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Miscellaneous Algorithms
Published in Nazmul Siddique, Hojjat Adeli, Nature-Inspired Computing, 2017
The Ising model is a mathematical model of ferromagnetism in statistical mechanics (Ising, 1925). It is named after German physicist Ernst Ising. The model consists of discrete variables that represent magnetic dipole moments of atomic spins. Spins can be in one of the two states +1 or −1, that is, σ ∈ {+1,−1}. Each spin σi is assumed to have a random bond Jij with other spin σj, allowing each spin to interact with its neighbors. Taking the summation of all pairs of spins yields H=−∑{i,j}NJijσiσj where {i,j} denotes the pairs of nearest neighbors.
An Unexpected Renaissance Age
Published in Alessio Plebe, Pietro Perconti, The Future of the Artificial Mind, 2021
Alessio Plebe, Pietro Perconti
Much more than one would expect, according to (Mehta and Schwab, 2014). They applied the renormalization group on the classical Ising model Ising (1925), showing that it is equivalent to an RBM neural model (see §4.1). The Ising model is made of units, organized in lattices, typically corresponding to atoms with two possible nuclear magnetic moments or ‘spins’. Neighboring units interact with each other, and can be subjected to an overall magnetic field. The application of the renormalization group to a Ising model can be intuitively described as the coalescence of a box of units into a single abstract unit with its own spin.
Optimization
Published in Victor A. Bloomfield, Using R for Numerical Analysis in Science and Engineering, 2018
The Ising model, named after the German physicist Ernst Ising, was developed as a statistical mechanical model of ferromagnetism. In its most common form it consists of discrete variables (usually ±1) that represent spins arranged on a lattice. Each spin can interact with its neighbors with an interaction energy J (taken = 1 in the calculation that follows). The 2D Ising model on a square lattice is one of the simplest statistical mechanical models to display a phase transition. (The 1D model does not, though it still has many interesting properties.)
Phase diagram of the square 2D Ising lattice with nearest neighbor and next-nearest neighbor interactions
Published in Phase Transitions, 2023
After Ising solved the 1D Ising model it took almost two decades before the square 2D Ising model was solved by Lars Onsager [4]. Several years before Onsager published his seminal article, Kramers and Wannier [5] already demonstrated that if the 2D Ising model with ferromagnetic isotropic nearest neighbor interactions has an order–disorder phase transition, the transition temperature is uniquely defined by the relation . In 1944 Onsager showed that the 2D Ising model indeed exhibits an order–disorder phase transition. Onsager derived an exact expression for the free energy per spin in the absence of an external magnetic field. Unfortunately, the free energy per spin does not give a full understanding of all the properties of the system. There are many interesting quantities such as the spontaneous magnetization, susceptibilities and correlation functions, that cannot be directly deduced from the free energy per spin. Despite the long history of Ising systems they still receive substantial attention [6–9]. To date more complicated 2D Ising lattices that involve interactions beyond nearest neighbor interactions or an external field have not yet been solved exactly.
Phase transitions in the Heisenberg model on a layered triangular lattice in a magnetic field
Published in Phase Transitions, 2021
Akai Murtazaev, Magomedzagir Badiev, Magomedsheykh Ramazanov, Magomed Magomedov
The investigation of the external magnetic field reveals the phase transition in the layered triangular Ising model to be of a second order in the range of 0 ≤ h ≤ 6 [30]. A further increase in the magnetic field removes the degeneracy of the ground state and smears the phase transition. In the two-dimensional Heisenberg model in the range of low magnetic fields at h = 2, the order parameters for the Z3 and O (2) symmetries demonstrate a behavior which is typical for a second-order and the Kosterlitz–Thoules (KT) transitions, respectively, at different transition temperatures [8]. In the high-field range 3 ≤ h ≤ 9, the order parameter for the Z3 symmetry shows the behavior of a second-order transition, while another order parameter, associated with the components perpendicular to the field, demonstrates divergence of the relaxation time from the asymptotic form pointing to the second-order transition rather than the KT transition. The results obtained for the quasi-two-dimensional Heisenberg model on a triangular lattice using Monte Carlo (MC) simulations in zero magnetic field show the presence of three Berezinsky–Kosterlitz–Thoules transitions with a finite temperature [31]. The first two transitions are associated with the violation of the discrete symmetry group Z6, and the third transition is associated with the quasi-long-range ordering of the transverse components. It is found that a magnetic field applied parallel to the anisotropy axis reduces the discrete symmetry to Z3.
Phase transitions and phase coexistence: equilibrium systems versus externally driven or active systems - Some perspectives
Published in Soft Materials, 2021
However, obtaining accurate information on the critical behavior of models for fluids exhibiting a vapor-liquid transition has been much more difficult and possible only at the beginning of the present century, [102] due to several reasons: (i) for off-lattice systems cluster algorithms are much less powerful than for the Ising model (ii) While in the Ising model there is the special symmetry that the internal energy is invariant against spin reversal while the order parameter changes its sign, these two quantities are in a sense orthogonal to each other, unlike fluids where no special symmetry exists, and energy density and particle number density are “coupled operators” (“field mixing” effect [103]). Therefore, it is a nontrivial task not only to locate the critical temperature, but also to locate the critical density (which in the lattice gas fluid, that corresponds to the Ising model, by symmetry is ) and the chemical potential and pressure at criticality. A consistent analysis [102] needs to consider the coupling between the three “scaling fields” and in order to correctly extract critical properties.