The project PDE Models for Nanotechnology started in 2013 and is currently scheduled to run till 2019. The subject are deterministic and stochastic partial differential equations (PDEs) with applications in nanoscience and nanotechnology. The project is funded by the START program of FWF.
In nanoscience and nanotechnology, there are many timely applications that give rise to multiscale problems in a natural manner. Therefore we also concern ourselves with the homogenization of deterministic and stochastic PDEs.
Noise, fluctuations, process variations, and uncertainty quantification in general are another area where stochastic PDEs are essential. In short, we go beyond deterministic models and consider more general model equations that include dependencies on random variables.
The mathematical aspects include
proving existence and uniqueness,
deterministic and stochastic homogenization problems, and
developing numerical methods, especially for stochastic PDE.
Important model equations in this context include, e.g.,
the Boltzmann transport equation and derived transport equations,
linear and nonlinear elliptic (Poisson) equations, and
systems such as the stochastic drift-diffusion-Poisson system.
Both deterministic and stochastic homogenization problems occur naturally in nanoscience and nanotechnology due to the often vastly different length scales of whole devices and their microscopic structure. Multiscale problems also arise when modeling materials with a periodic or random microscopic structure.
The applications include, e.g.,
nanowire bio- and gas sensors as more general structures than nanoscale transistors,
confined structures with large aspect ratios and hence posing multiscale problems, and
materials with a microscopic structure that is to be designed.
I believe that mathematical modeling and numerical simulations of real-world problems are an important contribution to the rational design of new devices and technologies.
The ultimate goals are
to provide new mathematical models for multiscale problems and uncertainty quantification,
to develop new simulation capabilities based on these mathematical models, and
to provide quantitative understanding for the development of new technologies.