Nonuniversal coarsening and universal distributions in far-from-equilibrium systems

Anomalous coarsening in far-from-equilibrium one-dimensional systems is investigated by applying simulation and analytic techniques to minimal hard-core particle (exclusion) models. They contain mechanisms of aggregated particle diffusion, with rates ϵ⪡1, particle deposition into cluster gaps, but suppressed for the smallest gaps, and breakup of clusters that are adjacent to large gaps. Cluster breakup rates vary with the cluster length x as kx^α. The domain growth law ⟨x⟩∼(ϵt)^z, with z=1∕(2+α) for α>0, is explained by a simple scaling picture involving the time for two particles to coalesce and a new particle to be deposited. The density of double vacancies, at which deposition and cluster breakup are allowed, scales as 1∕[t(ϵt)^z]. Numerical simulations for several values of α and ϵ confirm these results. A fuller approach is presented which employs a mapping of cluster configurations to a column picture and an approximate factorization of the cluster configuration probability within the resulting master equation. The equation for a one-variable scaling function explains the above average cluster length scaling. The probability distributions of cluster lengths x scale as P(x)=[1∕(ϵt)^z]g(y), with y≡x∕(ϵt)^z, which is confirmed by simulation. However, those distributions show a universal tail with the form g(y)∼exp(−y^3∕2), which is explained by the connection of the vacancy dynamics with the problem of particle trapping in an infinite sea of traps. The high correlation of surviving particle displacement in the latter problem explains the failure of the independent cluster approximation to represent those rare events.