History of the Lenz-Ising Model

1967/10/01 by Stephen G. Brush · 13 citations
Physics and Astronomy · #Theoretical and Computational Physics #Advanced Thermodynamics and Statistical Mechanics #Spectroscopy and Quantum Chemical Studies #Ising model #Physics #Lattice (music) #Statistical physics #Phase transition #Gravitational singularity #Square-lattice Ising model #Theoretical physics #Condensed matter physics #Quantum mechanics

paper · doi:10.1103/revmodphys.39.883

openalex publication_date 1967/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Many physico-chemical systems can be represented more or less accurately by a lattice arrangement of molecules with nearest-neighbor interactions. The simplest and most popular version of this theory is the so-called "Ising model," discussed by Ernst Ising in 1925 but suggested earlier (1920) by Wilhelm Lenz.Major events in the subsequent history of the Lenz-Ising model are reviewed, including early approximate methods of solution, Onsager's exact result for the two-dimensional model, the use of the mathematically equivalent "lattice gas" model to study gas-liquid and liquid-solid phase transitions, and recent progress in determining the singularities of thermodynamic and magnetic properties at the critical point. Not only is there a wide range of possible physical applications of the model, there is also an urgent need for the application of advanced mathematical techniques in order to establish its exact properties, especially in the neighborhood of phase transitions where approximate methods are unreliable.

Citations

• The Factorization of Linear Graphs
• Some theoretical aspects of melting
• Dimer problem in statistical mechanics-an exact result
• Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition
• Correlations and Spontaneous Magnetization of the Two-Dimensional Ising Model
• Some generalized order-disorder transformations
• Statistical hydrodynamics
• The effect of thermal agitation on atomic arrangement in alloys-III
• The effect of thermal agitaion on atomic arrangement in alloys—II

Cited by

• The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
• Statistical mechanics, three-dimensionality and NP-completeness
• 100 Glorious years of the Ising model
• Graph isomorphism problem
• Quantum simulations with cold trapped ions
• Record-breaking statistics near second-order phase transitions
• Stochastic functionals and fluctuation theorem for multikangaroo processes
• Ernst Ising—Physicist and Teacher
• Digitized-counterdiabatic quantum approximate optimization algorithm
• Statistical Mechanics and the Physics of Many-Particle Model Systems
• Turing-like patterns in an asymmetric dynamic Ising model
• Ising model [wikipedia]
• Square lattice Ising model [wikipedia]