Aspects of the statistical mechanics of gases

Abstract

The ‘raison d'etre' of this thesis may fairly be described as the paucity of useful exact calculations of the statistical mechanical properties of a gas of particles with known interaction between pairs* Formally this problem was solved by J.W. Gibbs in 1902 in terms of the canonical and grand-canonical partition functions, but it was not till 1936 that L. Tonks obtained explicitly the exact equation of state of a one—dimensional gas of non-interacting hard rods. Since then a number of calculations of the thermodynamic properties and distribution functions of classical one-dimensional continuum gases with simple interaction potential have been performed, using either combinatorial analysis or the techniques of statistical theory (Lenard, 1961; Edwards and Lenard, 1962; Prager, 1962; Kac, 1959). Following a re—statement in chapter 1 of the relevant results of Gibbs, it is shown in chapter 2 of this thesis that the statistical mechanical properties of any classical one—dimensional gas may be expressed in terms of the eigenvalues of a functional operator and the corresponding matrix elements of a related operator. This result is derived by the simple device of differentiating the canonical partition function of the gas with respect to the 'volume' (i.e, the length of the line on which the particles are confined). Although purely formal, this result has three significant corollaries : firstly, the 'ring approximation', normally derived by the rather ad hoc procedure of summing those terms in the virial expansion which correspond to potential bond diagrams of ring type (Mayer, 1950 )j can be obtained by a variational approximation; secondly, the distribution functions and their derivatives with respect to the mean particle density satisfy a simple relation; and finally, when the interaction potential satisfies a homogeneous linear differential equation of order p (say), explicit, exact results may be obtained in terms of an eigenvalue equation involving at most p variables# The last corollary ensures that the method may be used to obtain the properties of the one-dimensional plasma# This gas is considered in chapter 3 and explicit exact results obtained for both a system of equal and opposite charges and one of negative charges moving in a uniform neutralizing background of positive charge. In chapter k it is shown that the method of differentiating the canonical partition function is also capable of yielding useful exact results when the interaction potential consists of a repulsive hard core together with an interaction satisfying a homogeneous linear differential equation of finite order# It follows that all the exact results previously obtained for one—dimensional continuum gases with particular interactions may be derived by this technique. It is also shown in this chapter that the formal results of chapter 2 are applicable to the simple Tonk's gas of hard rods, even though the potential function appears to violate the differentiability condition originally imposed in the general derivation. It is therefore reasonable to suppose that the results are va4-id for any ’physical1 potential. In chapter 5 an attempt is made to consider gases of higher dimensionality by replacing the continuum by a lattice and transforming the grand—canonical partition function by a method used by S,F. Edwards (1959)« It is found that this technique is particularly appropriate when the interaction potential satisfies a decaying wave equation (the Coulomb potential is thereby included as a special case), for then the problem becomes mathematically equivalent to that of calculating the canonical partition function of a system with nearest—neighbour interaction. In one dimension such a problem may be solved exactly, but in two dimensions it reduces to one identical with that of calculating the lowest energy level of a one-dimensional quantum mechanical system of particles with Hook’s law attraction between first and second, second and third, third and fourth, etc., and with an applied external potential. Although this problem remains unsolved, it appears to be the obvious starting point for any possible further progress towards the exact calculation of the thermodynamic properties of two— or higher—dimensional gases, in particular plasmas. The form of the relation derived In chapter 2 between the distribution functions of a one—dimensional gas and their derivatives with respect to density suggests that it is applicable in any number of dimensions# In chapter 6 it is shown that this is in fact the case and new functions are defined which satisfy an even simpler relation. As the two—particle function of this set is the Ornstein-Zernike direct correlation function, it seems natural to term them the direct correlation functions# In view of the attractiveness of being able to predict statistical mechanical properties at one density in terms of those at an adjacent density, a closure of the relations by means of a superposition approximation is considered# The resulting equation for the two-particle direct correlation function is particularly straightforward to handle numerically.