Phase ordering kinetics of the long-range Ising model

Abstract

We use an efficient method that eases the daunting task of simulating dynamics in spin systems with long-range interaction. Our Monte Carlo simulations of the long-range Ising model for the nonequilibrium phase ordering dynamics in two spatial dimensions perform significantly faster than the standard Metropolis approach and considerably more efficiently than the kinetic Monte Carlo method. Importantly, this enables us to establish agreement with the theoretical prediction for the time dependence of domain growth, in contrast to previous numerical studies. This method can easily be generalized to applications in other systems.Generic models of statistical physics exhibiting a transition from disordered to ordered states have been proved to be instrumental for understanding the dynamics in diverse fields, from species evolution [1] to traffic flow [1], from economic dynamics [2] to rainfall dynamics [3]. An extensively used paradigm is the Ising model with nearest-neighbor (NNIM) interaction [4,5]. Even the complex neural dynamics of brain depends on similar underlying mechanisms [6]. The maximum entropy models obtained from experimental data upon mapping the spiking activities of the neurons onto spin variables are equivalent to Ising models [7]. However, it is believed that the neuron activities are effectively modelled by longdistance communications [6]. In nature, also many other intermolecular interactions are evidently long-range, e.g., electrostatic forces, polarization forces, etc. Hence, a more complete picture calls for employing models that consider long-range interactions.The simplest generic model system is the long-range Ising model (LRIM), which on a d-dimensional lattice is described by the Hamiltonian

Coexistence of coarsening and mean field relaxation in the long-range Ising chain

“…It should also be stressed that our result τ α (N ) ∼ N α is found as well at finite temperature for 1 < α ≤ 2, both in one dimension [14] and in higher dimension [23][24][25][26]. Based on our current understanding this seems a coincidence for a couple of reasons.…”

Understanding nonequilibrium scaling laws governing collapse of a polymer

“…The nonequilibrium phenomenon of coarsening in particle or spin systems is well understood [4,48] with current focus shifting towards more challenging scenarios like fluid mixtures [49,50]. Fundamentally, too, it is still developing as for example in computationally expensive long-range systems [51][52][53].…”

Understanding nonequilibrium scaling laws governing collapse of a polymer

“…The nonequilibrium phenomenon of coarsening in particle or spin systems is well understood [4,48] with current focus shifting towards more challenging scenarios like fluid mixtures [49,50]. Fundamentally, too, it is still developing as for example in computationally expensive long-range systems [51][52][53].…”