(B. Mehlig, J. Steif, B. Wennberg)
How do macroscopic phenomena (such as patterns of particles in random flows, or clusters in perco-lating systems) arise from microscopic laws and interactions in stochastic systems?
1. Particles in random flows (Mehlig, Ostlund). Our work on the dynamics of particles in random flows has raised questions of interest to the platform: first, we were led to study a random 3 x 3 matrix process. A mapping onto the quantum problem of interacting harmonic oscillators enabled us to employ algebraic methods. Our analysis needs to be complemented by a WKB approach. This would not only shed light on the problem at hand, it could also clarify open questions in WKB analysis of quantum systems. Second, we could diagonalise another Hamiltonian arising in this context by means of non-linear raising and lowering operators. We need to elucidate possible conncetions to the work on "follytons" by A. Laptev.
2. Interacting particle systems (Steif). We have studied properties of Gibbs measures in statistical mechanical models. These studies apply to systems in equilibrium, there is no dynamics in this context. By contrast, time evolution is introduced in the study of interacting particle systems. An example is dynamical percolation. We have shown that this model at criticality has exceptional times at which percolation can occur. We would like to extend these results to other critical models.
3. Kinetic theory (Wennberg). The Boltzmann equation is a bridge between the microscopic, reversible dynamics of a particle system, and its irreversible macroscopic behaviour. On the one hand, we are interested in obtaining rigorous results connecting microscopic models and kinetic equations, and on the other hand we study the Boltzmann equation in situations far from equilibrium, with applications in the study of suspended particle systems.