One Dimensional Phase-Ordering in the Ising Model with Space Decaying Interactions

Abstract

The study of the phase ordering kinetics of the ferromagnetic one-dimensional Ising model dates back to 1963 (R.J. Glauber, J. Math. Phys. 4, 294) for non conserved order parameter (NCOP) and to 1991 (S.J. Cornell, K. Kaski and R.B. Stinchcombe, Phys. Rev. B 44, 12263) for conserved order parameter (COP). The case of long range interactions J(r) has been widely studied at equilibrium but their effect on relaxation is a much less investigated field. Here we make a detailed numerical and analytical study of both cases, NCOP and COP. Many results are valid for any positive, decreasing coupling J(r), but we focus specifically on the exponential case, J exp (r) = e −r/R with varying R > 0, and on the integrable power law case, J pow (r) = 1/r 1+σ with σ > 0. We find that the asymptotic growth law L(t) is the usual algebraic one, L(t) ∼ t 1/z , of the corresponding model with nearest neighborg interaction (z NCOP = 2 and z COP = 3) for all models except J pow for small σ: in the non conserved case when σ ≤ 1 (z NCOP = σ + 1) and in the conserved case when σ → 0 + (z COP = 4β + 3, where β = 1/T is the inverse of the absolute temperature). The models with space decaying interactions also differ markedly from the ones with nearest neighbors due to the presence of many long-lasting preasymptotic regimes, such as an exponential mean-field behavior with L(t) ∼ e t , a ballistic one with L(t) ∼ t, a slow (logarithmic) behavior L(t) ∼ ln t and one with L(t) ∼ t 1/σ+1 . All these regimes and their validity ranges have been found analytically and verified in numerical simulations. Our results show that the main effect of the conservation law is a strong slowdown of COP dynamics if interactions have an extended range. Finally, by comparing the Ising model at hand with continuum approaches based on a Ginzburg-Landau free energy, we discuss when and to which extent the latter represent a faithful description of the former.

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

“…For instance, referring to nonconserved dynamics which proceeds through single spin flips, with nn the typical domain size grows algebraically as L(t) ∼ t 1/2 [12] whereas for WLR this happens to be true only for α > 2, while there is a non-universal α-dependent exponent for 1 < α ≤ 2. Nontrivial low temperature regimes also appear [13,14] and similar differences as α changes are observed in the aging properties [15,16].…”

“…At large times the uncorrelation hypothesis is surely not satisfied but the divergence of t * α (N ) signals that the dynamical behaviour of systems with α > 1 is radically different. As a matter of fact we know that for α > 1 relaxation is characterised by domain formation and coarsening [13][14][15][16].…”

“…Previous studies [13][14][15][16] have shown that with WLR interactions the relaxation phenomenology of the model is akin to the longly studied nn case. Once quenched, after a microscopic time, spin domains of opposite sign form, grow and compete: the phenomenon of coarsening.…”

“…As a first attempt to understand this topic, we focus on the one dimensional system, which is more suited for analytical approaches. Indeed, d = 1 is the only case where with nearest neighbor (nn) interaction the kinetics is amenable of an exact solution [12], and an analytic framework for the case α > 1 was also provided in [13,14]. Besides that, numerical simulations are obviously less demanding in d = 1.…”

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].…”