"The Ising model [...] is a mathematical model of ferromagnetism in statistical mechanics."
A model used to study the behavior of ferromagnetic materials.
Statistical mechanics: The study of the behavior of large systems of particles using probabilistic methods.
Thermodynamics: The study of the relationship between heat, temperature, and energy in a system.
Phase transitions: A sudden change in the properties of a system as it crosses a critical point, such as the melting of ice into water.
Magnetism: The study of magnetic fields, their interactions with magnetic materials, and the behavior of magnetic materials.
Ferromagnetism: A type of magnetism where different atoms in a magnetic material align their magnetic moments (spin) in a particular direction.
Ising model: A mathematical model used in statistical mechanics to study the behavior of magnetic systems.
Lattice: A regular, repeating geometric structure used in the Ising model to represent the crystal structure of magnetic materials.
Energy: A measure of the potential to do work, related to the configuration of magnetic spins in the Ising model.
Temperature: A measure of the average kinetic energy of the particles in a system, related to the random nature of the Ising model.
Magnetic field: A force field produced by a magnetic object that interacts with any other magnetic objects in its vicinity, affecting the behavior of magnetic spins in the Ising model.
Hamiltonian: A mathematical expression used in the Ising model to describe the energy of a given configuration of magnetic spins.
Mean field theory: An approximation used in the Ising model to simplify the calculation of the energy of a given configuration of magnetic spins.
Monte Carlo simulations: A computational method used to study the behavior of magnetic systems using the Ising model.
Critical phenomena: The behavior of a system at or near a critical point, such as the phase transition from a magnetized to a non-magnetized state.
Renormalization group theory: A method used to analyze the behavior of a system at different length scales, ultimately leading to a better understanding of phase transitions in the Ising model.
Classic Ising Model: The simplest form of the Ising Model, in which a two-dimensional lattice with fixed spins is modeled.
Quantum Ising Model: A quantum mechanical version of the Ising Model used in the study of magnetism in materials.
Directed Ising Model: An Ising Model on a directed graph, which is used to study the phenomenon of self-organization in complex systems.
Disordered Ising Model: An Ising Model with disorder, such as random couplings or random fields, which is used to study the effects of disorder on the system.
Long-range Ising Model: An Ising Model with interactions that decay as a power law as opposed to the usual exponential decay, which is used to study the effect of long-range interactions on the system.
Dynamic Ising Model: An Ising Model with dynamic couplings, which can change over time, used to study the evolution of systems under changing conditions.
Potts Model: A generalization of the Ising Model in which the spin states have more than two possible values, which is used to study systems with multiple phases.
XY Model: A model similar to the Ising Model in which the spins are allowed to point in any direction in the plane, used to study the properties of systems with continuous symmetry.
Kac-Potts Model: A variant of the Potts Model used to model phase transitions in systems with continuous symmetry.
Blume-Capel Model: A model similar to the Ising Model in which the spins can take on three possible values, used to study phase transitions in systems with tricritical behavior.
"The model consists of discrete variables [...] named after the physicists Ernst Ising and Wilhelm Lenz."
"The model consists of discrete variables that represent magnetic dipole moments of atomic 'spins' that can be in one of two states (+1 or −1)."
"The spins are arranged in a graph, usually a lattice, allowing each spin to interact with its neighbors."
"Neighboring spins that agree have a lower energy than those that disagree."
"The system tends to the lowest energy."
"Heat disturbs this tendency."
"The model allows the identification of phase transitions as a simplified model of reality."
"The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition."
"The Ising model was invented by the physicist Wilhelm Lenz (1920), who gave it as a problem to his student Ernst Ising."
"The one-dimensional Ising model was solved by Ising (1925) alone in his 1924 thesis; it has no phase transition."
"The two-dimensional square-lattice Ising model is usually solved by a transfer-matrix method, although there exist different approaches, more related to quantum field theory."
"In dimensions greater than four, the phase transition of the Ising model is described by mean-field theory."
"The Ising model for greater dimensions was also explored with respect to various tree topologies in the late 1970s."
"The solution to this model exhibited a new, unusual phase transition behavior, along with non-vanishing long-range and nearest-neighbor spin-spin correlations, deemed relevant to large neural networks as one of its possible applications."
"The Ising problem without an external field can be equivalently formulated as a graph maximum cut (Max-Cut) problem."
"The Ising problem can be solved via combinatorial optimization."
"The Ising model is a mathematical model of ferromagnetism in statistical mechanics."
"The model allows the identification of phase transitions as a simplified model of reality."
"The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition."