Ernst Ising 1900-1998

Kobe, S.

doi:10.1590/s0103-97332000000400003

Cited by 29 publications

(15 citation statements)

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“…The easiest and mandatory model is the spin- S Ising model [ 22 , 23 , 24 ], which can be written as

where J is the exchange interaction, H is the external field, and the first sum is over nearest-neighbor sites

on a lattice with N sites. Each site of the lattice has a spin S , taking values

for integer S , and

for half-integer S .…”

Section: Gibbs Phase Rule For Hamiltonian Modelsmentioning

confidence: 99%

Generalized Gibbs Phase Rule and Multicriticality Applied to Magnetic Systems

Dias

Lima

Plascak

2021

Entropy

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A generalization of the original Gibbs phase rule is proposed in order to study the presence of single phases, multiphase coexistence, and multicritical phenomena in lattice spin magnetic models. The rule is based on counting the thermodynamic number of degrees of freedom, which strongly depends on the external fields needed to break the ground state degeneracy of the model. The phase diagrams of some spin Hamiltonians are analyzed according to this general phase rule, including general spin Ising and Blume–Capel models, as well as q-state Potts models. It is shown that by properly taking into account the intensive fields of the model in study, the generalized Gibbs phase rule furnishes a good description of the possible topology of the corresponding phase diagram. Although this scheme is unfortunately not able to locate the phase boundaries, it is quite useful to at least provide a good description regarding the possible presence of critical and multicritical surfaces, as well as isolated multicritical points.

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“…The easiest and mandatory model is the spin- S Ising model [ 22 , 23 , 24 ], which can be written as

where J is the exchange interaction, H is the external field, and the first sum is over nearest-neighbor sites

on a lattice with N sites. Each site of the lattice has a spin S , taking values

for integer S , and

for half-integer S .…”

Section: Gibbs Phase Rule For Hamiltonian Modelsmentioning

confidence: 99%

Generalized Gibbs Phase Rule and Multicriticality Applied to Magnetic Systems

Dias

Lima

Plascak

2021

Entropy

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“…Among those techniques, two particular models on clustering are of interest: the Ising Model and the Potts Model, which are both known as Restricted Boltzmann machines (RBMs) (Goel, 2020). The Ising model is very famous as a model named after the physicist Ernst Ising: in 1925, Ernst who was a Lenz student, chose the model as the focus of his PhD dissertation (Kobe (2000), Kobe (1997), Ising (1925), Brush (1967), Ising et al (2017)). It is a NP-complete (Cipra, 2000) mathematical model of ferromagnetism in statistical mechanics ( Singh (2020), Giacomin & Mahfouf (2021), Cipra (1987), Glauber (1963), Pfeuty (1970), Brush (1967), Fredrickson & Andersen (1984), Stauffer et al (1993), Kadowaki & Nishimori (1998), Joya et al (2002), Aiyer et al (1990), Wen et al (2009), McCoy & Wu (2014)):…”

Section: On Clustering and The Random Potts Modelsmentioning

confidence: 99%

Potts Clustering with Complete Shrinkage: a novel clustering methodology with its Software Package pottscompleteshrinkage in Python

Alahassa¹

2021

Preprint

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We have modified the Potts Model Swendsen-Wang algorithm to insert some clusters constraints by applying a modified agglomerative clustering approach (Kurita, 1991). We have called the induced Potts Model, the Potts Clustering with Complete Shrinkage (PCCS), under the Python package pottscompleteshrinkage deployed on PyPi Index under its current release. In this approach, we deal with the increasing number of small clusters generated in a given partition by merging all small clusters of size ≤ h with their closest cluster in terms of minimal distance respectively, where h is an integer greater or equal to 2. The algorithm uses a technique in which distances of all pairs of observations are stored. Then the nearest cluster (with size ≥ h) is given by the cluster with the closest node in terms of minimal distance to the cluster to be merged using complete linkage. This approach is truly effective as it helps to control the clusters size, and we have found empirical evidence of Chi-Square and Gamma density curves for the constrained cluster size distribution of PCCS, when applied to some datasets taken from the multiple-output benchmark datasets available in the Mulan project website (Tsoumakas et al., 2020). We add a last framework based on Frequency of frequency distribution (FoF) to find the conditional bonds distribution given the clusters size constraints which results in an intractable distribution for large datasets, but its computation framework is a land of rich mathematical developments.

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“…Nevertheless, its invariant character is encoded in the conditioning, yielding a DLR measure that depends on ν. 7 Ising analyzed this model in one dimension in its thesis supervised by Lenz in 1922 [71]. The only contribution of that work to higher dimensions was the wrong interpretation that just as in d = 1, in higher dimension there is no phase transition.…”

Section: Example Of Phase Transition: Ferromagnetic 2d-ising Modelmentioning

confidence: 99%

Introduction to (generalized) Gibbs measures

Ny¹

2008

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Ernst Ising 1900-1998

Cited by 29 publications

References 0 publications

Generalized Gibbs Phase Rule and Multicriticality Applied to Magnetic Systems

Generalized Gibbs Phase Rule and Multicriticality Applied to Magnetic Systems

Potts Clustering with Complete Shrinkage: a novel clustering methodology with its Software Package pottscompleteshrinkage in Python

Introduction to (generalized) Gibbs measures

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