2020 Amick Lecture

Charles Amick Memorial Lectures in Applied Mathematics

Monday, February 10

Lecture One: Small Random Perturbation of Dynamical System: Reversible Model
Abstract: Dynamical systems that are perturbed by small random noises are known to exhibit metastable behavior. The average transition time between metastable states of a reversible diffusion process is described at the logarithmic scale by Arrhenius’ law. Eyring-Kramers formula classically provides a subexponential prefactor to this exponential estimate in the case of reversible diffusions. A scaling limit of the diffusion yields a Markov chain whose states are the metastable states, and whose jumps rates are described with the aid of Eyring-Kramers formula.

Tuesday, February 11

Lecture 2: Small Random Perturbation of Dynamical System: Irreversible Model
Abstract: In a more probabilistic language, Arrhenius’ law is a large deviation principle that can be established even for irreversible diffusions by the Freidlin-Wentzell theory. When the drift of this diffusion is gradient, the model is reversible and the metastable states correspond to the minima of the potential. Otherwise the quasi-potential of Freidlin and Wentzell plays the role of the potential. When the drift is certain perturbation of a gradient vector field, then metastability phenomena is expected to occur. Though this has not been rigorously verified except in dimension one. Some intersecting new behavior has been observed when the potential is a Hamiltonian vector field in dimension two.

Wednesday, February 12

Lecture Three: Metsatability and Condensation
Abstract:Zero Range Processes (ZRPs) are stochastic particle systems that are particularly tractable mathematically because of their simple interaction mechanism and explicit equilibrium states. If the particles reside on a periodic lattice of L many sites, we may recast ZRP as a random walk on a L-1-dimensional lattice (a discrete analog of models we have discussed in the previous lectures). When the interaction between particles is sufficiently attractive, particles tend to pile up at one site i.e., condensation occurs. As the number of particles N, and the number of sites L increase with N/L>\rho_c, for a critical density \rho_c, then particles condense at a unique random site x(t), for most the time t. After a suitable rescaling of the time, the location of the condensate would follow a macroscopic evolution that is given by a L\'evy process. The L\'evy measure of this process can be explicitly described in terms of the microscopic details of the underlying ZRP.