Nonconvex and random effects in complex energy landscapes

• Applicant: Professorin Dr. Maria G. Westdickenberg
• Subject Area: Mathematics
• Term: from 2015 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 273745410

In this proposal we apply variational methods and analytical techniques to deterministic and stochastically perturbed problems in applied analysis. Applied scientists have long confronted complex energy landscapes such as microstructure formation in materials or conformational changes of bio-molecules, often needing to rely on direct monte-carlo simulations or hybrid schemes with an ad hoc method for bridging scales. The questions coming from the applications require that we develop new tools in applied analysis to handle features of complex energy landscapes such as local energy minimizers, energy barriers, and metastability. Even in the deterministic setting, there are tough questions such as which states are dynamically accessible or how to measure the size of different regions of phase space. In the stochastic case, we need methods to quantitatively measure phenomena such as the competing effects of energy and entropy in spatially extended systems.We propose to study four specific subtopics related to the shape of the energy landscape, the timescales of the associated deterministic gradient flow, and the effect of random perturbations. We identify a natural model problem within each subtopic. Our models are nonlinear and generally chosen to be the simplest possible models in which one can find a given universal behavior of interest. Our goal is to find clean conditions and---whenever possible---elementary methods of proof. In addition, we emphasize developing tools---in particular, tools that are well-suited to generalization to more complicated problems.

• DFG Programme: Research Grants