Ann. Henri Poincar´e 19 (2018), 1115–1150
2018 The Author(s).
This article is an open access publication
published online February 24, 2018
Annales Henri Poincar´e
Convergence of Density Expansions of
Correlation Functions and the Ornstein–
Tobias Kuna and Dimitrios Tsagkarogiannis
Abstract. We prove absolute convergence of the multi-body correlation
functions as a power series in the density uniformly in their arguments.
This is done by working in the context of the cluster expansion in the
canonical ensemble and by expressing the correlation functions as the de-
rivative of the logarithm of an appropriately extended partition function.
In the thermodynamic limit, due to combinatorial cancellations, we show
that the coeﬃcients of the above series are expressed by sums over some
class of two-connected graphs. Furthermore, we prove the convergence of
the density expansion of the “direct correlation function” which is based
on a completely diﬀerent approach and it is valid only for some integral
norm. Precisely, this integral norm is suitable to derive the Ornstein–
Zernike equation. As a further outcome, we obtain a rigorous quantiﬁca-
tion of the error in the Percus–Yevick approximation.
Correlation functions of interacting particle systems provide important infor-
mation of the macroscopic as well as the microscopic properties of the system.
This was well captured already in the literature in the 30’s, see . Around
the same period, with the development of power series expansions by Mayer
and Mayer , a direct perturbative representation of correlation functions in
terms of integrals over conﬁgurations associated with a graphical expansion has
been suggested in , where the density expansion of the n-body correlation
function has been derived. However, being perturbative expansions around the
ideal gas, the density expansions of the correlation functions are not expected
to be valid at the densities of the liquid regime. So, one tries to develop a
theory of classical ﬂuids without using the density expansion formulas,.