Theoretical Statistics and Mathematics Unit, ISI Delhi
This is a P. C. Mahalanobis memorial lecture and is meant for a general audience.
Metastability is the phenomenon where a system, under the influence of a stochastic dynamics, moves between different subregions of its state space on different time scales. Metastability is encountered in a wide variety of stochastic systems. The challenge is to devise realistic models and to explain the experimentally observed universality that is displayed by metastable systems, both qualitatively and quantitatively.In statistical physics, metastability is the dynamical manifestation of a first-order phase transition. In this talk I give a brief historical account of metastability in this context. After that I describe the metastable behaviour of one particular model, namely, the Widom-Rowlinson model on a two-dimensional torus subject to a Metropolis stochastic dynamics. In this model, particles are randomly created and annihilated inside the torus as if the outside of the torus were an infinite reservoir with a given chemical potential. The particles are viewed as points carrying disks, and the energy of a particle configuration is equal to the volume of the union of the disks, called the "halo" of the configuration. Consequently, the interaction between the particles is attractive. We are interested in the metastable behaviour at low temperature when the chemical potential is supercritical. In particular, we start with the empty torus and are interested in the first time when we reach the full torus, i.e., the torus is fully covered by disks. In order to achieve the transition from empty to full, the system needs to create a sufficiently large droplet of overlapping disks, which plays the role of a "critical droplet'' that triggers the crossover. In the limit as the temperature tends to zero, we compute the asymptotic scaling of the average crossover time, show that the crossover time divided by its average is exponentially distributed, and identify the size and the shape of the critical droplet. It turns out that the critical droplet exhibits "surface fluctuations", which need to be understood in order to obtain a fine estimate of the crossover time.