FROM GELATION TO GAS-LIQUID TRANSITION AND SOME CONTRIBUTIONS TO BOSE-EINSTEIN CONDENSATION IN FINITE SYSTEMS

This work consists of four parts. In the first part, we have discussed the recent reformulation of the classical gelation theory of Flory and Stockmayer that has been furnished by Donoghue and Gibbs. We have carried out an analysis of the mean cluster size distribution in the light of the modern scaling theories. We have concluded that the "critical exponents" that describe the cluster size distribution near the gel point are not significantly different from the classical values. This indicates that so far as the 'critical' properties are concerned the Flory-Stockmayer theory and the Donoghue-Gibbs theory are essentially the same. This lead us to the conclusion that exclusion of intramolecular reaction is the predominant cause of the failure of both these two theories near the gel point. In order to extend the theory of Donoghue and Gibbs to include the formation of rings, we have derived a new recursion relation and have computed m(,nj), the mean number of clusters consisting of n structural units and j bonds. This distribution is in good agreement with the recent Monte Carlo results of Falk and Thomas.; The second part of this work is concerned with the reformulation of the Mayer theory of condensation. The mathematical similarity of Mayer's canonical partition function and Stockmayer's expression for total number of ways of forming a polymer system of M molecules from N monomers have been used to derive a set of recursion relations which have been used to compute the partition function exactly. We have used the available star integrals for Lennard-Jones fluid to compute the reducible cluster integrals. It has been shown by explicit numerical computation that a biomodality in the mean cluster size distribution can signal the onset of gas (--->) liquid transition in a finite system. We have also shown that only tree formation, i.e., chain branching without formation of rings, is sufficient to describe the cooperative essence of the gas liquid transition. We have also discussed the effect of successive addition of higher star integrals on pressure and the mean cluster size distribution. All of our results are compared with the Monte Carlo results of Hansen and Verlet.; The third part deals with two and three dimensional finite ideal Bose gases. The Kahn-Uhlenbeck quantum cluster expansion for canonical partition function has been used to compute several physical quantities exactly. We have derived an exact volume dependent cluster integrals for ideal Bose gas under different boundary conditions (b.c.). Numerical calculations have been carried out for periodic (p), anti-periodic (a.p.) and mixed b.c.s. Values of pressure and specific heat are calculated for several system sizes. For p.b.c., rapid convergence to infinite system result is obtained. In mean cluster size distribution, we have observed the appearance of a constant region equal to unity. This is observed for both two and three dimensional ideal Bose gases under p.b.c. This arises from the fact that canonical partition function Q(,N) becomes independent of N for N > l where l depends on the reduced volume (V/(lamda)('3)) of the system.; In the last part, we have applied our formalism to a realistic problem: the problem of specific heat anomaly of low density submonolayer He('4) films adsorbed on a graphite surface. We have modelled the system as a 2-d ideal Bose gas under a lateral harmonic potential which may arise from surface inhomogeneities. We have computed the cluster integrals by the use of a quasi-classical approximation. We have observed a bimodality in the mean cluster size distribution precisely at the temperature where the maximum is peaked. This indicates the occurrence of a spatial Bose condensation at that temperature.