Metastability of the Two-Dimensional Blume–Capel Model with Zero Chemical Potential and Small Magnetic Field

Landim, Cláudio; Lemire, Paul

doi:10.1007/s10955-016-1550-8

Cited by 30 publications

(29 citation statements)

References 23 publications

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“…The underlying idea is that, on the time‐scale at which the transition from a to b occurs, the evolution of

{false{X_{t}^{β} false}}_{t \in double-struckN}

can be represented by a 3‐state Markov chain with a complete graph as state space diagram whose states correspond to the 3 valleys/cycles around the stable configurations a, b , and c . Similar ideas have been successfully used to describe metastability and tunneling phenomena in.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 98%

“…Example of hard-core configuration on the 6 × 9 triangular grid that satisfies (19) [Color figure can be viewed at wileyonlinelibrary.com] 13 The configurations 1 and 2 corresponding to the initial configuration in Figure 12 We now describe in detail how to construct each of the paths (j) for j = 1, … , 2L. We distinguish 2 cases, depending on whether (a) j has a vertical bridge in column c 3j+2 or (b) not, see the 2 examples in Figure 13.…”

Section: Energy Reduction Algorithm By Columnsmentioning

confidence: 99%

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Tunneling of the hard‐core model on finite triangular lattices

Zocca

2018

Random Struct Algorithms

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We consider the hard‐core model on finite triangular lattices with Metropolis dynamics. Under suitable conditions on the triangular lattice sizes, this interacting particle system has 3 maximum‐occupancy configurations and we investigate its high‐fugacity behavior by studying tunneling times, that is, the first hitting times between these maximum‐occupancy configurations, and the mixing time. The proof method relies on the analysis of the corresponding state space using geometrical and combinatorial properties of the hard‐core configurations on finite triangular lattices, in combination with known results for first hitting times of Metropolis Markov chains in the equivalent zero‐temperature limit. In particular, we show how the order of magnitude of the expected tunneling times depends on the triangular lattice sizes in the low‐temperature regime and prove the asymptotic exponentiality of the rescaled tunneling time leveraging the intrinsic symmetry of the state space.

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“…The underlying idea is that, on the time‐scale at which the transition from a to b occurs, the evolution of

{false{X_{t}^{β} false}}_{t \in double-struckN}

Section: Proofs Of the Main Resultsmentioning

confidence: 98%

Section: Energy Reduction Algorithm By Columnsmentioning

confidence: 99%

Tunneling of the hard‐core model on finite triangular lattices

Zocca

2018

Random Struct Algorithms

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“…The celebrated Blume-Capel model [3,4,6] is a three-state model which has been originally introduced to study the He 3 -He 4 mixture low temperature properties. More recently, the model has received a lot of interest also in the mathematics literature, since it is a prototype to study the effects of multiple metastable states [12][13][14]24]. The Hamiltonian of the Blume-Capel setting is characterized by a nearest neighbor spin-spin interaction term, a chemical potential contribution, and, finally, a term describing the interaction with an external magnetic field.…”

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confidence: 99%

A lattice model approach to the morphology formation from ternary mixtures during the evaporation of one component

Cirillo

Colangeli

Moons

et al. 2019

Eur. Phys. J. Spec. Top.

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Stimulated by experimental evidence in the field of solution-born thin films, we study the morphology formation in a three state lattice system subjected to the evaporation of one component. The practical problem that we address is the understanding of the parameters that govern morphology formation from a ternary mixture upon evaporation, as is the case in the fabrication of thin films from solution for organic photovoltaics. We use, as a tool, a generalized version of the Potts and Blume-Capel models in 2D, with the Monte Carlo Kawasaki-Metropolis algorithm, to simulate the phase behaviour of a ternary mixture upon evaporation of one of its components. The components with spin +1, −1 and 0 in the Blume-Capel dynamics correspond to the electron-acceptor, electron-donor and solvent molecules, respectively, in a ternary mixture used in the preparation of the active layer films in an organic solar cell. Further, we introduce parameters that account for the ccmmms-morph˙preprint.tex -29 agosto 2018 1:15 arXiv:1808.09326v1 [cond-mat.stat-mech] 28 Aug 2018 relative composition of the mixture, temperature, and interaction between the species in the system. We identify the parameter regions that are prone to facilitate the phase separation. Furthermore, we study qualitatively the types of formed configurations. We show that even a relatively simple model, as the present one, can generate key morphological features, similar to those observed in experiments, which proves the method valuable for the study of complex systems. IntroductionMulti-state spin systems on a lattice have been widely studied in the Statistical Mechanics literature, we refer the reader for instance to [33] where efforts have been invested in connecting microscopic dynamics to dynamics at mesoscopic and/or continuum scales. The celebrated Blume-Capel model [3,4,6] is a three-state model which has been originally introduced to study the He 3 -He 4 mixture low temperature properties. More recently, the model has received a lot of interest also in the mathematics literature, since it is a prototype to study the effects of multiple metastable states [12][13][14]24]. The Hamiltonian of the Blume-Capel setting is characterized by a nearest neighbor spin-spin interaction term, a chemical potential contribution, and, finally, a term describing the interaction with an external magnetic field. In the Blume-Capel model the spin can take the values −1, 0, +1 and the three different interface pairs −+, 0−, and 0+ have different energetic costs. More precisely, the two pairs containing a zero have the same cost, whereas the −+ one has a larger cost (four times). Pairs with equal spins −−, ++, and 00 have no energetic cost.A different multi-state model is the Potts model [32,37] -a straightforward generalization of the Ising model in which the spin has q > 2 states and in the Hamiltonian the nearest neighbor interaction and the external field contribution are taken into account. In the standard version of the model, all the pairs of spin interact similarly, but in i...

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“…In particular, by Assertion 4.D in [7], On the other hand, for a configuration In particular, at the boundary B + the energy is minimized by configurations in R a . This means that σ t should attained B + at R a .…”

Section: )mentioning

confidence: 99%

“…We consider in this article the situation in which the length of the torus increases with the inverse of the temperature. The case in which L is fixed has been considered by Cirillo and Nardi [4], by us [7] and by Cirillo, Nardi and Spitoni [5].…”

Section: Introductionmentioning

confidence: 99%