NVT ensemble and the virial theorem

The examination of the equation of state of a two-dimensional model fluid (the hard disk system) was the very first application of the importance sampling Monte Carlo method in statistical mechanics (Metropolis et al., 1953), and since then the study of both atomic and molecular fluids by Monte Carlo simulation has been a very active area of research. Remember that statistical mechanics can deal well analytically with very dilute fluids (ideal gases), and it can also deal well with crystalline solids (making use of the harmonic approximation and perfect crystal lattice periodicity and symmetry), but the treatment of strongly correlated dense fluids (and their solid counterparts, amorphous glasses) is much more difficult. Even the description of short range order in fluids in a thermodynamic state far away from any phase transition is a non-trivial matter (unlike the lattice models discussed in the last chapter, where far away from phase transitions the molecular field approximation, or a variant thereof, is usually both good enough and easily worked out, and the real interest is generally in phase transition problems).

The discussion in this chapter will consider only symmetric particles; for the consideration of hard rods, spherocylinders, etc., the reader is referred elsewhere (Frenkel and Smit, 1996).