Short time behaviour of density correlation functions

Abstract

In this thesis the dynamical behaviour of the atoms in a fluid or gas is studied with time dependent correlation functions as the density-density correlation function and the velocity autocorrelation function. Theoretically it is not possible to calculate these correlation functions exactly for the whole time domain. An exact calculation is only possible for times small with respect to the duration of the collision (see Ch. 1), by using the moments expansion, and for times large with respect to the mean free time by solving the hydrodynamical equations. In chapter 2 a method is described, the Ursell expansion, which makes it possible to calculate the correlation functions for times up to the mean free time. Experimentally the density-density correlation function is known on this time scale from neutron scattering on noble gases with a low density. In the Ursell expansion the successive terms describe the effect of an increasing number of colliding particles. For times smaller than the mean free time the most dominant contribution to the correlation functions comes from those collisions in which not more as two particles, are involved. In chapter 2 a detailed expression for the two particle term is derived. It is shown, that due to an approximation for the static three particle correlation function, the moments of the two particle term do not agree completely with the exact raoments. Therefore for continuous potentials another expansion, the second derivative expansion, is derived; in this new expansion the two particle term has the exact moments. Chapter 3 gives the Ursell expansion for the case of a hard spheres interaction; the advantage of this interaction is that the mathematical expressions, that describes the collision, are very easy. Because the moments of the two particle term do not agree with the exact moraents, another expansion, the Ursell-2 expansion, will be derived. This expansion is only valid for hard spheres and reproduces the exact moments. At the end of chapter 3 the results of calculations on the hard spheres system are presented. It is shown that both expansions agree very well with molecular dynaraics calculations. Chapter 4 contains the results of calculations on a system with a Lennard-Jones interaction. It appears that both the Ursell expansion and the second derivative expansion agree very well with molecular dynamics calculations of the incoherent intermediate scattering function. The discrepancy between the theoretically calculated coherent intermediate scattering function and the experimental scattering function is substantial. This may be due to the large experimental error, which is of the same order of magnitude as the deviation of the correlation function from its ideal gas value.