Ising models
E46143
Ising models are mathematical models in statistical mechanics that describe systems of interacting binary variables (spins) on a lattice, widely used to study phase transitions, magnetism, and as a foundation for various probabilistic and machine learning models.
Aliases (4)
Statements (56)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical model
→
statistical mechanics model → |
| canInclude |
long-range interactions
→
|
| computationalComplexity |
NP-hard for general graphs
→
|
| describes |
interacting binary variables
→
spins on a lattice → |
| equivalentTo |
binary Markov random field
→
|
| exactSolution |
two-dimensional zero-field case
→
|
| exactSolutionBy |
Lars Onsager
→
|
| field |
condensed matter physics
→
machine learning → probability theory → statistical mechanics → |
| generalization |
Heisenberg model
→
Potts model → spin glass models → |
| HamiltonianForm |
H = -∑_{⟨i,j⟩} J_{ij} s_i s_j - ∑_i h_i s_i
→
|
| interactionParameter |
J_{ij}
→
|
| interactionType |
nearest-neighbor interaction
→
|
| localFieldParameter |
h_i
→
|
| namedAfter |
Ernst Ising
→
|
| noFiniteTemperatureTransition |
one dimension
→
|
| OnsagerSolutionYear |
1944
→
|
| parameter |
inverse temperature β
→
|
| partitionFunctionSymbol |
Z
→
|
| phaseTransitionDimension |
three dimensions
→
two dimensions → |
| probabilityDistributionForm |
P(s) ∝ exp(-βH(s))
→
|
| proposedBy |
Wilhelm Lenz
→
|
| relatedConcept |
Curie temperature
→
critical exponents → spontaneous magnetization → universality class → |
| relatedOptimizationFormulation |
quadratic unconstrained binary optimization
→
|
| representation |
graphical model
→
|
| spinValues |
+1
→
-1 → |
| spinVariableSymbol |
s_i
→
|
| typicalLattice |
one-dimensional lattice
→
three-dimensional lattice → two-dimensional lattice → |
| usedFor |
Boltzmann machine design
→
Markov random fields → combinatorial optimization → image denoising → probabilistic graphical modeling → quantum annealing formulations → study of critical phenomena → study of ferromagnetism → study of phase transitions → |
| usedIn |
computational biology
→
neuroscience network modeling → social interaction models → |
| variableType |
binary spin
→
|
| yearAnalyzedByIsing |
1924
→
|
| yearProposed |
1920
→
|
Referenced by (5)
| Subject (surface form when different) | Predicate |
|---|---|
|
Hopfield network
("Ising model")
→
Hopfield network ("Ising Hopfield model") → |
isRelatedTo |
|
Hendrik Anthony Kramers
("Kramers–Wannier duality")
→
Lars Onsager ("Crystal statistics. I. A two-dimensional model with an order-disorder transition") → |
notableWork |
|
Boltzmann machines
→
|
relatedTo |