Uniqueness of Gibbs State for Non-Ideal Gas in ℝd: The Case of Multibody Interaction

Abstract

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≤k≤m ₀<∞, their interaction is represented by a potential function Φ _k of a finite range. We introduce a stabilizing potential function $\Phi _{k_0}$, such that Φ(x ₁,..., $x_{k_0}$) grows sufficiently fast, when diam{x ₁,..., $x_{k_0}$} 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 ξ at sufficiently high temperature, such that for any χ>0, $\mathbb{E}e^{\chi \left| {\xi _V } \right|}$ ≤ C(V ₀)<∞ for all volumes V smaller than a certain fixed finite volume V ₀. 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. 1 for studying a gas with pair interaction.