Metastability

Abstract

In these lectures we will discuss Markov processes with a particular interest for a phenomenon called metastability. Basically this refers to the existence of two or more time-scales over which the system shows very different behaviour: on the short time scale, the systems reaches quickly a “pseudo-equilibrium” and remains effectively in a restricted subset of the available phase space; the particular pseudo-equilibrium that is reached will depend on the initial conditions. However, when observed on the longer time scale, one will occasionally observe transitions from one such pseudo-equilibrium to another one. In many cases (as we will see) there exists one particular time scale for each such pseudo-equilibrium; in other cases of interest, several, or even many, such distinct pseudo-equilibria exist having the same time scale of exit. Mathematically speaking, our interest is to derive the (statistical) properties of the process on these long time scales from the given description of the process on the microscopic time scale. In principle, our aim should be an effective model for the motion at the long time scale on a coarse grained state space; in fact, disregarding fast motion leads us naturally to consider a reduced state space that may be labeled in some way by the quasi-equilibria.The type of situation we sketched above occurs in many situations in nature. The classical example is of course the phenomenon of metastability in phase transitions: if a (sufficiently pure) container of water is cooled below freezing temperature, it may remain in the liquid state for a rather long period of time, but at some moment the entire container freezes extremely rapidly. In reality, this moment is of course mostly triggered by some slight external perturbation. Another example of the same phenomenon occurs in the dynamics of large bio-molecules, such as proteins. Such molecules frequently have several possible spatial conformations, transitions between which occur sporadically on often very long time scales. Another classical example is metastability in chemical reactions. Here reactants oscillate between several possible chemical compositions, sometimes nicely distinguished by different colours. This example was instrumental in the development of stochastic models for metastability by Eyring, Kramers and others [21, 30]. Today, metastable effects are invoked to explain a variety of diverse phenomena such as changes in global climate systems both on earth (ice-ages) and on Mars (liquid water presence), structural transitions on eco- and oeco systems, to name just a few examples. KeywordsMarkov ChainMarkov ProcessInvariant MeasureDirichlet ProblemSmall EigenvalueThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.