We use variational (VMC) and diffusion (DMC) Monte Carlo methods to study the ground state properties of boson or fermion fluids such as the He four or the Jellium. Through path integral Monte Carlo (PIMC) methods we determine the finite temperature properties (both in the canonical and in the grand-canonical ensembles). We study properties like the structure, the pressure, the internal energy, and various other thermodynamic quantities, the superfluid fraction, the fluid phase coexistence.

The Monte Carlo method is exact only for boson fluids. For fermion fluids, the yet unsolved "sign problem" requires the formulation of some approximation in the numerical calculation. So that even computationally we are still unable to extract exact statistical mechanical properties for fermions.

Another limitation of the path integral Monte Carlo is that, whereas it is able to describe the molecules formation from the constituent atomic species, it is unable to describe the atom formation from the constituent electrons and nuclei, unless for an highly diluted system. This is due to the fact that since the mass of an electron is three orders of magnitude smaller than the one of the nucleous the degeneracy temperature of the electons is three orders of magnitude bigger than the one of the nuclei, at a given density. Therefore it is very unlikely that an electron, with a world-line with many particle exchanges, will bind to a nucleous, which has a world-line with many less particle exchanges.

In simulations of systems at room temperature or below and densities n such that T < T_F ∝ n^2/3ℏ²/mk_B, electrons are to a good approximation degenerate, and in most cases the nuclei can be treated as classical. In those cases, there exists an effective potential between the nuclei due to the electrons. Knowing this potential, one could solve most problems of chemical structure with simulation. But it needs to be computed very accurately because the natural electronic energy scale is the Hartree, me⁴/ℏ²), or Rydberg and chemical energies are needed to better than k_BT. This requires an accuracy of one part in 10³ for a hydrogen atom. Higher relative accuracy is needed for heavier atoms. If T>T_F for fixed nuclei spatial positions it is necessary to perform a PIMC on the electrons component and use the result to perform a classical Monte Carlo (MC) calculation on the nuclei.