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The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics.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).
Detailed balance, which is a requirement of reversibility, states that if you observe the system for a long enough time, the system goes from state to with the same frequency as going from to . In equilibrium, the probability of observing the system at state A is given by the Boltzmann weight , e − E A / T {\displaystyle e^{-E_{A}/T}} .
The problem of the critical slowing-down affecting local processes is of fundamental importance in the study of second-order phase transitions (like ferromagnetic transition in the Ising model), as increasing the size of the system in order to reduce finite-size effects has the disadvantage of requiring a far larger number of moves to reach thermal equilibrium.
In d=2, the two-dimensional critical Ising model's critical exponents can be computed exactly using the minimal model,. In d=4, it is the free massless scalar theory (also referred to as mean field theory). These two theories are exactly solved, and the exact solutions give values reported in the table.
The Wolff algorithm, [1] named after Ulli Wolff, is an algorithm for Monte Carlo simulation of the Ising model and Potts model in which the unit to be flipped is not a single spin (as in the heat bath or Metropolis algorithms) but a cluster of them.
In statistical mechanics, the two-dimensional square lattice Ising model is a simple lattice model of interacting magnetic spins. The model is notable for having nontrivial interactions, yet having an analytical solution. The model was solved by Lars Onsager for the special case that the external magnetic field H = 0. [1]
A Boltzmann machine (also called Sherrington–Kirkpatrick model with external field or stochastic Ising model), named after Ludwig Boltzmann, is a spin-glass model with an external field, i.e., a Sherrington–Kirkpatrick model, [1] that is a stochastic Ising model. It is a statistical physics technique applied in the context of cognitive ...
The Ising model can then be viewed as the case = of the -state Potts model, whose parameter can vary continuously, and is related to the central charge of the Virasoro algebra. In the critical limit, connectivities of clusters have the same behaviour under conformal transformations as correlation functions of the spin operator.