ABSTRACT
Statistical Mechanics provides exact microscopic expressions and well-defined procedures to obtain thermodynamic properties like specific heat, free energy, entropy and others. These expressions can be easily obtained in terms of partition function although the latter is not accessible experimentally. On the other hand, experiments can provide detailed information about molecular arrangements in space in forms of multi-particle joint probability distribution functions, like radial distribution function, universally denoted by g(r) (where r is the separation distance between two particles). Thermodynamic functions can be obtained in terms of g(r) or other distribution functions. Thanks to extensive neutron scattering experiments, quantitative information about molecular arrangements even at small separations, like the nearest-neighbor distances, are now available for many liquids and polymers. Study of distribution functions requires a different theoretical development. We again start with partition function but derive equations that need to be solved for the distribution functions. In this chapter, we discuss the distribution functions and discuss why they are so powerful in the study of condensed matter physics and chemistry. We also discuss celebrated equations like Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY), Yvon-Born-Green (YBG), Born-Green (BG), Percus-Yevick (PY), hypernetted chain (HNC) equations and discuss how these are solved to obtain radial distribution function. An essential ingredient of the study of distribution function is direct correlation function c(r) that plays a key role in the development of the liquid state theory. It is introduced by the Ornstein-Zernike equation and offers a fruitful approach (than YBG) to the solution for g(r).
