Ising Model - Definition

Definition

Given a graph Λ (for example, a d-dimensional lattice), per each lattice site j ∈ Λ there is a discrete variable σj such that σj ∈{+1, −1}. A spin configuration, σ = (σj)j∈Λ is an assignment of spin value to each lattice site.

For any two adjacent sites i, j ∈Λ one has an interaction Jij, and a site i ∈ Λ has an external magnetic field hi. The energy of a configuration σ is given by the Hamiltonian Function

H(\sigma) = - \sum_{<i~j>} J_{ij} \sigma_i \sigma_j -\sum_{j} h_j\sigma_j

where the first sum is over pairs of adjacent spins (every pair is counted once). indicates that sites i and j are nearest neighbors. The configuration probability is given by the Boltzmann distribution with inverse temperature β ≥0:

P_\beta(\sigma) ={e^{-\beta H(\sigma)} \over Z_\beta}, \,

where β = (kBT)-1

and the normalization constant

Z_\beta = \sum_\sigma e^{-\beta H(\sigma)} \,

is the partition function. For a function f of the spins ("observable"), one denotes by

the expectation (mean value) of f.

The configuration probabilities represent the probability of being in a state with configuration σ in equilibrium.

Read more about this topic:  Ising Model

Famous quotes containing the word definition:

    “I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

    “It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
    —François, Duc De La Rochefoucauld (1613–1680)

    “According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
    —Ana Castillo (b. 1953)