On the use of “small external fields” in the problem of symmetry breakdown in statistical mechanics

Abstract

We study the infinite system equilibrium states in the statistical mechanics of classical lattice gases. We show that breakdown of the translation invariance occurs if and only if the derivative dP(Φ + λΨ) dλ is discontinuous at λ = 0 for some Ψ. In this formula, P is the pressure, Φ the translation invariant interaction of the system, and Ψ a “small external field” from a suitable class of nontranslation invariant interactions. In an appendix we show that an Ising ferromagnet in a nonvanishing magnetic field has only one equilibrium state.