This research contains several projects that are united by the common theme: interaction of a physical system with the environment. While a system of interest is often relatively simple, the environment it interacts with is often complicated and requires a judicious choice of a mathematical model. A model must be rich enough to capture essential features of the underlying phenomena; at the same time, the model needs to be mathematically tractable. In some of the models under investigation, the environment manifests itself as static disorder. This is in particular the case in models of activated carbon. One of the challenges in this particular investigation is to understand appearance of graphene sheets in activated carbon production. Graphene, a material that consists of just one layer of carbon atoms forming two-dimensional sheets, demonstrates a plethora of remarkable properties just beginning to be employed in engineering. This work, which has both theoretical and strong applied potential, will be done in collaboration with scientists working in industry. The long-term goal is to understand statistical mechanics in the presence of noise and disorder, which will enable study of more complicated systems relevant for applications in life sciences. The project will involve training of students through involvement in the research.
In the majority of systems under study in this project, interaction with the environment takes the form of noise, modeled as a random perturbation, varying in time. Both classical and quantum systems will be modeled this way. In the case of classical systems, the project studies the qualitative impact of noise in systems with magnetic and time-dependent forces and, more generally, bifurcations resulting from the presence of noise. Of particular interest are also systems of large number of interacting particles. Previous work on one such system has possible applications for design of systems of task-performing micro-robots; that research will be continued in the present project. The theory of open quantum systems is much less developed; in this context the project investigates simple quantum optical systems, as well as the basic phenomenon of quantum Brownian motion. Particularly interesting physically is the case in which the strength of the interaction of the Brownian particle with the environment depends on its position. Mathematical models of this situation are not well understood and the project aims to improve the existing theory. More involved systems will also be studied, especially in the limit of small particle masses. The research on diffusive and open systems will be done in close collaboration with physics groups, both experimental and theoretical.
