Analytic and clustering properties of thermodynamic functions and distribution functions for classical lattice and continuum systems

Abstract

Our most complete results concern the Ising spin system with purely ferromagnetic interactions in a magnetic fieldH (or the corresponding lattice gas model with fugacityz=const. exp(−2mHβ) wherem is the magnetic moment of each spin). We show that, in the limit of an infinite lattice, (i) the free energy per site and the distribution functionsn _s (x ₁, ...,x _s ; β,z) are analytic in the two variables β andH if the reciprocal temperature β>0 and the complex numberH is not a limit point of zeros of the grand partition function ξ, and (ii) the Ursell functionsu _s (x ₁, ...,x _s ; β,z) tend to 0 as Δ _s ≡Max _{i, j} |x _i −x _j | → ∞ if β>0 and ReH≠0; in particular, if the interaction potential vanishes for separations exceeding some fixed cutoff value λ, then |u _s |<C exp [(−2 βm |ReH|+ε) Δ _s /λ] where ε is any small positive number andC is independent of Δ _s . One consequence of the result (i) is that a phase transition can occur as β is varied at constantH only ifH is a limit point of zeros of ξ (which can happen only if ReH=0); this supplements Lee and Yang's result that the same condition is necessary for a phase transition whenH is varied at constant β.

For a lattice or continuum gas with non-negative interaction potential (corresponding, in the lattice case, to an Ising antiferromagnet), similar results are shown to hold provided β>0 and the complex fugacityz is less than the radius of convergence of the Mayerz expansion; for the continuum gas, however,n _s andu _s must be replaced by their values integrated over small volumes surrounding each of the pointsx ₂, ...,x _s .

It is shown that the pressurep is analytic in both β andz, if it is analytic inz at fixed β over a suitable range of values of β andz, and further that, except for continuum systems without hard cores,p,n _s andu _s have convergent Maclaurin expansions in β for small enoughz.