  In this article we review a less known unperturbative and powerful many-body
method in the framework of classical statistical mechanics and then we show how
it works by means of explicit calculations for a nontrivial classical model.
The formalism of two-time Green functions in classical statistical mechanics is
presented in a form parallel to the well known quantum counterpart, focusing on
the spectral properties which involve the important concept of spectral
density. Furthermore, the general ingredients of the classical spectral density
method (CSDM) are presented with insights for systematic nonperturbative
approximations to study conveniently the macroscopic properties of a wide
variety of classical many-body systems also involving phase transitions. The
method is implemented by means of key ideas for exploring the spectrum of
elementary excitations and the damping effects within a unified formalism.
Then, the effectiveness of the CSDM is tested with explicit calculations for
the classical $d$-dimensional spin-$S$ Heisenberg ferromagnetic model with
long-range exchange interactions decaying as $r^{-p}$ ($p>d$) with distance $r$
between spins and in the presence of an external magnetic field. The analysis
of the thermodynamic and critical properties, performed by means of the CSDM to
the lowest order of approximation, shows clearly that nontrivial results can be
obtained in a relatively simple manner already to this lower stage. The basic
spectral density equations for the next higher order level are also presented
and the damping of elementary spin excitations in the low temperature regime is
studied. The results appear in reasonable agreement with available exact ones
and Monte Carlo simulations and this supports the CSDM as a promising method of
investigation in classical many-body theory.
