01-185 V. Beltsky, E.A. Pechersky

Uniqueness of Gibbs State for Non-Ideal Gas in ${\bf R}^d$: the Case of Multibody Interaction (58K, LaTeX 2e) May 21, 01

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Abstract. We study the question of existence and uniqueness of non-ideal gas in ${\mathbb R}^d$ with multi-body interactions among its particles. For each $k$-tuple of the gas particles, $2\leq k\leq m_0<\infty$, their interaction is represented by a potential function $\Phi_k$ of a finite range. We introduce a {\em stabilizing} potential function $\Phi_{k_0}$, such that $\Phi(x_1, \ldots, x_{k_0})$ grows sufficiently fast, when diam$\{x_1, \ldots, x_{k_0}\}$ shrinks to $0$. Our results hold under the assumption that at least one of the potential functions is stabilizing, that causes a sufficiently strong repulsive force due to the stabilizing potential. We prove that {\it (i)} for any temperature there exists at least one Gibbs field, and {\it (ii)} there exists exactly one Gibbs filed $\xi$ at low enough temperature, such that ${\mathbb E} e^{\chi |\xi_V|}<\infty$ for any real $\chi>0$ and any small volume $V$. The proofs use the criterion of the uniqueness of Gibbs field in non-compact case developed in (\cite{DP}), and the technique employed in (\cite{PZh}) for studying a gas with pair interaction.

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