Investigation of the Conditions of the Asymptotic Existence of the Configuration Integral of the Gibbs Distribution
Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 4, pp. 626-643
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Let $V$ be a cube of dimension $\nu$, with volume $|V|$. Let ${{|V|}/{N \to\nu}}$, $N\to\infty$. Let ${\mathbf x}=(x_1,\cdots ,x_N)$, $x_i\in V$, $i=1,\cdots ,N$,
$$
Q(V,N)=\int_V\dotsi\int_V\exp\{-\beta U({\mathbf x})\}\,dx_1\dots dx_N,
$$
where
$$
U({\mathbf x})=\sum_{1\leqq i\leqq N}\Phi(|x_i-x_j|).
$$
The conditions on $\Phi(y)$, which are sufficient and in some sense necessary for the existence of the finite limit
$$
\lim_{N\to\infty}\frac1N\log\frac1{{N!}}Q(V,N)
$$
are given.
@article{TVP_1964_9_4_a3,
author = {R. L. Dobru\v{s}in},
title = {Investigation of the {Conditions} of the {Asymptotic} {Existence} of the {Configuration} {Integral} of the {Gibbs} {Distribution}},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {626--643},
publisher = {mathdoc},
volume = {9},
number = {4},
year = {1964},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1964_9_4_a3/}
}
TY - JOUR AU - R. L. Dobrušin TI - Investigation of the Conditions of the Asymptotic Existence of the Configuration Integral of the Gibbs Distribution JO - Teoriâ veroâtnostej i ee primeneniâ PY - 1964 SP - 626 EP - 643 VL - 9 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TVP_1964_9_4_a3/ LA - ru ID - TVP_1964_9_4_a3 ER -
%0 Journal Article %A R. L. Dobrušin %T Investigation of the Conditions of the Asymptotic Existence of the Configuration Integral of the Gibbs Distribution %J Teoriâ veroâtnostej i ee primeneniâ %D 1964 %P 626-643 %V 9 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/item/TVP_1964_9_4_a3/ %G ru %F TVP_1964_9_4_a3
R. L. Dobrušin. Investigation of the Conditions of the Asymptotic Existence of the Configuration Integral of the Gibbs Distribution. Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 4, pp. 626-643. http://geodesic.mathdoc.fr/item/TVP_1964_9_4_a3/