Exploring the applications of fractional calculus: Hierarchically built semiflexible polymers

The emergence of self-organization in complex systems–Preface

“…The Tsallis entropy and distribution represent an attempt to build a non-standard statistical mechanics for complex systems, and superstatistics proved to be compatible with systems whose complexity originates from non-homogeneity of some physical parameter in the medium supporting the anomalous transport. It is not yet clear if this picture is in agreement with other modeling approaches, such as the fractional one (see [8,9] and the comment to papers [24,25] reported below). Paper [22] illustrates the application of kinetic theory and hydrodynamic equations to a granular binary mixture, which is an interesting prototype of complexity.…”

“…In papers [24] and [25] the complex behaviour is associated with the emergence of fractional derivative operators [6]. These two papers discuss two alternative approaches in the derivation and interpretation of fractional operators: in [24] the trapping mechanism is modeled, while, in [25], the heterogeneity of the system is considered.…”

“…This result is even more interesting as it has been depicted in the framework of a complex biological system. In recent years, the derivation of fractional operators in the context of disordered, non-homogeneous media has attracted some interest (see, e.g., [8,9]), and paper [25] surely gives an important contribution in shedding light into the relationship between complexity and fractional calculus.…”

“…In paper [25] the topological complexity is explicitly investigated by considering the interplay between structure and dynamics in semiflexible, hierarchically-built fractal polymers under external pulling forces. A Langevin modeling approach is here used with the introduction of a random relaxation parameter.…”

“…These two papers discuss two alternative approaches in the derivation and interpretation of fractional operators: in [24] the trapping mechanism is modeled, while, in [25], the heterogeneity of the system is considered. In paper [24] the authors study "the kinetics of subdiffusionlimited growth and dissolution of nanoprecipitates in disordered solids on the base of subdiffusion equations with fractional derivatives" and compare their analytical results with Monte Carlo stochastic simulations.…”