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

Abstract

We study the kinetics after a low temperature quench of the one-dimensional Ising model with long range interactions between spins at distance rr decaying as r^{-\alpha}r−α. For \alpha =0α=0, i.e. mean field, all spins evolve coherently quickly driving the system towards a magnetised state. In the weak long range regime with \alpha >1α>1 there is a coarsening behaviour with competing domains of opposite sign without development of magnetisation. For strong long range, i.e. 0<\alpha <10<α<1, we show that the system shows both features, with probability P_\alpha (N)Pα(N) of having the latter one, with the different limiting behaviours \lim _{N\to \infty}P_\alpha (N)=0limN→∞Pα(N)=0 (at fixed \alpha<1α<1) and \lim _{\alpha \to 1}P_\alpha (N)=1limα→1Pα(N)=1 (at fixed finite NN). We discuss how this behaviour is a manifestation of an underlying dynamical scaling symmetry due to the presence of a single characteristic time \tau _\alpha (N)\sim N^\alphaτα(N)∼Nα.

Domain Statistics in the Relaxation of the One-Dimensional Ising Model with Strong Long-Range Interactions

“…In this paper we have considered the phase-ordering kinetics of the one-dimensional Ising model with long-range interactions decaying with distance as J(r) ∝ r −α , with α ≤ 1. In a previous paper [6] it was shown that, in this process, the statistical non-equilibrium ensemble splits into two different kinds of trajectories, those which develop coarsening domains, and those which do not so and behave similarly to mean field. Restricting the observation on the former ones, in the present study we have performed numerical simulations to compute the probability distribution p α (s, t, N) of the domain sizes, from which the growth-law of the average domains size s α (t, N) can be inferred.…”

“…In a previous article [6] we have shown that for α ≤ 1, the presence of configurations with coarsening domains is a stochastic phenomenon which may occur (or not), depending on the different dynamical realisations, with a given probability π α (t, N) plotted in Fig. 1 and defined as the fraction of trajectories that at time t = 1 presents at least two domains.…”

“…Instead, in a finite system, it occurs only when the largest domains have grown comparable to the system's size, which we will denote in the following as a border finite-size effect. In the case of interactions of the form J(r) ∝ r −α [4][5][6], which will be considered in this paper, the phenomenology described above is observed in the so-called weak long-range case (WLR), namely when α > d, where d is the spatial physical dimension. This guarantees that some foundations of usual statistical mechanics, such as additivity and extensivity, are preserved [7].…”

“…The evolution of systems with strong long-range interactions (SLR), i.e. with an algebraic decay of J(r) and α ≤ d, was recently studied in one dimension in [6]. There it was shown that, for a given system size, the non-equilibrium ensemble contains a fraction of trajectories along which coarsening domains are unexpectedly displayed, whereas the remaining ones behave similarly to the mean field, as shown in Fig.…”

Asymptotic States of Ising Ferromagnets with Long-range Interactions

“…In the first case, sometimes denoted as strong long-range regime, some fundamental physical properties, such as additivity, are spoiled, causing major differences with respect to the kinetics of the short-range case. In particular, it was shown [19] that, at least in d = 1, a mean-field like dynamics dominates in the thermodynamic limit L → ∞. In this case, spins tend to align with the sign of global magnetization, the symmetry is soon broken, and metastable states such as those described before for short-range systems cannot be sustained.…”