Fluctuations in Classical and Quantum Fields

The basic subject of this project proposal is the theoretical investigation of fluctuating nonequilibrium systems involving many degrees of freedom. I hope to pursue this project in the group of Prof. Mehran Kardar in the Department of Physics at the Massachusetts Institute of Technology (MIT). We plan to study problems related to growing surfaces, biological systems, and fluctuating quantum fields. These topics match the interests of Prof. Kardar very closely, and, hence, Prof. Kardar`s group at MIT would provide an excellent academic environment for this project. The research proposal consists of three main sections describing the research goals in detail: Section 3 of this proposal discusses problems which can be addressed on basis of a recently introduced multiscale method for the analysis of stochastic lattice models. An exciting direction for future research is to develop, on basis of this method, a first-principles multiscale theory for the analysis of surface growth, connecting quantum mechanics to the large-scale properties of surface morphologies. In the context of quantum dot formation, such a multiscale theory would relate atomistic kinetics directly to continuum elasticity, which could lead to an improved understanding of the role played by fluctuations during the self-organization of nanostructures. Similar ideas could find fruitful applications in biological systems, for example in the fields of epidemiology and tumor growth. Section 4 is concerned with the interplay between growth and order. We start by constructing a general mathematical model connected to several fundamental problems in statistical mechanics. Applying these ideas to surface growth, we would like to investigate the meaning of "roughness" in amorphous growth in the presence of an order parameter describing lattice deformations. Moreover, this prototype model most likely permits the formation of domain walls, which leads us to the question of how particles or lines slide on growing surfaces, a problem with far-reaching connections to the advection of passive scalars in turbulent flows. Section 5 deals with problems related to the Casimir force generated by fluctuations in the quantum electromagnetic field in the presence of bounding surfaces. From the point of view of applications, the Casimir force has profound implications for nanoscale devices in static as well as dynamic setups. Our basic aim is to investigate the Casimir effect for different surface morphologies and shapes, so as to provide a guide for future designs of nanodevices. This is likely to necessitate the development of new methods for path integral quantization. The above research aims have direct relevance for research conducted in Austria in nanotechnology and on the optical properties of quantum dots. The research proposal concludes with a summary of the various goals and a tentative project timetable.