In statistical mechanics, the Griffiths inequality, sometimes also called Griffiths–Kelly–Sherman inequality or GKS inequality, named after Robert B. Griffiths, is a correlation inequality for ferromagnetic spin systems.
The inequality was proved by Griffiths for Ising ferromagnets with two-body interactions,[1] then generalised by Kelly and Sherman to interactions involving an arbitrary number of spins,[2] and then by Griffiths to systems with arbitrary spins.
be a configuration of (continuous or discrete) spins on a lattice Λ.
If A ⊂ Λ is a list of lattice sites, possibly with duplicates, let
Assign an a-priori measure dμ(σ) on the spins; let H be an energy functional of the form where the sum is over lists of sites A, and let be the partition function.
As usual, stands for the ensemble average.
The system is called ferromagnetic if, for any list of sites A, JA ≥ 0.
The system is called invariant under spin flipping if, for any j in Λ, the measure μ is preserved under the sign flipping map σ → τ, where In a ferromagnetic spin system which is invariant under spin flipping, for any list of spins A.
Observe that the partition function is non-negative by definition.
Proof of first inequality: Expand then where nA(j) stands for the number of times that j appears in A.
Now, by invariance under spin flipping, if at least one n(j) is odd, and the same expression is obviously non-negative for even values of n. Therefore, Z<σA>≥0, hence also <σA>≥0.
For the second Griffiths inequality, double the random variable, i.e. consider a second copy of the spin,
Then Introduce the new variables The doubled system
is invariant under spin flipping because
with positive coefficients The first Griffiths inequality applied to
[6] The Ginibre inequality is an extension, found by Jean Ginibre,[4] of the Griffiths inequality.
Let (Γ, μ) be a probability space.
For functions f, h on Γ, denote Let A be a set of real functions on Γ such that.
for every f1,f2,...,fn in A, and for any choice of signs ±, Then, for any f,g,−h in the convex cone generated by A, Let Then Now the inequality follows from the assumption and from the identity