Metastability for serial and parallel dynamics

Metastability is an ubiquitous phenomenon in nature. It appears in a plethora of diverse fields including physics, chemistry, biology, computer science, climatology and economics. Metastability is best described as a dynamical phenomenon that occurs when a system is close to a first order phase transition. After changing some thermodynamic parameters, the system remains for a considerable (random) time in the old phase, the metastable state, before suddenly making a transition to the new phase, the stable state. In other words, on a short time scale, the system behaves as if it was in equilibrium, while, on a long time scale, it moves between different regions of the state space. At low temperature, this motion is preceded by the appearance of a critical mesoscopic configuration of the system via a spontaneous fluctuation or some external perturbation. Thus, when the system is initiated in the metastable phase, it starts its long transition towards the stable phase. In particular, it must overcome an energy barrier to reach the stable state starting from the metastable state. Formally, an Hamiltonian or energy function and the associated dynamics characterized the detailed evolution of the system. Moreover, it is possible to define an equilibrium measure based on the Hamiltonian, for example the Gibbs measure. If the dynamics satisfies the detailed balance condition, then it is reversible with respect to this equilibrium measure. There are three interesting questions that are typically investigated in metastability. The first is the study of the transition time from the set of metastable states to the set of the stable states, i.e., the time necessary to arrive at the equilibrium phase. The second issue is the identification of the so-called critical configurations that the system creates in order to reach equilibrium. The third question concerns the study of the typical paths that the system follows with high probability during the transition from the metastable state to the stable state. The thesis is organized in five chapters. In the first two chapters, we present the different approaches and results on asynchronous (serial) and synchronous (parallel) dynamics, in Chapters 2 and 3 respectively. In particular, at the end of the third chapter, we present one of the novelties of this work: an estimate of the mixing time and of the spectral gap, and the computation of the prefactor for the mean transition time also in the case of a series of degenerate metastable states. Next, we study three models evolving under different dynamics. In Chapter 4, we examine the Ising model on the hexagonal lattice with a serial non-conservative dynamics, Glauber dynamics. In particular, we prove some model-dependent results that together with the results of Chapter 2 yield the desired metastability theorems. Indeed, we identify the metastable states proving the recurrence property and computing the maximal stability level. In addition to this, we give a geometrical description of the critical configurations and we show how these are related to polyiamonds. The characterization of the shape of the critical configurations allows us to estimate the mean transition time via potential-theoretic approach. In Chapter 5.1, we describe a particular Probabilistic Cellular Automata model to represent the metastable behavior of a system subject to parallel dynamics. In particular, using our model-independent results at the end of Chapter 3, we show the behavior of the mixing time and the spectral gap, and we find a rigorous estimate of the expected hitting time. In addition, we identify the metastable states proving recurrence property and classifying the configurations according their stability level. Finally, in Chapter 6, we study the Blume-Capel model evolving under a serial conservative dynamics, the Kawasaki dynamics. These results are still quite heuristic since this is an ongoing project. We present an heuristic study of the phase-diagram and we explain the behavior of the system showing which should be the stable and the metastable states. We suggest an idea based on the crucial role of the boundary to estimate the stability level, to prove the recurrence property and to show the asymptotic behavior of transition time.