Griffiths inequality

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.

  1. ^ Griffiths, R.B. (1967). "Correlations in Ising Ferromagnets. I". J. Math. Phys. 8 (3): 478–483. Bibcode:1967JMP.....8..478G. doi:10.1063/1.1705219.
  2. ^ Kelly, D.J.; Sherman, S. (1968). "General Griffiths' inequalities on correlations in Ising ferromagnets". J. Math. Phys. 9 (3): 466–484. Bibcode:1968JMP.....9..466K. doi:10.1063/1.1664600.
  3. ^ Griffiths, R.B. (1969). "Rigorous Results for Ising Ferromagnets of Arbitrary Spin". J. Math. Phys. 10 (9): 1559–1565. Bibcode:1969JMP....10.1559G. doi:10.1063/1.1665005.
  4. ^ Ginibre, J. (1970). "General formulation of Griffiths' inequalities". Comm. Math. Phys. 16 (4): 310–328. Bibcode:1970CMaPh..16..310G. doi:10.1007/BF01646537. S2CID 120649586.