Fractional Derivatives in Static and Dynamic Scaling

Abstract

A derivative or integral of fractional order is usually defined by analytically continuing a suitable definition of the derivative or integral of integer order n ∈ N to real or complex values of n. My purpose here is to give some of the definitions and then to review brief- ly recent applications of fractional derivatives and integrals in physics with emphasis on static and dynamic scaling. Derivatives of fractional order have recently emerged in physics as generators of time evolutions in ergodic theory [1, 2, 3, 4, 5, 6], and as a tool for classifying phase transitions in thermodynamics by generalizing the classification scheme of Ehrenfest [1, 7, 8, 9, 10]. Given the fact that a fractional integral is, loosely speaking, a convolution operator with a power law kernel it is perhaps not too surprising that fractional integrals and derivatives are useful tools in scaling theory.