Metastability

Many stochastic systems — chemical reactions, glasses, magnetic spin models, certain Markov chain Monte Carlo samplers — spend long stretches of time fluctuating around a particular equilibrium before, eventually, being driven by noise to a different one. The mathematical study of this phenomenon — the timescales between transitions, the typical paths followed, the limiting laws of the waiting times — is metastability.

Quantitative statements typically take the form of an Eyring–Kramers formula: the mean time to escape an energy well of depth \(h\) at temperature \(T\) is, to leading order, \(C \exp(h/T)\), with a prefactor \(C\) determined by the local curvature of the potential at the well and at the saddle. Behind the scenes one usually wants log-Sobolev or Poincaré inequalities adapted to the multi-well geometry.

I am interested in metastability for diffusion processes and continuous-spin lattice systems, particularly in how metastable timescales depend on the geometry of the underlying potential landscape and on parameters of the physical model.