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Mechanical and thermodynamic systems of a very few atoms ( N < 100) interacting through central-forces are considered in the light of their role in an ideal model for " precipitation ". We discuss the stability of such systems at low energies and present detailed numerical investigations of the potential-energy surfaces of " clusters " of up to thirteen atoms interacting through Lennard-Jones and Morse potentials. Our main results comprise what we believe to be an almost exhaustive survey of the distinct minima available to systems of this size for the potentials used. Each such stable structure obtained is vibrationally and rotationally analysed and its symmetry is examined.The most striking feature of these results is the extreme sensitivity of the number of possible stable configurations to the range and softness of the pair potential. Thus, of no fewer than 988 minima for 13 Lennard-Jones atoms, only some 36 are supported by the (a = 3) Morse potential. The minima available are also classified geometrically and it is shown that non-crystallographic configurations predominate in structures of both greatest and least binding energy.A preliminary account of the statistical mechanics of cluster systems based on the rigid-rotor/harmonic oscillator approximation is given.

Abstract. We present here a probabilistic approach to the generation of new polynomials in two discrete variables. This extends our earlier work on the 'classical' orthogonal polynomials in a previously unexplored direction, resulting in the discovery of an exactly soluble eigenvalue problem corresponding to a bivariate Markov chain with a transition kernel formed by a convolution of simple binomial and trinomial distributions. The solution of the relevant eigenfunction problem, giving the spectral resolution of the kernel, leads to what we believe to be a new class of orthogonal polynomials in two discrete variables. Possibilities for the extension of this approach are discussed.

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