Continuous limit of discrete systems with long-range interaction

Discrete systems with long-range interactions are considered. Continuous medium models as continuous limit of discrete chain system are defined. Long-range interactions of chain elements that give the fractional equations for the medium model are discussed. The chain equations of motion with long-range interaction are mapped into the continuum equation with the Riesz fractional derivative. We formulate the consistent definition of continuous limit for the systems with long-range interactions. In this paper, we consider a wide class of long-range interactions that give fractional medium equations in the continuous limit. The power-law interaction is a special case of this class. PACS: 45.05.+x; 45.50.-j; 45.10.Hj