Cluster Monte Carlo and dynamical scaling for long-range interactions

Weigel, Martin; Kenna, Ralph; Berche, Bertrand

doi:10.1140/epjst/e2016-60338-3

Cited by 11 publications

(10 citation statements)

References 43 publications

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“…Naturally, simulations of such systems with longrange interaction are computationally far more expensive than its short-range counterpart. For equilibrium studies, the advent of various collective updates based on the Swendsen-Wang cluster algorithm [8] allows one to perform efficient Monte Carlo (MC) simulations [9][10][11]. Conversely, for understanding the nonequilibrium ordering kinetics following a quench from the high-temperature disordered phase into the ordered phase below the critical temperature T c , one is restricted to use only local moves, viz., single spin flips.…”

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confidence: 99%

“…When simulating a long-range interacting system using periodic boundary conditions (via minimum-image convention), one encounters strong finite-size effects. We circumvent this problem by using Ewald summation [11,24,27] for calculating the effective interaction J(r ij ). To prepare an initial configuration that mimics a hightemperature paramagnetic phase (T ≫ T c ) we choose a square lattice having linear dimension L with randomly 50% up and 50% down spins.…”

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Phase ordering kinetics of the long-range Ising model

2019

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We use an efficient method that eases the daunting task of simulating dynamics in spin systems with long-range interaction. Our Monte Carlo simulations of the long-range Ising model for the nonequilibrium phase ordering dynamics in two spatial dimensions perform significantly faster than the standard Metropolis approach and considerably more efficiently than the kinetic Monte Carlo method. Importantly, this enables us to establish agreement with the theoretical prediction for the time dependence of domain growth, in contrast to previous numerical studies. This method can easily be generalized to applications in other systems.Generic models of statistical physics exhibiting a transition from disordered to ordered states have been proved to be instrumental for understanding the dynamics in diverse fields, from species evolution [1] to traffic flow [1], from economic dynamics [2] to rainfall dynamics [3]. An extensively used paradigm is the Ising model with nearest-neighbor (NNIM) interaction [4,5]. Even the complex neural dynamics of brain depends on similar underlying mechanisms [6]. The maximum entropy models obtained from experimental data upon mapping the spiking activities of the neurons onto spin variables are equivalent to Ising models [7]. However, it is believed that the neuron activities are effectively modelled by longdistance communications [6]. In nature, also many other intermolecular interactions are evidently long-range, e.g., electrostatic forces, polarization forces, etc. Hence, a more complete picture calls for employing models that consider long-range interactions.The simplest generic model system is the long-range Ising model (LRIM), which on a d-dimensional lattice is described by the Hamiltonian

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Phase ordering kinetics of the long-range Ising model

2019

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“…( 2). The resampling procedures in the extended cluster algorithms for long-range interactions and for quantum spin systems [21][22][23][24] can be understood as specific cases of the clock method.…”

Section: Which Requests An O(n ) Computation In the Metropolis Algorithmmentioning

confidence: 99%

“…Making use of the particular feature that each bond is treated independently in the cluster-update scheme [20], Luijten and Blöte [21] applied an efficient sampling procedure to place occupied bonds, instead of visiting each bond sequentially and throwing a random number to decide its status. The Luijten-Blöte cluster algorithm has O(1) complexity [21][22][23], and has been generalized to quantum systems [24]. Recently, an EC algorithm was proposed for long-range soft-sphere systems [25].…”

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Clock Monte Carlo methods

2019

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We propose the clock Monte Carlo technique for sampling each successive chain step in constant time. It is built on a recently proposed factorized transition filter and its core features include its O(1) computational complexity and its generality. We elaborate how it leads to the clock factorized Metropolis (clock FMet) method, and discuss its application in other update schemes. By grouping interaction terms into boxes of tunable sizes, we further formulate a variant of the clock FMet algorithm, with the limiting case of a single box reducing to the standard Metropolis method. A theoretical analysis shows that an overall acceleration of O(N κ ) (0 ≤ κ ≤ 1) can be achieved compared to the Metropolis method, where N is the system size and the κ value depends on the nature of the energy extensivity. As a systematic test, we simulate long-range O(n) spin models in a wide parameter regime: for n = 1, 2, 3, with disordered algebraically decaying or oscillatory Ruderman-Kittel-Kasuya-Yoshida-type interactions and with and without external fields, and in spatial dimensions from d = 1, 2, 3 to mean-field. The O(1) computational complexity is demonstrated, and the expected acceleration is confirmed. Its flexibility and its independence from the interaction range guarantee that the clock method would find decisive applications in systems with many interaction terms.

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“…While cluster updates for short-ranged interactions are standard textbook knowledge, this is less the case for methods for systems with long-ranged interactions. Flores-Sola et al review the available approaches and propose a new single-cluster update for spin systems with power-law interactions [4]. A recent new addition in the simulational toolbox that is particularly well suited for massively parallel architectures, population annealing, is discussed by Barash et al [5] with applications to the simulation of first-order phase transitions.…”

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Recent advances in phase transitions and critical phenomena

Bachmann

Bittner

Fytas

et al. 2017

Eur. Phys. J. Spec. Top.

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Abstract. Phase transitions and critical phenomena are of ubiquitous importance from the femtometre scale in quantum chromodynamics to galaxy formation in the universe, from the folding, adsorption or denaturation of bio-polymers to the magnetisation effects in storage media, from percolation in complex social networks to fragmentation transitions in atomic nuclei. The present issue discusses a cross section of the current research on phase transitions and critical phenomena in condensed-matter physics, with a focus on soft and hard matter systems as well as the most important methods used for studying such problems.The study of phase transitions is by now a quite mature subject. Early notions akin to modern ideas of phase transitions are already present in ancient Greek philosophical texts, for instance in Aristotle's theory of the elements. Still, it was only in the late 18th and early 19th century that the advent of the steam engine necessitated a profound theoretical description. This clarified the conversion of heat from and to mechanical work, helped establish an abstract notion of energy, and finally resulted in the formulation of the laws of thermodynamics. The evaporation and condensation of water in the steam engine are prototypic examples of first-order phase transitions. As the temperature or pressure are increased, this system eventually reaches a critical point at which liquid and vapor become indistinguishable. The transition is no longer discontinuous but becomes continuous there. The character of this special point was a

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Cluster Monte Carlo and dynamical scaling for long-range interactions

Cited by 11 publications

References 43 publications

Phase ordering kinetics of the long-range Ising model

Phase ordering kinetics of the long-range Ising model

Clock Monte Carlo methods

Recent advances in phase transitions and critical phenomena

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