Coarsening and percolation in a disordered ferromagnet

Corberi, Federico; Cugliandolo, Leticia F.; Insalata, Ferdinando; Picco, Marco

doi:10.1103/physreve.95.022101

Cited by 28 publications

(58 citation statements)

References 54 publications

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“…It is well known, in fact, that even a tiny amount of such randomness may change radically the kinetics of coarsening systems [28], both in one dimension [29] and higher dimensions [30,31,32]. The situation becomes even more complex if disorder introduces frustration [33,34]. In particular, in the case σ ≤ 0, the one-dimensional Edwards-Anderson model with algebraically decaying coupling constants, shows different behaviors depending on the exponent σ [35,36,37,38].…”

Section: Discussionmentioning

confidence: 99%

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

Corberi

Lippiello²,

Politi

2019

J Stat Phys

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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.

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Section: Discussionmentioning

confidence: 99%

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

Corberi

Lippiello²,

Politi

2019

J Stat Phys

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“…[14] the scaling properties of the pair connectedness function were studied for random and clean Ising models evolving with kinetic Monte Carlo dynamics with non-conserved order parameter. It was shown in this article that the data for g(r, t) can be collapsed onto the same master curve in the percolation ‡ Note that for bond percolation on the square lattice, the same quantity A c /L regime by rescaling the distance r by the characteristic length G (t) obtained from the excess energy, Eq.…”

Section: Pair Connectedness Functionmentioning

confidence: 99%

“…The role played by the fact that there are two large clusters in competition in the magnetic models compared to the single leading cluster of the percolation problem is also discussed. Moreover, we elaborate upon the understanding of the problem as one with an effective lattice spacing d (t) [14].…”

Section: Introductionmentioning

confidence: 99%

Critical percolation in the dynamics of the 2D ferromagnetic Ising model

Blanchard¹,

Cugliandolo²,

Picco³

et al. 2017

J. Stat. Mech.

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Abstract. We study the early time dynamics of the 2d ferromagnetic Ising model instantaneously quenched from the disordered to the ordered, low temperature, phase. We evolve the system with kinetic Monte Carlo rules that do not conserve the order parameter. We confirm the rapid approach to random critical percolation in a time-scale that diverges with the system size but is much shorter than the equilibration time. We study the scaling properties of the evolution towards critical percolation and we identify an associated growing length, different from the curvature driven one. By working with the model defined on square, triangular and honeycomb microscopic geometries we establish the dependence of this growing length on the lattice coordination. We discuss the interplay with the usual coarsening mechanism and the eventual fall into and escape from metastability.arXiv:1705.06508v3 [cond-mat.stat-mech]

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“…[30] and in another solvable mode in Ref. [24], promoting this feature to a rather generic property. Accordingly, although we have restricted our attention here to a temperature quench, we expect a similar scenario to be observed if other kind of quenches, e.g., a quench in the parameter r, were studied, as well as if other observables beyond the order parameter variance were considered.…”

Section: Discussionmentioning

confidence: 87%

“…In this work we consider the fluctuations dynamics in the Gaussian model, a standard statistical mechanical model which may be regarded as the simplest Ginzburg-Landau theory for the description of the disordered phase of Ising-like systems [21]. In this approach, the probability distribution of the order parameter variance S, the observable we focus on, displays a singular point in S = S c both in and out of equilibrium [4,22,[22][23][24][25][26][27][28][29][30]. This fact is associated to the so-called condensation of fluctuations [22], a phenomenon whereby certain fluctuations are realized by a condensation mechanism in which a single degree of freedom becomes macroscopically populated, similarly to what happens in usual condensation phenomena, e.g., in the Bose gas.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of fluctuations in the Gaussian model with conserved dynamics

Corberi¹,

Mazzarisi²,

Gambassi³

2019

J. Stat. Mech.

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We study the dynamics of the fluctuations of the variance s of the order parameter of the Gaussian model, following a temperature quench of the thermal bath. At each time t, there is a critical value sc(t) of s such that fluctuations with s > sc(t) are realized by condensed configurations of the systems, i.e., a single degree of freedom contributes macroscopically to s. This phenomenon, which is closely related to the usual condensation occurring on average quantities, is usually referred to as condensation of fluctuations. We show that the probability of fluctuations with s < inft[sc(t)], associated to configurations that never condense, after the quench converges rapidly and in an adiabatic way towards the new equilibrium value. The probability of fluctuations with s > inft[sc(t)], instead, displays a slow and more complex behavior, because the macroscopic population of the condensing degree of freedom is involved. P (S, t) = Dϕ P ([ϕ], t) δ(S − S[ϕ]).(1)

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Coarsening and percolation in a disordered ferromagnet

Cited by 28 publications

References 54 publications

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

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

Critical percolation in the dynamics of the 2D ferromagnetic Ising model

Dynamics of fluctuations in the Gaussian model with conserved dynamics

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