Relaxation height in energy landscapes : an application to multiple metastable states

The study of systems with multiple (not necessarily degenerate) metastable states presents subtle difficulties from the mathematical point of view related to the variational problem that has to be solved in these cases. We introduce the notion of relaxation height in a general energy landscape and we prove sufficient conditions which are valid even in presence of multiple metastable states. We show how these results can be used to approach the problem of multiple metastable states via the use of the modern theories of metastability. We finally apply these general results to the Blume-Capel model for a particular choice of the parameters ensuring the existence of two multiple, and not degenerate in energy, metastable states.