Physica A 187 (1992) 456-474 ~ A North-Holland Statistical geometry of hard particles on a sphere S. Prestipino Giarritta Dottorato di Ricerca in Fisica, UniversiM degli Studi di Messina, C.P. 50, 98166 S. Agata, Messina, Italy M. Ferrario and P.V. Giaquinta Istituto di Fisica Teorica, Universitd degli Studi di Messina, C.P. 50, 98166 S. Agata, Messina, Italy Received 14 January 1992 Revised manuscript received 14 April 1992 We present a Monte Carlo study of a two-dimensional system of hard particles embedded on the surface of a sphere. Thermodynamic and structural evidence of an ordering phase transition is found at high densities in spite of the frustration induced on the hexagonal covering by the peculiar topology of the host surface. The nature of this transition is analyzed and contrasted with the fluid-solid transition occurring in a flat geometry. 1. Introduction The classical Landau argument about the first-order character of the melting transition is supported in three dimensions by the evidence that solids show lattice order (of low symmetry) while liquids visit mostly disordered states (the liquid phase is one of high symmetry). In two dimensions (2D) only quasi-long- range translational order may survive in the "lattice", as first shown by Peierls [1] and Landau [2] within the harmonic approximation, and by Mermin under more general hypotheses on the shape of the interaction potential [3]. If u(R) represents the deviation from equilibrium of the atom oscillating about site R, then (lu(R) - u(R')] 2 > ~lnlR- R'I, as ]R-R'I --~o~ . (1.1) Obviously, this fact is of little importance for finite systems [4]; however, it may be crucial if we are interested in the destiny of ideally infinite systems. 0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved