Thermodynamic limit in the elastic triangle: Padé approximants

The equilibrium properties of a triangular pile of N rows of particles, interconnected by linear springs and subjected to gravity are explored. A preliminary algebraic investigation of the small N value case suggests that the particle displacement field can be represented by low order Padé approximants. [1,1] Padé approximant representations are introduced for the horizontal and vertical displacement field components. Each representation contains three polynomials in the particle position coordinates. In the vertical displacement case the polynomials are successfully fitted to numerical solutions of the equilibrium equations. The resulting expression exhibits a thermodynamic limit with 1/N corrections. This procedure fails in the horizontal case. It is suspected that a higher order Padé approximant is required.