology one employs systems with many random particles and interactions. [>> more]

The theory of large deviations, a branch of probability theory, provides tools for the description of the asymptotic decay rate of a small probability, as a certain parameter diverges or shrinks to zero. Examples are large times, low temperatures, large numbers of stochastic quantities, or an approximation parameter. [>> more]

Kinetic equations describe the rate at which a system or mixture changes its chemical properties. Such equations are often non-linear, because interactions in the material are complex and the speed of change is dependent on the system size as well as the strength of the external influences. [>> more]

At WIAS, a number of models of the natural sciences are examined that live from a close interlinking of the random and spatial components, i.e., in which phenomena are observed that mainly arise from the interaction of random influences and spatiality. These include interacting systems of paths or graphs or other structures in space that are used to describe condensation or coagulation processes, or spatial branching processes in random media, to name just a few examples. The aim is to develop mathematical methods for the macroscopic description of such systems. [>> more]

Many questions about processes and phenomena are described by random operators, most notably by the random Schrödinger operator (Anderson operator) and the Laplace operator with random weights (conductances) on the edges between nearest neighbours. [>> more]

Many physical phenomena can be described by suitable functionals, whose critical points play the role of equilibrium solutions. Of particular interest are local and global minimizers: a soap bubble minimizes the surface area subject to a given volume and an elastic body minimizes the stored elastic energy subject to given boundary conditions. [>> more]

Archive

Further mathematical research topics where the institute has expertise in:

Large random interacting particle systems are mostly defined either by coupled systems of stochastic differential equations or via Hamiltonian functions. [>> more]