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)
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.