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. In this context, we define the term "particle systems" very broadly and also include point processes with percolation properties and random graph structures as well as Gibbs interactions. It also includes random movements of these particles, such as those that occur in spatial models for communication. At WIAS, many macroscopic properties of these particle systems are investigated that arise from the microscopic rules, such as phase transitions (condensation, percolation, crystallization) and critical properties such as rescaling limits. [>> 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. This probabilistic theory is also indispensable in the treatment of a number of models in statistical physics, as it makes them accessible for analysis using variational techniques. Both theory and sophisticated applications in physics and chemistry are being investigated at WIAS. [>> more]

Systems with many random components distributed in space (points, edges, graphs, trajectories, etc.) with many short- or long-range interactions are examined at the WIAS for their macroscopic properties. Particular attention is paid to the formation of particularly large structures in the system or other phase transitions. [>> 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. At WIAS, methids from the calcuus of variations are applied and further developed to solve problems in physics and technology such as continuum mechanics, quantum mechanics, and optimal control. [>> more]

Archive

Further mathematical research topics where the institute has expertise in:

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]

Spectra of random operators are used to describe physical processes and phenomena. Our research in this area focuses on the mathematical investigation of concentration phenomena of spectral states at the edge of the spectrum using probabilistic methods (Feynman-Kac formula, large deviations) and their effects on electrical conduction properties of a metal alloy described by the model. These investigations are combined with the analysis of the analogous probabilistic models (random walks in random potential) and thus lead to further results of independent interest. [>> more]