Specific Heat and Entropy Bounds for Ising Spin Systems

Abstract

For quite general Ising spin systems of arbitrary dimensionality and with ferromagnetic interactions of arbitrary range, an entropy lower bound is obtained from Griffiths' inequalities and convexity relations. The bound is valid for nonnegative magnetic fields which may vary from site to site. The specific heat computed from the bound is shown to be an upper bound for the exact specific heat. If the magnetic field is taken to be zero, the entropy bound is shown to be tighter than that obtained previously by truncating a cumulantlike entropy expansion at second order. The present lower bound achieves equality with the exact entropy of the nearest-neighbor, N-spin, Ising ring as N → ∞ for zero field and positive temperatures. An interpretation of the entropy bound is that if in the truncated expansion only binary terms arising from primary correlation are retained, the result is, nevertheless, an overestimate of the order.