Nature’s Patterns and the Fractional Calculus

Bruce J. West, US Army Research Office, Cary, US

Table of Content:

Chapter 1: Complexity Science

1.1 It started with physics

1.2 Complexity

1.3 Measures of size

1.4 Allometry heuristics

1.5 Overview

Chapter 2: Empirical Allometry

2.1 Living networks

2.2 Physical networks

2.3 Natural history

2.4 Sociology

2.5 Summary

Chapter 3 Statistics, Scaling and Simulation

3.1 Interpreting fluctuations

3.2 Phenomenological distributions

3.3 Are ARs universal?

3.4 Summary

Chapter 4: Models & Derivations of ARs

4.1 Optimization principles

4.2 Scaling and allometry

4.3 Stochastic differential equations

4.4 Fokker-Planck equations

4.5 Summary

Chapter 5: Complex and Strange Kinetics

5.1 Fractional thinking

5.2 Fractional rate equations

5.3 Fractional Poisson process

5.4 A closer look at complexity

5.5 Recapitulation

5.6 Appendix

Chapter 6: Fractional Probability Calculus

6.1 Fractional Fokker-Planck equation

6.2 Fully fractional phase space equations

6.3 Entropy entails allometry

6.4 Statistics of allometry parameters

6.5 Discussion and conclusions

6.6 Epilogue