Relationships between power-law long-range interactions and fractional mechanics

Lattice fractional calculus

“…A scheme to derive lattice differential operators of integer orders from the discrete velocities and associated Maxwell-Boltzmann distributions that are used in lattice hydrodynamics has been suggested in the articles [51,52]. In this paper to formulate a lattice fractional calculus, we use other approach that is based on models of physical lattices with long-range inter-particle interactions and its continuum limit that are suggested in [41,42,25] (see also [43][44][45][46][47][55][56][57][58][59]). …”

Fermi-Pasta-Ulam chains with harmonic and anharmonic long-range interactions

“…Fractional differentiation is an extension of the order of differentiation from an integer to a real number. We also introduce Riesz derivatives, which are the most useful representation applicable to a model with long-range interaction [36]. In addition, we present fractional integration, which is the inverse operation of fractional differentiation.…”

A Fractional Micro-Macro Model for Crowds of Pedestrians based on Fractional Mean Field Games

“…Long range interactions have been considered in the schooling of fish, flocking of birds and control of multi-agent systems and effects of long range interactions that dominating system's phase transition have just received a lot of attention recently. Based on obtained results in [16], we can say that long range interactions in micro scale are connected with the fractional dynamics in macro scale.…”