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

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We study the question of existence and uniqueness of non-ideal gas in ℝd with multi-body interactions among its particles. For each k-tuple of the gas particles, 2≤km0<∞, their interaction is represented by a potential function Φk of a finite range. We introduce a stabilizing potential function Φk0, such that Φ(x1,…, xk0) grows sufficiently fast, when diam{x1,…, xk0} 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, 𝔼eχ|ξv|C(V0)<∞ for all volumes V smaller than a certain fixed finite volume V0. 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.

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