A central focus of my work is the development of multi-scale methods for dynamic problems.
Three aspects of this subject are:
- Dynamics of discrete (atomic) models. This
fascinating field can be traced back to a very
innocent-looking computer experiment by Fermi, Pasta, and Ulam
in 1953: they considered an chain model of 64 vibrating
identical atoms linked end to end by springs. The hallmark of
their experiment was to consider nonlinear
springs. The results were very surprising; they are
beautifully described in the article The
Real Scientific Hero of 1953 in the New York Times, by Steven
Strogatz. There has been significant progress in the analysis
of the Fermi-Pasta-Ulam chain over the last few years, but
many problems remain open.
For solid-solid phase transitions, the situation is even more interesting
(to me): the springs are not only nonlinear, but also
non-monotone. This makes many mathematical tools
break down, and the analysis becomes even more
challenging. Yet, such an analysis might reveal a kinetic
relation, which relates the velocity of a phase boundary
to an applied force. In Engineering, kinetic relations are
often phenomenologically assumed. Can a mathematical analysis
reveal kinetic relations in a rigorous manner?
- Gradient flows.
I am generally interested in nonequilibrium problems and the derivation of thermodynamic quantities such as entropy from microscopic models. A starting point is here the analysis of the over-damped limit of an evolution, that is, a context where the damping is so strong that inertial effects can be neglected. Then the behaviour of many materials can be described by the desire to reduce their energy. We call an evolution a gradient flow when, loosely speaking, a dynamical system moves at any time in the direction of the steepest descent of the energy. Here a fascinating problem is to derive rigorously the effective energy and evolution equation for a given microscopic model. For example, it is known that the diffusion equation can be interpreted as gradient flow of the entropy in the Wasserstein metric. How can we link this description directly to an underlying model of Brownian motion? This link shows that entropy is indeed macroscopically the driving force, and explains why the Wasserstein metric occurs. The argument combines probabilistic and analytic techniques, namely Large Deviation Principles and Gamma-convergence.
- Mathematical chemistry. The dynamics of atoms and molecules pose challenging mathematical problems. Many problems can be described as a potential energy landscape with many wells, separated by barriers. One then often wants to find a trajectory joining a given initial configuration with a given final one. For example, the two configurations can be different conformational states of a molecule. The dynamics is complicated: Typically, the trajectories will jostle around in the well belonging to the initial configuration, before a rare spontaneous fluctuation occurs and the trajectory crosses the barrier and reaches the next valley of the energy landscape. How can we compute these rare events?
I joined the Department of Mathematical
Sciences at the University
of Bath from the Max
Planck Institute for Mathematics in the Sciences in Leipzig,
Germany. In Leipzig, I was head of the Emmy Noether Group
`Mathematical Analysis of the Static and Dynamic Behaviour of
Materials with Phase Transitions and Microstructures'. Before
that, I was a postdoc at the California Institute of
Technology, Division of
Engineering and Applied Science.
More details can be found in the cv.