Baxter, Rodney James

### Description

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...[Show more] 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.

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