Author	West, Bruce J. author
Title	Physics of Fractal Operators [electronic resource] / by Bruce J. West, Mauro Bologna, Paolo Grigolini
Imprint	New York, NY : Springer New York, 2003
Connect to	http://dx.doi.org/10.1007/978-0-387-21746-8
Descript	X, 354 p. 23 illus. online resource

This text describes the statistcal behavior of complex systems and shows how the fractional calculus can be used to model the behavior. The discussion emphasizes physical phenomena whose evolution is best described using the fractional calculus, such as systems with long-range spatial interactions or long-time memory. The book gives general strategies for understanding wave propagation through random media, the nonlinear response of complex materials, and the fluctuations of heat transport in heterogeneous materials

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 Greenโs 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 Lรฉvy 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 Lรฉvy 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

1. Physics
2. Dynamics
3. Ergodic theory
4. Quantum physics
5. Statistical physics
6. Dynamical systems
7. Physics
8. Statistical Physics
9. Dynamical Systems and Complexity
10. Quantum Physics
11. Dynamical Systems and Ergodic Theory