Uniqueness of Gibbs state for non-ideal gas in Rd:: The case of multibody interaction

We study the question of existence and uniqueness of non-ideal gas in R-d with multi-body interactions among its particles. For each k-tuple of the gas particles, 2 less than or equal to k less than or equal to m(0) less than or equal to infinity, their interaction is represented by a potential function Phi(k) of a finite range. We introduce a stabilizing potential function Phi(ko), such that grows sufficiently fast, when diam{x(1,)..., x(ko)} shrinks to 0. Our results hold under the assumption that at least one of the potential functions is stabilizing. which causes a sufficiently strong repulsive force. We prove that (i) for any temperature there exists at least one Gibbs field, and (ii) there exists exactly one Gibbs Field zeta at sufficiently high temperature, such that for any x > 0, Ee(infinity\xiV\) less than or equal to C(V-0) < ∞, for all volumes V smaller than a certain fixed finite volume V., The proofs use the criterion of the uniqueness of Gibbs field in non-compact case developed in ref. 4, and the technique employed in ref. I for studying a gas with pair interaction.