We consider a periodic Ising chain with nearest-neighbour and $r$-th neighbour interaction and quench it from infinite temperature to zero temperature. The persistence probability $P(t)$, measured as the probability that a spin remains unflipped upto time $t$, is studied by computer simulation for suitable values of $r$. We observe that as time progresses, $P(t)$ first decays as $t^{-0.22}$ (-the {\em first} regime), then the $P(t)-t$ curve has a small slope (in log-log scale) for some time (-the {\em second} regime) and at last it decays nearly as $t^{-3/8}$ (-the {\em third} regime). We argue that in the first regime, the persistence behaviour is the usual one for a two-dimensional system, in the second regime it is like that of a non-interacting (`zero-dimensional') system and in the third regime the persistence behaviour is like that of a one dimensional Ising model. We also provide explanations for such behaviour.
We obtain the persistence exponents for an antiferromagnetic Ising system in which the magnetisation is kept constant. This system belongs to Model C (system with non-conserved order parameter with a conserved density) and is expected to have persistence exponents different from that of Model A (system with no conservation) but independent of the conserved density. Our numerical results for both local persistence at zero temperature and global persistence at the critical temperature however indicate that the exponents are dependent on the conserved magnetisation in both two and three dimension…
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.
The spatial distribution of persistent spins at zero-temperature in the pure two-dimensional Ising model is investigated numerically. A persistence correlation length, $ξ(t)\sim t^Z$ is identified such that for length scales $r<<ξ(t)$ the persistent spins form a fractal with dimension $d_f$; for length scales $r>>ξ(t)$ the distribution of persistent spins is homogeneous. The zero-temperature persistence exponent, $θ$, is found to satisfy the scaling relation $θ= Z(2-d_f)$ with $θ=0.209\pm 0.002, Z=1/2$ and $d_f\sim 1.58$.
We study persistence in one-dimensional ferromagnetic and anti-ferromagnetic nearest-neighbor Ising models with parallel dynamics. The probability P(t) that a given spin has not flipped up to time t, when the system evolves from an initial random configuration, decays as P(t) \sim 1/t^theta_p with theta_p \simeq 0.75 numerically. A mapping to the dynamics of two decoupled A+A \to 0 models yields theta_p = 3/4 exactly. A finite size scaling analysis clarifies the nature of dynamical scaling in the distribution of persistent sites obtained under this dynamics.
The zero-temperature Glauber dynamic is used to investigate the persistence probability $P(t)$ in the randomic two-dimensional ferromagnetic Ising model on a Voronoi-Delaunay tessellation. We consider the coupling factor $J$ varying with the distance $r$ between the first neighbors to be $J(r) \propto e^{-αr}$, with $α\ge 0$. The persistence probability of spins flip, that does not depends on time $t$, is found to decay to a non-zero value $P(\infty)$ depending on the parameter $α$. Nevertheless, the quantity $p(t)=P(t)-P(\infty)$ decays exponentially to zero over long times. Furthermore, the …
We present very accurate numerical estimates of the time and size dependence of the zero-temperature local persistence in the $2d$ ferromagnetic Ising model. We show that the effective exponent decays algebraically to an asymptotic value $θ$ that depends upon the initial condition. More precisely, we find that $θ$ takes one universal value $0.199(2)$ for initial conditions with short-range spatial correlations as in a paramagnetic state, and the value $0.033(1)$ for initial conditions with the long-range spatial correlations of the critical Ising state. We checked universality by working with …
One-dimensional non-equilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and nearest neighbour spin exchanges exhibiting a parity-conserving (PC) phase transition on the level of kinks are investigated here numerically from the point of view of the underlying spin system. The dynamical persistency exponent $Θ$ and the exponent $lambda$ characterising the two-time autocorrelation function of the total magnetization under non-equilibrium conditions are reported. It is found that the PC transition has strong effect: the process becomes non-Markovi…
We study the statistical properties of the sum $S_t=\int_{0}^{t}dt' σ_{t'}$, that is the difference of time spent positive or negative by the spin $σ_{t}$, located at a given site of a $D$-dimensional Ising model evolving under Glauber dynamics from a random initial configuration. We investigate the distribution of $S_{t}$ and the first-passage statistics (persistence) of this quantity. We discuss successively the three regimes of high temperature ($T>T_{c}$), criticality ($T=T_c$), and low temperature ($T<T_{c}$). We discuss in particular the question of the temperature dependence of the pers…
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…
We investigate both the local and global persistence behaviour in ANNNI (axial next-nearest neighour Ising) model. We find that when the ratio $κ$ of the second neighbour interaction to the first neighbour interaction is less than 1, $P(t)$, the probability of a spin to remain in its original state upto time $t$ shows a stretched exponential decay. For $κ> 1$, $P(t)$ has a algebraic decay but the exponent is different from that of the nearest neighbour Ising model. The global persistence behaviour shows similar features. We also conduct some deeper investigations in the dynamics of the ANNNI m…
A system's configurational state can be manipulated using dynamic variation of control parameters, such as temperature, pressure, or magnetic field; for finite-duration driving, excess work is required above the equilibrium free-energy change. Minimum-work protocols in multidimensional control-parameter space have potential to significantly reduce work relative to one-dimensional control. By numerically minimizing a linear-response approximation to the excess work, we design protocols in control-parameter spaces of a 2D Ising model that efficiently drive the system from the all-down to all-up …
We measure the persistence exponent in a phase separating two-dimensional spin system with non-conserved dynamics quenched in a region with four coexisting stripe phases. The system is an Ising model with nearest neighbor, next-to-the-nearest neighbor and plaquette interactions. Due the particular nature of the ground states, the order parameter is defined in terms of blocks of spins. Our estimate of the persistence exponent, $θ=0.42$, differs from those of the two-dimensional Ising and four state Potts models. Our procedure allows the study of persistence properties also at finite temperature…
We examine persistence in one dimensional Ising model under zero temperature Glauber dynamics for random initial states with unequal fraction of up and down spins. We find the persistence exponent varies continuously with the fraction of up spins in the initial state. Apparently this feature has been overlooked in the studies so far.
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 persistence P(t) of the magnetization of a d' dimensional manifold (i.e., the probability that the manifold magnetization does not flip up to time t, starting from a random initial condition) in a d-dimensional spin system at its critical point. We show analytically that there are three distinct late time decay forms for P(t) : exponential, stretched exponential and power law, depending on a single parameter ζ=(D-2+η)/z where D=d-d' and η, z are standard critical exponents. In particular, our theory predicts that the persistence of a line magnetization decays as a power law in the…
A ``persistence'' exponent theta has been extensively used to describe the nonequilibrium dynamics of spin systems following a deep quench: for zero-temperature homogeneous Ising models on the d-dimensional cubic lattice, the fraction p(t) of spins not flipped by time t decays to zero like t^[-theta(d)] for low d; for high d, p(t) may decay to p(infinity)>0, because of ``blocking'' (but perhaps still like a power). What are the effects of disorder or changes of lattice? We show that these can quite generally lead to blocking (and convergence to a metastable configuration) even for low d, and t…
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.
We investigate the persistence properties of critical d-dimensional systems relaxing from an initial state with non-vanishing order parameter (e.g., the magnetization in the Ising model), focusing on the dynamics of the global order parameter of a d'-dimensional manifold. The persistence probability P(t) shows three distinct long-time decays depending on the value of the parameter ζ= (D-2+η)/z which also controls the relaxation of the persistence probability in the case of a disordered initial state (vanishing order parameter) as a function of the codimension D = d-d' and of the critical expon…
For the two dimensional kinetic Ising model at finite temperature, the local mean magnetisation $M_t=t^{-1}\int_{0}^tσ(t')\d t'$, simply related to the fraction of time spent by a given spin in the positive direction, has a limiting distribution, singular at $\pm m_0(T)$, the Onsager spontaneous magnetization. The exponent of this singularity defines the persistence exponent $θ$. We also study first passage exponents associated to persistent large deviations of $M_t$, and their temperature dependence.
The global persistence exponent $θ_g$ is calculated for the two-dimensional Blume-Capel model following a quench to the critical point from both disordered states and such with small initial magnetizations. Estimates are obtained for the nonequilibrium critical dynamics on the critical line and at the tricritical point. Ising-like universality is observed along the critical line and a different value $θ_g =1.080(4)$ is found at the tricritical point.
We investigate the global persistence properties of critical systems relaxing from an initial state with non-vanishing value of the order parameter (e.g., the magnetization in the Ising model). The persistence probability of the global order parameter displays two consecutive regimes in which it decays algebraically in time with two distinct universal exponents. The associated crossover is controlled by the initial value m_0 of the order parameter and the typical time at which it occurs diverges as m_0 vanishes. Monte-Carlo simulations of the two-dimensional Ising model with Glauber dynamics d…
We investigate the dynamical behaviour of the Ising model under a zero temperature quench with the initial fraction of up spins $0\leq x\leq 1$. In one dimension, the known results for persistence probability are verified; it shows algebraic decay for both up and down spins asymptotically with different exponents. It is found that the conventional finite size scaling is valid here. In two dimensions however, the persistence probabilities are no longer algebraic; in particular for $x\leq 0.5$, persistence for the up (minority) spins shows the behaviour $P_{min}(t) \sim t^{-γ}\exp(-(t/τ)^δ)$ wit…
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 $α=1/2$ …
The Glauber dynamics of the pure and weakly disordered random-bond 2d Ising model is studied at zero-temperature. A single characteristic length scale, $L(t)$, is extracted from the equal time correlation function. In the pure case, the persistence probability decreases algebraically with the coarsening length scale. In the disordered case, three distinct regimes are identified: a short time regime where the behaviour is pure-like; an intermediate regime where the persistence probability decays non-algebraically with time; and a long time regime where the domains freeze and there is a cessatio…
Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlevé VI distribution
Université Paris-Saclay（巴黎萨克雷大学）；CEA Saclay（法国原子能和替代能源委员会萨克雷中心）；CNRS（法国国家科学研究中心）；Sorbonne université（索邦大学）；ENS Paris-Saclay（巴黎萨克雷高等师范学院）；The University of Hong Kong（香港大学）
In this paper a new approach to solving the Ising-Onsager problem in external magnetic field is investigated. The expression for free energy on one Ising spin in external field both for the twodimensional and threedimensional Ising model with interaction of the nearest neighbors are derived. The representations of free energy being expressed by multidimensional integrals of Gauss type with the appropriate dimensionality are shown. Possibility of calculating the integrals and the critical indices on the base of the derived representations for free energy is investigated.
In this letter we study the local persistence of the two--dimensional Blume-- Capel Model by extension of the concept of Glauber dynamics. We verify that for any value of the ratio $α=D/J$ between anisotropy $D$ and exchange $J$ the persistence shows a power law behavior. In particular for $α<0$ we find a persistence exponent $θ_{l}=0.2096(13)$, \textit{i.e.} in the Ising universality class. For $α>0$ ($α\neq 1$) we observe the occurrence of blocking.
We demonstrate the large scale effects of the interplay between shape and hard core interactions in a system with left- and right-pointing arrowheads ~$\textless ~~ \textgreater$~ on a line, with reorientation dynamics. This interplay leads to the formation of two types of domain wall, diffusive ($A$) and static ($B$). The correlation length in the equilibrium state diverges exponentially with increasing arrowhead density, with an ordered state of like orientations arising in the limit. In time, the approach to the ordered state is described by a coarsening process governed by the kinetics of …
The scaling behaviour of the persistence probability in the critical dynamics is investigated with both the heat-bath and the Metropolis algorithm for the two-dimensional Ising model and Potts model. Special attention is drawn to the dependence on the initial magnetization. The global persistence exponent is measured. Universality is confirmed.
We study phase ordering dynamics in the three-dimensional nearest-neighbor Ising model, following rapid quenches from infinite to zero temperature. Results on various aspects, viz., domain growth, persistence, aging and pattern, have been obtained via the Glauber Monte Carlo simulations of the model on simple cubic lattice. These are analyzed via state-of-the-art methods, including the finite-size scaling, and compared with those for quenches to a temperature above the roughening transition. Each of these properties exhibit remarkably different behavior at the above mentioned final temperature…
Inferring the microscopic mechanisms of opinion dynamics using a kinetic Ising model
Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）；LIP6（LIP6实验室）；Complex Systems Institute of Paris île-de-France (ISC-PIF, UAR3611)（巴黎-法兰西岛复杂系统研究所）；Centre d’Analyse et de Mathématique Sociales (CAMS, UMR8557)（社会分析与数学中心）；Université Paris-Saclay（巴黎萨克雷大学）；CEA（法国原子能和替代能源委员会）
We study the dynamics of ordering in ferromagnets via Monte Carlo simulations of the Ising model, employing the Glauber spin-flip mechanism, in space dimensions $d=2$ and $3$. Results for the persistence probability and the domain growth are discussed for quenches to various temperatures ($T_f$) below the critical one ($T_{c}$), from different initial temperatures $T_{i} \geq T_{c}$. In long time limit, for $T_{i} > T_{c}$, the persistence probability exhibits power-law decay with exponents $θ\simeq 0.22$ and $\simeq 0.18$ in $d=2$ and $3$, respectively. For finite $T_i$, the early time behavi…
We study the probability distribution $Q(n,t)$ of $n(t)$, the fraction of spins unflipped till time $t$, in a Ising chain with ferromagnetic interactions. The distribution shows a peak at $n=n_{max}$ and in general is non-Gaussian and asymmetric in nature. However for $n>n_{max}$ it shows a Gaussian decay. A data collapse can be obtained when $Q(n,t)/L^α$ versus $(n-n_{max})L^β$ is plotted with $α\sim 0.45$ and $β\sim 0.6$. Interestingly, $n_{max}(t)$ shows a different behaviour compared to $<n(t)> = P(t)$, the persistence probability which follows the well-known behaviour $P(t)\sim t^{-θ}$. A…
We study the first passage time properties of an integrated Brownian curve both in homogeneous and disordered environments. In a disordered medium we relate the scaling properties of this center of mass persistence of a random walker to the average persistence, the latter being the probability P_pr(t) that the expectation value <x(t)> of the walker's position after time t has not returned to the initial value. The average persistence is then connected to the statistics of extreme events of homogeneous random walks which can be computed exactly for moderate system sizes. As a result we obtain a…
We investigate the dynamics of a two dimensional axial next nearest neighbour Ising (ANNNI) model following a quench to zero temperature. The Hamiltonian is given by $H = -J_0\sum_{i,j=1}^L S_{i,j}S_{i+1,j} - J_1\sum_{i,j=1} [S_{i,j} S_{i,j+1} -κS_{i,j} S_{i,j+2}]$. For $κ<1$, the system does not reach the equilibrium ground state but slowly evolves to a metastable state. For $κ> 1$, the system shows a behaviour similar to the two dimensional ferromagnetic Ising model in the sense that it freezes to a striped state with a finite probability. The persistence probability shows algebraic decay he…
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…
The zero temperature quenching dynamics of the ferromagnetic Ising model on a densely connected small world network is studied where long range bonds are added randomly with a finite probability $p$. We find that in contrast to the sparsely connected networks and random graph, there is no freezing and an initial random configuration of the spins reaches the equilibrium configuration within a very few Monte Carlo time steps in the thermodynamic limit for any $p \ne 0$. The residual energy and the number of spins flipped at any time shows an exponential relaxation to equilibrium. The persistence…
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…
Randomness is known to affect the dynamical behaviour of many systems to a large extent. In this paper we investigate how the nature of randomness affects the dynamics in a zero temperature quench of Ising model on two types of random networks. In both the networks, which are embedded in a one dimensional space, the first neighbour connections exist and the average degree is four per node. In the random model A, the second neighbour connections are rewired with a probability $p$ while in the random model B, additional connections between neighbours at Euclidean distance $l ~ (l >1)$ are introd…
We recast the persistence probability for the spin located at the origin of a half-space arbitrarily $m$-magnetized Glauber-Ising chain as a Fredholm Pfaffian gap probability generating function with a sech-kernel. This is then spelled out as a tau-function for a certain Painlevé VI transcendent, the persistence exponent $θ(m)/2$ emerging as an asymptotic decay rate. Using a known yet remarkable correspondence that relates Painlevé equations to Bonnet surfaces, the persistence probability also acquires a geometric meaning in terms of the mean curvature of the latter, and even a topological one…
The idea that the dynamics of a spin is determined by the size of its neighbouring domains was recently introduced (S. Biswas and P. Sen, Phys. Rev. E {\bf 80}, 027101 (2009)) in a Ising spin model (henceforth, referred to as model I). A parameter $p$ is now defined to modify the dynamics such that a spin can sense domain sizes up to $R = pL/2$ in a one dimensional system of size $L$. For the cutoff factor $p$ \to 0$, the dynamics is Ising like and the domains grow with time $t$ diffusively as $ t^{1/z}$ with $z=2$, while for $p=1$, the original model I showed ballistic dynamics with $z \simeq…
We study the thermodynamics of the spin-$S$ two-dimensional quantum Heisenberg antiferromagnet on the square lattice with nearest ($J_1$) and next-nearest ($J_2$) neighbor couplings in its collinear phase ($J_2/J_1>0.5$), using the pure-quantum self-consistent harmonic approximation. Our results show the persistence of a finite-temperature Ising phase transition for every value of the spin, provided that the ratio $J_2/J_1$ is greater than a critical value corresponding to the onset of collinear long-range order at zero temperature. We also calculate the spin- and temperature-dependence of the…
The dynamical evolution of a recently introduced one dimensional model in \cite{biswas-sen} (henceforth referred to as model I), has been made stochastic by introducing a parameter $β$ such that $β=0$ corresponds to the Ising model and $β\to \infty$ to the original model I. The equilibrium behaviour for any value of $β$ is identical: a homogeneous state. We argue, from the behaviour of the dynamical exponent $z$,that for any $β\neq 0$, the system belongs to the dynamical class of model I indicating a dynamic phase transition at $β= 0$. On the other hand, the persistence probabilities in a syst…
We study the relaxation of the bi-dimensional kinetically constrained spiral model. We show that due to the reversibility of the dynamic rules any unblocked state fully decorrelates in finite times irrespectively of the system being in the unjammed or the jammed phase. In consequence, the evolution of any unblocked configuration occurs in a different sector of phase space from the one that includes the equilibrium blocked equilibrium configurations at criticality and in the jammed phase. We argue that such out of equilibrium dynamics share many points in common with coarsening in the one-dimen…
