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Abstract
In the two separate parts of this thesis, some special exactly solvable models are defined and analysed. In Part I it is proved rigorously that the coherent potential approximation is exact for the effective conductivity of a wide class of hierarchical models made of spherical grains of various conductivities. In Part II the existence of continuum fluid models that exhibit a discontinuity in the pressure versus density isotherms is established by explicit construction of a class of examples. Part II also incorporates a broad study of discontinuities in thermodynamic variables.
The basic models of Part I are constructed as follows: the starting material 0 in the hierarchy is chosen arbitrarily, otherwise material j = 1, 2, ..., consists of equisized spheres, say j-spheres, of arbitrary conductivities embedded in material (j - 1). The distribution of the j-spheres must satisfy a mild homogeneity condition and their radius r(,j) must, asymptotically, increase faster than expotentially with j. The minimum spacing, 2s(,j), between the j-spheres is such that s(,j)/r(,j) diverges. On the basis of these and other ancillary conditions it is established that the coherent potential approximation becomes exact for the effective conductivity (or the dielectric constant or the magnetic permeability) of material j (--->) (INFIN). The model composites and the proof of realizability may be generalised to allow non-spherical grains. By introducing ensembles of composites the homogeneity condition and the spacing condition can be relaxed somewhat. Another related approximation, the iterated dilute limit approximation is also proved to be realizable.
In Part II, one-dimensional continuum models are defined in which classical particles interact through many-body potentials meeting conditions sufficient to ensure a proper thermodynamic limit. An exact analysis proves that for certain ranges of parameter values the pressure versus density isotherms are discontinuous. Extended models are described which exhibit various other unobserved, but thermodynamically allowed, anomalous first-order transitions. A classification of discontinuities is made and a convexity lemma is established which leads to a proof that certain types of discontinuity are thermodynamically forbidden.
