Dynamics of the Statistical Mechanics Models

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Proposal: DMS-9800860 Principal Investigator: Senya Shlosman Abstract: Shlosman will study the behavior of a system subject to stochastic dynamics in the situation where the initial state of the system is far from equilibrium. The typical example is the Glauber dynamics of the model of statistical mechanics, which model is taken with parameters close to (but different from) those of the phase transition region. For a long time the system behaves as if it were in the "wrong" equilibrium, until it finally undergoes the transition to the "true" (and unique) one. The (non-equilibrium!) states of the model before transition are what should be considered the metastable states of the system. Shlosman shows that such family of states is a smooth continuation of the curve of equilibrium phases through the critical point. The transition happens because of the creation and growth of "droplets" of the equilibrium phase inside the non-equilibrium phase. The PI will study the lifetime of these metastable states. It is possible to write an explicit expression describing it. The formula is quite interesting and contains, in particular, a term corresponding to the surface energy of the Wulff droplet. The explanation is that the shape of the critical droplet is given by the Wulff construction. This settles a controversy on the subject that has existed in the literature. The subject of this research project is the rigorous study of the phenomenon of metastability. A good example of the metastable state of matter is supercooled water (i.e., water cooled below its freezing temperature, yet remaining liquid). This state of water can indeed be observed in nature. The characteristic feature of supercooled water is that it cannot stay liquid for a very long time: after a while, it suddenly freezes. The concept of metastable state is well known in physics, where it plays an important role. What has hitherto been lacking is the rigorous understanding of this phenomenon on the level of microscopic statistical mechan ics. As one famous scientist put it, "The concepts of the metastable state and of fractional dimension are useful, provided they are not taken too seriously." Shlosman's research is aimed precisely at providing the tools to understand metastability one hundred percent "seriously." To this end, the concept of a "critical droplet" (which is a microscopic piece of ice in the case of water) is developed. Shlosman will study the process through which such a droplet is created and examine the process of growth of this droplet, until it eventually absorbs the whole sample (by coalescing with other growing droplets). The overall theory looks quite complete and satisfactory.