Persistence and Life Time Distribution in Coarsening Phenomena

We investigate the life time distribution P (τ, t) in one dimensional and two dimensional coarsening processes modelled by Ising-Glauber dynamics at zero temperature. The life time τ is defined as the time that elapses between two successive flips in the time interval (0, t) or between the last flip and the observation time t. We calculate P (τ, t) averaged over all the spins in the system and over several initial disorder configurations. We find that asymptotically the life time distribution obeys a scaling ansatz: P (τ , t) = t−1φ(ξ), where ξ = τ/t. The scaling function φ(ξ) is singular at ξ = 0 and 1, mainly due to slow dynamics and persistence. An independent life time model where the life times are sampled from a distribution with power law tail is presented, which predicts analytically the qualitative features of the scaling funtion. The need for going beyond the independent life time models for predicting the scaling function for the Ising-Glauber systems is indicated. Coarsening phenomenon [1], is a simple example of a dynamical process which is slow, and which becomes slower as time proceeds. This phenomenon is found in several nonequilibrium systems, e.g. phase separation in binary alloys, grain growth, growth of soap bubbles, magnetic bubbles etc. A striking feature of the coarsening phenomenon is the dynamical scale invariance. The domain structure at different times are statistically similar to each other but for a rescaling length L(t). The rescaling length can be taken as the typical linear size of a domain and it increases with time as t. It was soon realized that the dynamical scaling exponent z is not adequate to describe completely the coarsening dynamics, see below. An interesting question asked in this context concerns the persistence probability P0(t), that a local order parameter does not change its sign until the observation time t. In other words, P0(t) refers to the fraction of the total volume that remains unswept by the domain walls until time t, in the context of three dimensional coarsening phenomenon. This quantity, exhibits, asymptotically, a power law decay, P0(t) ∼ t , where θ is called the persistence exponent, and it is independent of z, the dynamical scaling exponent. The persistence phenomenon has a long history. The earliest question on persistence was perhaps asked and answered by W. A. Whitworth in 1878 and later by J. Bertrand in 1887, see Feller [2]. The question relates to two canditates P and Q, who poll p votes and q votes respectively, with canditate P winning the election by a margin of x = p− q votes.