  We study self-consistent approximations such as the $\Phi$-derivable and
virial approaches to dilute strongly interacting systems in equilibrium. We
consider a system of non-relativistic fermions of one kind interacting via a
pair potential Thermodynamical quantities are expressed in terms of various
spectral functions. We review the $\Phi$ derivable approximation scheme
demonstrating the exact conservation of the Noether and the Botermans-Malfliet
fermion number densities, and then for $\Phi$ described by the tadpole and
sandwich diagrams we show the coincidence of these two number densities. As
examples of test pair potentials, we consider the Yukawa central
nucleon-nucleon potentials within Walecka, CD Bonn, and Reid parameterizations,
and the corresponding classical Lennard-Jones potentials. Expressions for the
second and third virial coefficients are derived and analyzed for $\Phi$
described by the tadpole and sandwich diagrams. Next, we focus on the virial
approach to the equation of state. Classical, semiclassical, and purely quantum
approaches are studied in detail. Then, different extrapolations of the virial
equation of state are considered including the van~der~Waals form and
excluded-volume models. We derive the expression for the second virial
coefficient using the effective range approximation for the scattering
amplitude and compare the result with the purely quantum result using the
experimental phase shifts. Attention is focused on the problem of anomalously
large value of the nucleon-nucleon scattering length appearing due to the
presence of the quasi-bound state in nucleon-nucleon scattering, which can be
destroyed in the matter because of the action of the Pauli blocking. We present
results for the second virial coefficient subtracting this term. We discuss the
validity of such a procedure to describe the equation of state of the nuclear
matter in the virial limit.
