We study the kinetics after a low temperature quench of the one-dimensional Ising model with long range interactions between spins at distance $r$ decaying as $r^{-α}$. For $α=0$, i.e. mean field, all spins evolve coherently quickly driving the system towards a magnetised state. In the weak long range regime with $α>1$ there is a coarsening behaviour with competing domains of opposite sign without development of magnetisation. For strong long range, i.e. $0<α<1$, we show that the system shows both features, with probability $P_α(N)$ of having the latter one, with the different limiting behaviours $\lim _{N\to \infty}P_α(N)=0$ (at fixed $α<1$) and $\lim _{α\to 1}P_α(N)=1$ (at fixed finite $N$). We discuss how this behaviour is a manifestation of an underlying dynamical scaling symmetry due to the presence of a single characteristic time $τ_α(N)\sim N^α$.
We report a study of nonequilibrium relaxation in a two-dimensional random field Ising model at a nonzero temperature. We attempt to observe the coarsening from a different perspective with a particular focus on three dynamical quantities that characterize the kinetic coarsening. We provide a simple generalized scaling relation of coarsening supported by numerical results. The excellent data collapse of the dynamical quantities justifies our proposition. The scaling relation corroborates the recent observation that the average linear domain size satisfies different scaling behavior in differen…
If quenched to zero temperature, the one-dimensional Ising spin chain undergoes coarsening, whereby the density of domain walls decays algebraically in time. We show that this coarsening process can be interrupted by exerting a rapidly oscillating periodic field with enough strength to compete with the spin-spin interaction. By analyzing correlation functions and the distribution of domain lengths both analytically and numerically, we observe nontrivial correlation with more than one length scale at the threshold field strength.
We investigate a nonequilibrium coarsening dynamics of a one-dimensional Ising spin system with chirality. Only spins at domain boundaries are updated so that the model undergoes a coarsening to either of equivalent absorbing states with all spins + or -. Chirality is imposed by assigning different transition rates to events at down (+-) kinks and up (-+) kinks. The coarsening is characterized by power-law scalings of the kink density $ρ\sim t^{-δ}$ and the characteristic length scale $ξ\sim t^{1/z}$ with time $t$. Surprisingly the scaling exponents vary continuously with model parameters, whi…
We introduce a driven version of the 1D kinetically constrained spin chain. In its original undriven version, this model shows anomalous coarsening following a quench to a low temperature, with an equilibration time that diverges as exp(1/T^2) for T\to 0. We show that driving of constant rate gamma_dot interrupts coarsening and stabilises the chain in a state analogous to that of a coarsening chain of age 1/gamma_dot. We present an analytical theory for this steady state, and demonstrate it to be in excellent agreement with our simulation results.
Via Monte Carlo simulations we study pattern and aging during coarsening in nonconserved nearest neighbor Ising model, following quenches from infinite to zero temperature, in space dimension $d=3$. The decay of the order-parameter autocorrelation function is observed to obey a power-law behavior in the long time limit. However, the exponent of the power-law, estimated accurately via a state-of-art method, violates a well-known lower bound. This surprising fact has been discussed in connection with a quantitative picture of the structural anomaly that the 3D Ising model exhibits during coarsen…
Coarsening and persistence of Ising spins on a ladder is examined under voter dynamics. The density of domain walls decreases algebraically with time as $t^-{1/2}$ for sequential as well as parallel dynamics. The persistence probability decreases as $t^{-θ_{s}}$ under sequential dynamics, and as $t^{-θ_{p}}$ under parallel dynamics where $θ_{p} = 2 θ_{s} \approx .88$. Numerical values of the exponents are explained. The results are compared with the voter model on one and two dimensional lattices, as well as Ising model on a ladder under zero-temperature Glauber dynamics.
Through large-scale numerical simulations, we study the phase ordering kinetics of the $2d$ Ising Model after a zero-temperature quench from a high-temperature homogeneous initial condition. Analysing the behaviour of two important quantities -- the winding angle and the pair-connectedness -- we reveal the presence of a percolating structure in the pattern of domains. We focus on the pure case and on the random field and random bond Ising Model.
We investigate the zero-temperature coarsening dynamics of a chain of Ising spins with a nearest-neighbor ferromagnetic and an nth-neighbor antiferromagnetic interactions. For sufficiently large antiferromagnetic interaction, the ground state consists of $n$ consecutive up spins followed by n down spins, etc. We show that the asymptotic coarsening into this ground state is governed by a multispecies reactive gas of elementary excitations. The basic elementary excitations are identified and each decays at a different power-law rate in time. The dominant excitations are domains of n+1 spins whic…
We quantify the effect of system size in the kinetics of domain growth in Ising model with 50:50 composition in two spatial dimensions. Our estimate of the exponent, $α=0.334\pm0.004$, for the power law growth of linear domain size, from Monte Carlo simulation using small systems of linear dimensions L=16, 32, 64, and 128, is in excellent agreement with the prediction of Lifshitz-Slyozov (LS) theory, $α=1/3$. We find that the LS exponent sets in very early and continues to be true until average size of domains reaches three quarters of equilibrium limit.
We study the low-temperature coarsening of an Ising chain subject to spin-exchange dynamics and a small driving force. This dynamical system reduces to a domain diffusion process, in which entire domains undergo nearest-neighbor hopping, except for the shortest domains -- dimers -- which undergo long-range hopping. This system is characterized by two independent length scales: the average domain length L(t)~t^{1/2} and the average dimer hopping distance l(t)~ t^{1/4}. As a consequence of these two scales, the density C_k(t) of domains of length k does not obey scaling. This breakdown of scalin…
We examine an opinion formation model, which is a mixture of Voter and Ising agents. Numerical simulations show that even a very small fraction ($\sim 1\%$) of the Ising agents drastically changes the behaviour of the Voter model. The Voter agents act as a medium, which correlates sparsely dispersed Ising agents, and the resulting ferromagnetic ordering persists up to a certain temperature. Upon addition of the Ising agents, a logarithmically slow coarsening of the Voter model ($d=2$), or its active steady state ($d=3$), change into an Ising-type power-law coarsening.
We investigate the laws of coarsening of a two-dimensional system of Ising spins evolving under single-spin-flip irreversible dynamics at low temperature from a disordered initial condition. The irreversibility of the dynamics comes from the directedness, or asymmetry, of the influence of the neighbours on the flipping spin. We show that the main characteristics of phase ordering at low temperature, such as self-similarity of the patterns formed by the growing domains, and the related scaling laws obeyed by the observables of interest, which hold for reversible dynamics, are still present when…
We investigate the nonequilibrium dynamics following a quench to zero temperature of the non-conserved Ising model with power-law decaying long-range interactions $\propto 1/r^{d+σ}$ in $d=2$ spatial dimensions. The zero-temperature coarsening is always of special interest among nonequilibrium processes, because often peculiar behavior is observed. We provide estimates of the nonequilibrium exponents, viz., the growth exponent $α$, the persistence exponent $θ$, and the fractal dimension $d_f$. It is found that the growth exponent $α\approx 3/4$ is independent of $σ$ and different from $α=1/2$ …
We consider the zero temperature coarsening in the Ising model in two dimensions where the spins interact within the Moore neighbourhood. The Hamiltonian is given by $H = - \sum_{<i,j>}{S_iS_j} - κ\sum_{<i,j'>}{S_iS_{j'}}$ where the two terms are for the first neighbours and second neighbours respectively and $κ\geq 0$. The freezing phenomena, already noted in two dimensions for $κ=0$, is seen to be present for any $κ$. However, the frozen states show more complicated structure as $κ$ is increased; e.g. local anti-ferromagnetic motifs can exist for $κ>2$. Finite sized systems also show the exi…
We study the coarsening dynamics of the three-dimensional random field Ising model using Monte Carlo numerical simulations. We test the dynamic scaling and super-scaling properties of global and local two-time observables. We treat in parallel the three-dimensional Edward-Anderson spin-glass and we recall results on Lennard-Jones mixtures and colloidal suspensions to highlight the common and different out of equilibrium properties of these glassy systems.
We study the dynamics of a class of two dimensional stochastic processes, depending on two parameters, which may be interpreted as two different temperatures, respectively associated to interfacial and to bulk noise. Special lines in the plane of parameters correspond to the Ising model, voter model and majority vote model. The dynamics of this class of models may be described formally in terms of reaction diffusion processes for a set of coalescing, annihilating, and branching random walkers. We use the freedom allowed by the space of parameters to measure, by numerical simulations, the persi…
Coarsening kinetics in spin systems with long-range interactions: from voter to Ising
Università di Salerno（萨莱诺大学）；INFN Sezione di Napoli, Gruppo collegato di Salerno（意大利国家核物理研究所那不勒斯分部萨莱诺协作组）；Università della Campania（坎帕尼亚大学）；Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche（意大利国家研究委员会复杂系统研究所）；INFN Sezione di Firenze（意大利国家核物理研究所佛罗伦萨分部）
We study the early time dynamics of bimodal spin systems on $2d$ lattices evolving with different microscopic stochastic updates. We treat the ferromagnetic Ising model with locally conserved order parameter (Kawasaki dynamics), the same model with globally conserved order parameter (nonlocal spin exchanges), and the voter model. As already observed for non-conserved order parameter dynamics (Glauber dynamics), in all the cases in which the stochastic dynamics satisfy detailed balance, the critical percolation state persists over a long period of time before usual coarsening of domains takes o…
Using Monte Carlo simulations, we have studied the relaxation of energy of the three-dimensional Ising spin-glass model in aging process. Our finite-size-scaling analysis on the isothermal energy decay after the quench suggests strongly that the energy relaxes by coarsening of domain walls in agreement with the droplet theory. We have also evaluated relaxation times required for spin configurations in small systems, which are regarded as isolated droplets, to flip globally. The results support the fundamental assumption of the droplet theory that coarsening of domain walls, in case of spin gla…
In a recent paper [P. Mayer et al., Phys. Rev. Lett. 93, 115701 (2004)] it was shown, by means of experiments, theory and simulations, that coarsening systems display dynamic heterogeneity analogous to that of glass formers. Here, we present a detailed analysis of dynamic heterogeneities in the Glauber-Ising chain. We discuss how dynamic heterogeneity in Ising systems must be measured through connected multi-point correlation functions. We show that in the coarsening regime of the Ising chain these multi-point functions reveal the growth of spatial correlations in the dynamics, beyond what can…
We consider the coarsening properties of a kinetic Ising model with a memory field. The probability of a spin-flip depends on the persistence time of the spin in a state. The more a spin has been in a given state, the less the spin-flip probability is. We numerically studied the growth and persistence properties of such a system on a two dimensional square lattice. The memory introduces energy barriers which freeze the system at zero temperature. At finite temperature we can observe an apparent arrest of coarsening for low temperature and long memory length. However, since the energy barriers …
We study the evolution of spin clusters on two dimensional slices of the $3d$ Ising model in contact with a heat bath after a sudden quench to a subcritical temperature. We analyze the evolution of some simple initial configurations, such as a sphere and a torus, of one phase embedded into the other, to confirm that their area disappears linearly in time and to establish the temperature dependence of the prefactor in each case. Two generic kinds of initial states are later used: equilibrium configurations either at infinite temperature or at the paramagnetic-ferromagnetic phase transition. We …
We study the domain number and size distributions in the one-dimensional Ising and $q$-state Potts models subject to zero-temperature Glauber dynamics. The survival probability of a domain, $S(t)\sim t^{-ψ}$, and an unreacted domain, $Q_1(t)\sim t^{-δ}$, are characterized by two independent nonrtrivial exponents. For the Ising case, we find $ψ=0.126$ and $δ=1.27$ using numerical simulations. We develop an independent interval approximation (IIA) that predicts the qualitative behavior of the domain distribution and provides good estimates for the exponents. Exact results for the domain distribu…
We investigate the life time distribution in one and two dimensional coarsening processes modelled by Ising - Glauber dynamics at zero temperature. We find that the life time distribution obeys a scaling ansatz, asymptotically. An independent life time model where the life times are sampled from a distribution with power law tail is presented, which predicts analytically the qualitative features of the scaling function.
By studying numerically the phase-ordering kinetics of a two-dimensional ferromagnetic Ising model with quenched disorder -- either random bonds or random fields -- we show that a critical percolation structure forms in an early stage and is then progressively compactified by the ensuing coarsening process. Results are compared with the non-disordered case, where a similar phenomenon is observed, and interpreted within a dynamical scaling framework.
We investigate the statistics of the mean magnetisation, of its large deviations and persistent large deviations in simple coarsening systems. We consider more specifically the case of the diffusion equation, of the Ising chain at zero temperature and of the two dimensional voter model. For the diffusion equation, at large times, the mean magnetisation has a limit law, which is studied analytically using the independent interval approximation. The probability of persistent large deviations, defined as the probability that the mean magnetisation was, for all previous times, greater than some le…
Using the density-matrix renormalization-group method we study the two-dimensional Ising model in strip geometry. This renormalization scheme enables us to consider the system up to the size 300 x infinity and study the influence of the bulk magnetic field on the system at full range of temperature. We have found out the crossover in the behavior of the correlation length on the line of coexistence of the excited states. A detailed study of scaling of this line is performed. Our numerical results support and specify previous conclusions by Abraham, Parry, and Upton based on the related bubble …
We discuss different approaches for studying the influence of disorder in the three-dimensional Ising model. From the theoretical point of view, renormalisation group calculations provide quite accurate results. Experiments carried out on crystalline mixtures of compounds lead to measurements as accurate as three digits on the values of critical exponents. Numerically, extensive Monte Carlo simulations then pretend to be of comparable accuracy. Life becomes complicated when details are compared between the three approaches.
We study three kinetic models with constraint, namely the Symmetrically Constrained Ising Chain, the Asymmetrically Constrained Ising Chain, and the Backgammon Model. All these models show glassy behavior and coarsening. We apply to them the Stillinger and Weber decomposition, and find that they share the same configurational entropy, despite of their different nonequilibrium dynamics. We conclude therefore that the Stillinger and Weber decomposition is not relevant for this type of models.
Following quenches from random initial configurations to zero temperature, we study aging during evolution of the ferromagnetic (nonconserved) Ising model towards equilibrium, via Monte Carlo simulations of very large systems, in space dimensions $d=2$ and $3$. Results for the two-time autocorrelations, obtained by using different acceptance probabilities for the spin-flip trial moves, are in agreement with each other. We demonstrate the scaling of this quantity with respect to $\ell/\ell_w$, where $\ell$ and $\ell_w$ are the average domain sizes at $t$ and $t_w$ $(\leqslant t)$, the observati…
We study the dynamical behavior of a square lattice Ising model with exchange and dipolar interactions by means of Monte Carlo simulations. After a sudden quench to low temperatures we find that the system may undergo a coarsening process where stripe phases with different orientations compete or alternatively it can relax initially to a metastable nematic phase and then decay to the equilibrium stripe phase through nucleation. We measure the distribution of equilibration times for both processes and compute their relative probability of occurrence as a function of temperature and system size.…
We study the fate of the 2d kinetic q-state Potts model after a sudden quench to zero temperature. Both ground states and complicated static states are reached with non-zero probabilities. These outcomes resemble those found in the quench of the 2d Ising model; however, the variety of static states in the q-state Potts model (with q>=3) is much richer than in the Ising model, where static states are either ground or stripe states. Another possibility is that the system gets trapped on a set of equal-energy blinker states where a subset of spins can flip ad infinitum; these states are similar t…
Using extensive Monte Carlo simulations we study aging properties of two disordered systems quenched below their critical point, namely the two-dimensional random-bond Ising model and the three-dimensional Edwards-Anderson Ising spin glass with a bimodal distribution of the coupling constants. We study the two-times autocorrelation and space-time correlation functions and show that in both systems a simple aging scenario prevails in terms of the scaling variable $L(t)/L(s)$, where $L$ is the time-dependent correlation length, whereas $s$ is the waiting time and $t$ is the observation time. The…
The physical analysis of generic phase coexistence in the North-East-Center Toom model was originally given by Bennett and Grinstein. The gist of their argument relies on the dynamics of interfaces and droplets. We revisit the same question for a specific totally asymmetric kinetic Ising model on the square lattice. This nonequilibrium model possesses the remarkable property that its stationary-state measure in the absence of a magnetic field coincides with that of the usual ferromagnetic Ising model. We use both analytical arguments and numerical simulations in order to make progress in the q…
The coarsening exponents describing the growth of long-range order in systems quenched from a disordered to an ordered phase are discussed in terms of the decay rate, omega(k), for the relaxation of a distortion of wavevector k applied to a topological defect. For systems described by order parameters with Z(2) (`Ising'), and O(2) (`XY') symmetry, the appropriate defects are domain walls and vortex lines respectively. From omega(k) ~ k^z, we infer L(t) ~ t^(1/z) for the coarsening scale, with the assumption that defect relaxation provides the dominant coarsening mechanism. The O(2) case requir…
We investigate the motion of a tagged spin in a ferromagnetic Ising chain evolving under Kawasaki dynamics. At equilibrium, the displacement is Gaussian, with a variance growing as $A t^{1/2}$. The temperature dependence of the prefactor $A$ is derived exactly. At low temperature, where the static correlation length $ξ$ is large, the mean square displacement grows as $(t/ξ^2)^{2/3}$ in the coarsening regime, i.e., as a finite fraction of the mean square domain length. The case of totally asymmetric dynamics, where $(+)$ (resp. $(-)$) spins move only to the right (resp. to the left), is also co…
We study magnetoelastic properties of a spin-1/2 Ising-Heisenberg diamond chain, whose elementary unit cell consists of two decorating Heisenberg spins and one nodal Ising spin. It is assumed that each couple of the decorating atoms including the Heisenberg spins harmonically vibrates perpendicularly to the chain axis, while the nodal atoms involving the Ising spins are placed at rigid positions when ignoring their lattice vibrations. An effect of the magnetoelastic coupling on a ground state and finite-temperature properties is particularly investigated close to a triple coexistence point dep…
The two-dimensional Ising model with fixed magnetization is studied using Monte Carlo techniques. At the coexistence line, the macroscopic, extensive droplet of minority spins becomes thermally unstable by breaking up into microscopic clusters. Intriguing finite--size effects as well as singularities of thermal and cluster properties associated with the transition are discussed.
We analyse the out of equilibrium behavior of an Ising spin chain with an asymmetric kinetic constraint after a quench to a low temperature T. In the limit T\to 0, we provide an exact solution of the resulting coarsening process. The equilibration time exhibits a `glassy' divergence \teq=\exp(const/T^2) (popular as an alternative to the Vogel-Fulcher law), while the average domain length grows with a temperature dependent exponent, \dbar ~ t^{T\ln 2}. We show that the equilibration time \teq also sets the timescale for the linear response of the system at low temperatures.
We suggest that coarsening dynamics can be described in terms of a generalized random walk, with the dynamics of the growing length $L(t)$ controlled by a drift term, $μ(L)$, and a diffusive one, ${\cal D}(L)$. We apply this interpretation to the one dimensional Ising model with a ferromagnetic coupling constant decreasing exponentially on the scale $R$. In the case of non conserved (Glauber) dynamics, both terms are present and their balance depend on the interplay between $L(t)$ and $R$. In the case of conserved (Kawasaki) dynamics, drift is negligible, but ${\cal D}(L)$ is strongly dependen…
In this paper we consider the Glauber dynamics for a disordered ferromagnetic Ising model, in the region of phase coexistence. It was conjectured several decades ago that the spin autocorrelation decays as a negative power of time [Huse and Fisher, Phys. Rev. B, 1987]. We confirm this behavior by establishing a corresponding lower bound in any dimensions $d \geqslant 2$, together with an upper bound when $d=2$. Our approach is deeply connected to the Wulff construction for the dilute Ising model. We consider initial phase profiles with a reduced surface tension on their boundary and prove that…
In this paper, we study perturbations of the $d$-dimensional Ising model for $d\geq 2$, including long range ones to which the Pirogov-Sinai theory is not applicable. We show that the uniqueness of the equilibrium state of the Ising model at high temperature and the coexistence of equilibrium states at low temperature are preserved by spin-flip symmetric perturbations.
Coarsening in the long-range Persistent Voter Model
Universidade Federal do Rio Grande do Sul（南里奥格兰德联邦大学）；Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos（国家科学与技术研究所 - 复杂系统）；Università di Salerno（萨莱诺大学）；INFN Sezione di Napoli, Gruppo collegato di Salerno（意大利国家核物理研究院那不勒斯分部，萨莱诺协作组）；Universidade Federal Fluminense（弗鲁米嫩塞联邦大学）
While a large number of studies have focused on the nonequilibrium dynamics of a system when it is quenched instantaneously from a disordered phase to an ordered phase, such dynamics have been relatively less explored when the quench occurs at a finite rate. Here we study the slow quench dynamics in two paradigmatic models of classical statistical mechanics, {viz.}, one-dimensional kinetic Ising model and mean-field zero-range process, when the system is annealed slowly to the critical point. Starting from the time evolution equations for the spin-spin correlation function in the Ising model a…
We show by means of experiments, theory and simulations, that the slow dynamics of coarsening systems displays dynamic heterogeneity similar to that observed in glass-forming systems. We measure dynamic heterogeneity via novel multi-point functions which quantify the emergence of dynamic, as opposed to static, correlations of fluctuations. Experiments are performed on a coarsening foam using Time Resolved Correlation, a recently introduced light scattering method. Theoretically we study the Ising model, and present exact results in one dimension, and numerical results in two dimensions. For al…
We present evidence for a deep connection between the zero-temperature coarsening of the two-dimensional kinetic Ising model (KIM) and critical continuum percolation. In addition to reaching the ground state, the KIM can also fall into a variety of topologically distinct metastable stripe states. The probability to reach a stripe state that winds a times horizontally and b times vertically on a square lattice with periodic boundary conditions equals the corresponding exactly-solved critical percolation crossing probability P_{a,b} for a spanning path with winding numbers a and b.
We introduce and analyze a quantum spin/Majorana chain with a tricritical Ising point separating a critical phase from a gapped phase with order-disorder coexistence. We show that supersymmetry is not only an emergent property of the scaling limit, but manifests itself on the lattice. Namely, we find explicit lattice expressions for the supersymmetry generators and currents. Writing the Hamiltonian in terms of these generators allows us to find the ground states exactly at a frustration-free coupling. These confirm the coexistence between two (topologically) ordered ground states and a disorde…
Properties of the two dimensional Ising model with fixed magnetization are deduced from known exact results on the two dimensional Ising model. The existence of a continuous phase transition is shown for arbitrary values of the fixed magnetization when crossing the boundary of the coexistence region. Modifications of this result for systems of spatial dimension greater than two are discussed.
Using the density-matrix renormalization-group method we study the surface critical behaviour of the magnetization in Ising strips in the subcritical region. Our results support the prediction that the surface magnetization in the two phases along the pseudo-coexistence curve also behaves as for the ordinary transition below the wetting temperature for the finite value of the surface field.
Nonequilibrium behavior and dynamic phase transition properties of a kinetic Ising model under the influence of periodically oscillating random-fields have been analyzed within the framework of effective field theory (EFT) based on a decoupling approximation (DA). Dynamic equation of motion has been solved for a simple cubic lattice ($q=6$) by utilizing a Glauber type stochastic process. Amplitude of the sinusoidally oscillating magnetic field is randomly distributed on the lattice sites according to bimodal and trimodal distribution functions. For a bimodal type of amplitude distribution, it …
The spin-1/2 transverse field two-leg Ising ladder with nearest-neighbor exchange and plaquette four-spin interaction $J_{4}$ is studied analytically and numerically with the density matrix renormalization group approach. The quantum phase diagram in the transverse field $B$ versus $J_{4}$ plane has been obtained. There are three different phases: a paramagnetic (PM) phase for high values of the transverse field and, for low values of $B$, a ferromagnetic (FM) ordered phase for small $J_{4}$ and a dimerized-rung (DR) phase for large negative values of $J_{4}$. All phases are separated by quant…
We study the zero temperature coarsening dynamics in an Ising chain in presence of a dynamically induced field that favors locally the `-' phase compared to the `+' phase. At late times, while the `+' domains still coarsen as $t^{1/2}$, the `-' domains coarsen slightly faster as $t^{1/2}\log (t)$. As a result, at late times, the magnetization decays slowly as, $m(t)=-1 +{\rm const.}/{\log (t)}$. We establish this behavior both analytically within an independent interval approximation (IIA) and numerically. In the zero volume fraction limit of the `+' phase, we argue that the IIA becomes asympt…
We present detailed analytical studies on the zero temperature coarsening dynamics in an Ising spin chain in presence of a dynamically induced field that favors locally the `-' phase compared to the `+' phase. We show that the presence of such a local kinetic bias drives the system into a late time state with average magnetization m=-1. However the magnetization relaxes into this final value extremely slowly in an inverse logarithmic fashion. We further map this spin model exactly onto a simple lattice model of granular compaction that includes the minimal microscopic moves needed for compacti…
