Correlation inequality in statistical mechanics
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. Informally, it says that in ferromagnetic spin systems, if the 'a-priori distribution' of the spin is invariant under spin flipping, the correlation of any monomial of the spins is non-negative; and the two point correlation of two monomial of the spins is non-negative.
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.[3] A more general formulation was given by Ginibre,[4] and is now called the Ginibre inequality.
Definitions
Let
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
be the product of the spins in A.
Assign an a-priori measure dμ(σ) on the spins;
let H be an energy functional of the form
![{\displaystyle H(\sigma )=-\sum _{A}J_{A}\sigma _{A}~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22d1fe2d7c52ba6913104ce90de807fd6493c52d)
where the sum is over lists of sites A, and let
![{\displaystyle Z=\int d\mu (\sigma )e^{-H(\sigma )))](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9ac26cd2937f90c7ee1aec91854eafc825bfc6)
be the partition function. As usual,
![{\displaystyle \langle \cdot \rangle ={\frac {1}{Z))\sum _{\sigma }\cdot (\sigma )e^{-H(\sigma )))](https://wikimedia.org/api/rest_v1/media/math/render/svg/7302df33c913d4a9c262d1796e338e57c70fa9e8)
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
![{\displaystyle \tau _{k}={\begin{cases}\sigma _{k},&k\neq j,\\-\sigma _{k},&k=j.\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c02b247b1783eec2beaba2928b22307026c337fe)
Statement of inequalities
First Griffiths inequality
In a ferromagnetic spin system which is invariant under spin flipping,
![{\displaystyle \langle \sigma _{A}\rangle \geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/870e3f40058144c0f0b0ef0263dd8c67402064aa)
for any list of spins A.
Second Griffiths inequality
In a ferromagnetic spin system which is invariant under spin flipping,
![{\displaystyle \langle \sigma _{A}\sigma _{B}\rangle \geq \langle \sigma _{A}\rangle \langle \sigma _{B}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3919b9d17dbeec5c8bd90de9ccb023b957dc9ee)
for any lists of spins A and B.
The first inequality is a special case of the second one, corresponding to B = ∅.
Proof
Observe that the partition function is non-negative by definition.
Proof of first inequality: Expand
![{\displaystyle e^{-H(\sigma )}=\prod _{B}\sum _{k\geq 0}{\frac {J_{B}^{k}\sigma _{B}^{k)){k!))=\sum _{\{k_{C}\}_{C))\prod _{B}{\frac {J_{B}^{k_{B))\sigma _{B}^{k_{B))}{k_{B}!))~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3676d7c90ebc1c083a2b1bbf4b3c795678dc45e2)
then
![{\displaystyle {\begin{aligned}Z\langle \sigma _{A}\rangle &=\int d\mu (\sigma )\sigma _{A}e^{-H(\sigma )}=\sum _{\{k_{C}\}_{C))\prod _{B}{\frac {J_{B}^{k_{B))}{k_{B}!))\int d\mu (\sigma )\sigma _{A}\sigma _{B}^{k_{B))\\&=\sum _{\{k_{C}\}_{C))\prod _{B}{\frac {J_{B}^{k_{B))}{k_{B}!))\int d\mu (\sigma )\prod _{j\in \Lambda }\sigma _{j}^{n_{A}(j)+k_{B}n_{B}(j)}~,\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01b88b3af5a558017d87ae44f43e4a82c5f303d0)
where nA(j) stands for the number of times that j appears in A. Now, by invariance under spin flipping,
![{\displaystyle \int d\mu (\sigma )\prod _{j}\sigma _{j}^{n(j)}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546ced6bae49da84fe3e550f86ecfeccfcca17d2)
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.
Proof of second inequality. For the second Griffiths inequality, double the random variable, i.e. consider a second copy of the spin,
, with the same distribution of
. Then
![{\displaystyle \langle \sigma _{A}\sigma _{B}\rangle -\langle \sigma _{A}\rangle \langle \sigma _{B}\rangle =\langle \langle \sigma _{A}(\sigma _{B}-\sigma '_{B})\rangle \rangle ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0abdfdbf73bb0f8f8520a10754dc5161dabb335d)
Introduce the new variables
![{\displaystyle \sigma _{j}=\tau _{j}+\tau _{j}'~,\qquad \sigma '_{j}=\tau _{j}-\tau _{j}'~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59ed01d44e5ab8e7bc50374a5c73a852033d1328)
The doubled system
is ferromagnetic in
because
is a polynomial in
with positive coefficients
![{\displaystyle {\begin{aligned}\sum _{A}J_{A}(\sigma _{A}+\sigma '_{A})&=\sum _{A}J_{A}\sum _{X\subset A}\left[1+(-1)^{|X|}\right]\tau _{A\setminus X}\tau '_{X}\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42a78f2b066a5afb2621416d89eb7d720e374005)
Besides the measure on
is invariant under spin flipping because
is.
Finally the monomials
,
are polynomials in
with positive coefficients
![{\displaystyle {\begin{aligned}\sigma _{A}&=\sum _{X\subset A}\tau _{A\setminus X}\tau '_{X}~,\\\sigma _{B}-\sigma '_{B}&=\sum _{X\subset B}\left[1-(-1)^{|X|}\right]\tau _{B\setminus X}\tau '_{X}~.\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f551dad3777f3261341a434630f2edd25da29c7a)
The first Griffiths inequality applied to
gives the result.
More details are in [5] and.[6]
Extension: Ginibre inequality
The Ginibre inequality is an extension, found by Jean Ginibre,[4] of the Griffiths inequality.
Formulation
Let (Γ, μ) be a probability space. For functions f, h on Γ, denote
![{\displaystyle \langle f\rangle _{h}=\int f(x)e^{-h(x)}\,d\mu (x){\Big /}\int e^{-h(x)}\,d\mu (x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/014e5e82e38c2ad9e9e758a5f01f19fa05a6559e)
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 ±,
![{\displaystyle \iint d\mu (x)\,d\mu (y)\prod _{j=1}^{n}(f_{j}(x)\pm f_{j}(y))\geq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2e4301b4baf1b7b7f1354dd5ae1259b4ab702fa)
Then, for any f,g,−h in the convex cone generated by A,
![{\displaystyle \langle fg\rangle _{h}-\langle f\rangle _{h}\langle g\rangle _{h}\geq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e237195ee25891855cbaaa2c843ff99b8e20ed6)
Proof
Let
![{\displaystyle Z_{h}=\int e^{-h(x)}\,d\mu (x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c46a800829b46539b65f22d07fca771bd7ea5d43)
Then
![{\displaystyle {\begin{aligned}&Z_{h}^{2}\left(\langle fg\rangle _{h}-\langle f\rangle _{h}\langle g\rangle _{h}\right)\\&\qquad =\iint d\mu (x)\,d\mu (y)f(x)(g(x)-g(y))e^{-h(x)-h(y)}\\&\qquad =\sum _{k=0}^{\infty }\iint d\mu (x)\,d\mu (y)f(x)(g(x)-g(y)){\frac {(-h(x)-h(y))^{k)){k!)).\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc491c25c6d2a9ef3881677f358388f2fe0c9b48)
Now the inequality follows from the assumption and from the identity
![{\displaystyle f(x)={\frac {1}{2))(f(x)+f(y))+{\frac {1}{2))(f(x)-f(y)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/423349f2cd4c73e2206fb62d9501bd2d85d4fd43)
Examples