https://en.wikipedia.org/wiki/Ising_model
The Ising model 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). The spins are arranged in a periodical 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 but heat disturbs this tendency, thus creating the possibility of different structural phases. 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.
# Spins on the 2D lattice 10x10 # # - + + + + - + + - + # + - + + - - + + + - # + + - + - - + - + + # - + - + + + + - + - # - - - - - + + - + + # + - + - + - + - - - # - + + - - + - + + - # + + - + - + + - - + # + - - + - + + - + - # + - - - - - - + + +
The energy of a spin configuration S = {S_i} is given by the Hamiltonian function
H(S) = -J \sum_ij S_i S_j,
where the sum is over pairs of adjacent spins (nearest neighbors),
S_i is a spin configuration at a site i,
J is an interaction [J has dimensions of energy],
β = 1 / (k_B T) [k_B T has dimensions of energy],
k_B is the Boltzmann constant.
If J > 0 then the interaction is called 'ferromagnetic'.
If J < 0 then the interaction is called 'antiferromagnetic'.
If J = 0 then the spins are 'noninteracting'.
Z(β) = \sum_S exp(-β H(S)) is the partition function.
The configuration probability is given by the Boltzmann distribution
P_β(S) = exp(-β H(S)) / Z(β).
The expectation (mean) value of a function F of the spins is
<F>_β = \sum_S F(S) P_β(S).
The mean energy is <H>_β.
Magnetization M(S) = \sum_i S_i.
The mean magnetization is <M>_β.
Problem: for a 10x10 lattice there is 2^100 ~ 10^30 possible configurations. We are not able to calculate Z(β), P_β(S), and mean values. This motivates the reason for the Ising model to be simulated using MC methods.
In the ferromagnetic case there is a phase transition.
The transition temperature is T_c = 2.2692 [1944 Onsager].
In the ferromagnetic case there is a continuous phase transition.
The transition temperature is T_c = 4.5116 (the critical point).