Grid Size:
Temperature = 100 K
Update steps : 150
Modify the temperature to see the system undergo a phase transition.
The Ising model is used to describe the behavior of interacting spins arranged on a crystalline lattice
(in practice, often a square grid).
It consists of N spins arranged on a lattice, each of which has
a state +1 or -1 (up or down) that can be flipped to the opposite state under the influence of
neighboring spins or thermal excitation. This model, first proposed as a problem by Wilhelm Lenz to
his student Ernst Ising in 1920, was in fact the onset of a very active topic of research in
mathematical physics, with applications to the understanding of phase transitions.
Grid Size:
Temperature = 100 K
Update steps : 150
Modify the temperature to see the system undergo a phase transition.
The one-dimensional Ising model was solved analytically by Ernst Ising in his 1924 Ph.D. thesis, where he
discovered that this simple model does not exhibit a phase transition. However, the 2D version,
solved analytically by Lars Onsager in 1944, shows a transition (can you find at which temperature in the simulation above ?).
We know that the three-dimensional model can also undergo a phase transition, but its complete analytical solution
has not yet been found, and proving that such an analytical solution actually exists is still a current
research topic (see
this source).
This model provides a simple, yet very powerful, explanation of transition behaviors in interacting-lattice systems,
such as the Curie Temperature of demagnetization for ferromagnetic materials.
For theoretical details on the Ising model, see this introductory pdf on the one and two-dimensional case, these video lectures (in French) on the d-dimensional case by Hugo Duminil-Copin, and for even more details, this advanced course on the Ising and Pott models by the same author.