Persistence in systems with conserved order parameter

Abstract

We consider the low-temperature coarsening dynamics of a one-dimensional Ising ferromagnet with conserved Kawasaki-like dynamics in the domain representation. Domains diffuse with sizedependent diffusion constant, D(l) ∝ l γ with γ = −1. We generalize this model to arbitrary γ, and derive an expression for the domain density, N (t) ∼ t −φ with φ = 1/(2 − γ), using a scaling argument. We also investigate numerically the persistence exponent θ characterizing the power-law decay of the number, Np(t), of persistent (unflipped) spins at time t, and find Np(t) ∼ t −θ where θ depends on γ. We show how the results for φ and θ are related to similar calculations in diffusionlimited cluster-cluster aggregation (DLCA) where clusters with size-dependent diffusion constant diffuse through an immobile 'empty' phase and aggregate irreversibly on impact. Simulations show that, while φ is the same in both models, θ is different except for γ = 0. We also investigate models that interpolate between symmetric domain diffusion and DLCA.

Persistence and first-passage properties in nonequilibrium systems

“…The low temperature coarsening dynamics of the one-dimensional Ising ferromagnet with conserved Kawasaki-like dynamics was also studied in Ref. [155]. In these models, the domains of size l diffuse with a size-dependent diffusion constant D(l) ∝ l γ , with γ = −1.…”

“…In Ref. [155] the authors generalized the original model to arbitrary γ and, by using a scaling argument to compute the size distribution of domains, showed that the domain density decreases algebraically as N (t) ∼ t −1/(2−γ) . The persistence probability was shown, numerically, to decay as a power law as Q(t) ∼ t −θ , where θ depends on γ.…”

“…We refer the reader to Ref. [155] for a more detailed discussion of this model and its relation to so called diffusion-limited cluster-cluster aggregation (DLCA) model.…”

Local persistence in the directed percolation universality class

“…Persistence was also studied in various other contexts, including random walks and Gaussian processes [9,[38][39][40], fluctuating interfaces and surface growth [41][42][43][44][45][46][47], systems with shear flow [48][49][50], models with algebraic long-range interactions [51,52], and disordered systems [53][54][55][56]. Moreover, various authors have improved and generalized the concept of persistence in different ways [57][58][59][60][61][61][62][63][64][65][66][67][68]. Some predictions were confirmed experimentally and successfully applied to financial data [64,[69][70][71][72][73][74].…”

Universality of Persistence Exponents in Two-Dimensional Ostwald Ripening

“…Measurements of persistence exponents in thin layers of twisted nematic liquid crystal, which provide a physical realization of a quenched thermodynamic system characterized by an Ising-like nonconserved order parameter have been carried out [13]. However, beyond computer simulations of Ising ferromagnets with conserved Kawasaki-like dynamics [14], no measurements of persistence were performed in coarsening systems characterized by conserved dynamics and controllable volume fraction [15].…”