Correlation Inequalities for Coupled Oscillators

Abstract

Correlations in a system of N classical, coupled oscillators are studied, with a view toward obtaining a more complete understanding of Griffiths‐type inequalities. The potential energy is assumed to be U = ½ Σ_iΣ_jJ_ijx_ix_j − ΣH_kkx_k, with − ∞ ≤ x_i ≤ ∞ and J_ij = J_ji. The N × N matrix with elements {J_ij} is assumed positive definite. Sufficiency conditions for the correlation functions to satisfy Griffiths‐type inequalities are found to be: (i) K_ij ≥ 0 for all i, j, where K = J⁻¹ and (ii) Σ_jK_ljH_j ≥ 0 for all l. The class of systems obeying (i) and (ii) contains those for which J_ij ≤ 0, i ≠ j, and H_k ≥ 0 for all k; these are direct analogs of Ising ferromagnets. It is proved that a necessary and sufficient condition for Griffiths‐type inequalities to hold for arbitrary {H_k ≥ 0} is simply (i) above. The sufficiency conditions (i) and (ii) are broader than those available to date for Ising models (only the sufficiency condition of ferromagnetic coupling is known). The necessary and sufficient condition (i) has no known Ising counterpart at present.