Physics of Fractal Operators

1 Non-differentiable processes.- 1.1 Classical mechanics.- 1.2 Langevin equation.- 1.3 Comments on the physics of the fractional calculus.- 1.4 Commentary.- 2 Failure of traditional models.- 2.1 Fractals; geometric and otherwise.- 2.2 Generalized Weierstrass function.- 2.3 Fractional operators.- 2.4 Intervals of the generalized Weierstrass function.- 2.5 Commentary.- 3 Fractional dynamics.- 3.1 Elementary properties of fractional derivatives.- 3.2 The generalized exponential functions.- 3.3 Parametric derivatives.- 3.4 Commentary.- 4 Fractional Fourier transforms.- 4.1 A brief review of Fourier analysis.- 4.2 Linear fields.- 4.3 Fourier transforms in the fractional calculus.- 4.4 Generalized Fourier transform.- 4.5 Commentary.- 5 Fractional Laplace transforms.- 5.1 Solving differential equations.- 5.2 Generalized exponentials.- 5.3 Fractional Green’s functions.- 5.4 Commentary.- 6 Fractional randomness.- 6.1 Ordinary random walk.- 6.2 Continuous-time random walk.- 6.3 Fractional random walks.- 6.4 Fractal stochastic time series.- 6.5 Evolution of probability densities.- 6.6 Langevin equation with Lévy statistics.- 6.7 Commentary.- 7 Fractional Rheology.- 7.1 History and definitions.- 7.2 Fractional relaxation.- 7.3 Path integrals.- 7.4 Commentary.- 8 Fractional stochastics.- 8.1 Fractional stochastic equations.- 8.2 Memory kernels.- 8.3 The continuous master equation.- 8.4 Back to Langevin.- 9 The ant in the gurge metaphor.- 9.1 Lévy statistics and renormalization.:.- 9.2 An ad hoc derivation.- 9.3 Fractional eigenvalue equation.- 9.4 Fractional stochastic oscillator.- 9.5 Fractional propagation-transport equation.- 9.6 Commentary.- 10 Appendix.- 10.1 Special functions.- 10.2 Fractional derivatives.- 10.3 Mellin transforms.