STATISTIC MECHANICS FOR LINEAR DEFECT-MEDIATED MELTING

The melting of elemental solids is modelled as a dislocation pair-mediated transition on a lattice. Statistical mechanics of linear defects is used to obtain the relation between melting temperature and shear modulus. It is derived theoretically that the phase transition is the first order and the formula for latent heat is also derived.

DETERMINANTAL IDENTITIES ON INTEGRABLE MAPPINGS

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds of transformations on q×q matrices: the inversion of the q×q matrix and an (involutive) permutation of the entries of the matrix. In a case where the permutation is a particular elementary transposition of two entries, it is shown that the iteration of this group of birational transformations yield algebraic elliptic curves in the parameter space associated with the (homogeneous) entries of the matrix. It is also shown that the successive iterated matrices do have remarkable factorization properties which yield introducing a series of canonical polynomials corresponding to the greatest common factor in the entries. These polynomials do satisfy a simple nonlinear recurrence which also yields algebraic elliptic curves, associated with biquadratic relations. In fact, these polynomials not only satisfy one recurrence but a whole hierarchy of recurrences. Remarkably these recurrences are universal: they are independent of q, the size of the matrices. This study provides examples of infinite dimensional integrable mappings.