Fractional kinetics in space : anomalous transport models

299 Pages·2018·13.405 MB·English

by  Vladimir Vasilʹevich Uchaĭkin, Renat Sibatov

FRACTIONAL
KINETICS
IN SPACE
Anomalous Transport Models

FRACTIONAL
KINETIC
IN SPACE
Anomalous Transport Models
Vladimir Uchaikin
Renat Sibatov
Ulyanovsk State University, Russia
1
World Scientific
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Library of Congress Cataloging-in-Publication Data
Names: Uchaikin, V. V. (Vladimir Vasilevich), author. I Sibatov, Renat, author.
Title: Fractional kinetics in space : anomalous transport models I by Vladimir V. Uchaikin
(Ulyanovsk State University, Russia), Renat Sibatov (Ulyanovsk State University, Russia).
Description: New Jersey : World Scientific, 2018. I Includes bibliographical references.
Identifiers: LCCN 20170410181 ISBN 9789813225428 (hardcover: alk. paper)
Subjects: LCSH: Fractional calculus. I Cosmic physics.
Classification: LCC QC20.7.F75 U34 2018 j DDC 523.0l--dc23
LC record available at https://lccn.loc.gov/2017041018
British Library Cataloguing-in-Publication Data
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Printed in Singapore
"The Cosmos is all that is or was or ever will be. Our feeblest
contemplations of the Cosmos stir us - there is a tingling in the
spine, a catch in the voice, a faint sensation, as if a distant
memory, of falling from a height. We know we are approaching
the greatest of mysteries."
Carl Sagan

Contents
Overview 1
1. Mathematical prelude 9
1.1 Self-similar structures 9
1.2 Self-similar processes . 12
1.3 Generalized (Levy) statistics . 14
1.4 Fractional Brownian motion . 23
1.5 Bifractional equation and random walks 28
1.6 Tempered fractional processes . . . . . 36
2. Nonlocal diffusion models in hydrodynamics 39
2.1 The Navier-Stokes equation 39
2.2 Einstein's diffusion equation 40
2.3 Ballistic correlations .... 43
2.4 The Schonfeld eddy interpretation 47
2.5 Monin's hypothesis ......... 51
2.6 The Heisenberg-Kolmogorov-Weizsacker formalism 55
2.7 Tchen's interpretation ................ 58
3. Interstellar medium 61
3.1 Interstellar medium . ...... 61
3.2 Physics of anomalous diffusion . 66
3.3 Walking magnetic field lines . 70
3.4 Diffusion in molecular clouds 73
3.5 Fractional diffusion in plasma 78
3.6 Radio-signals from pulsars . 82
vii
viii Fractional Kinetics in Space
3.7 Astrophotography and image processing 87
4. Solar system scales 91
4.1 Subdiffusion in photosphere 91
4.2 Diffusion on a granular surface 95
4.3 Solar flares and solar wind . . . 101
4.4 Solar cosmic rays . . . . . . . . 105
4.5 Tempered superdiffusion parallel to magnetic field 111
5. From classic to fractional models of cosmic ray transport 119
5.1 Ginzburg-Syrovatsky model ...... . 119
5.2 Collisionless Vlasov equation . . . . . . 124
5.3 Cosmic rays in turbulent magnetic field 128
5.4 Appearance of fractional derivatives in CR phenomenology 135
5.5 Nonlocal diffusion model: physical basement . 142
5.6 Percolation of cosmic rays 164
6. Acceleration of cosmic rays 169
6 .1 Acceleration mechanisms . . . . . . . . . . . 169
6.2 Two kinds of momentum diffusion operator 173
6.3 Superdiffusive shock acceleration . . . . . . 176
6.4 Subdiffusive shock acceleration ...... . 179
6.5 Anomalous second order Fermi acceleration 180
6.6 Stochastic acceleration in Earth's magnetotail 187
7. Nonlocal relativistic diffusion model 189
7.1 Relativistic speed-limit requirement and CR transport 189
7.2 NoRD model . . . . . . . . . . 194
7.3 One-dimensional NoRD theory 202
7.4 On the energy spectrum ... . 211
7.5 Cosmic ray anisotropy .... . 218
7.6 Milne problem for a fractal half-space 224
8. Cosmological scales 233
8.1 Two kinds of large-scale statistics 233
8.2 Correlation functions and power spectra 236
8.3 Fractional Ornstein-Zernike equation 241
8.4 Pencil beam processes . . . . . . . . . . 244
Contents ix
8.5 The large-scale structure formation . . . 246
9. Conclusion: Invitation to fractional cosmology 253
9 .1 Fractional methodology 253
9.2 Turbulence ...... . 255
9.3 Fractional burgulence .. 256
9.4 Visible subsystem as an open system 258
9.5 Complexity and fractionality 264
Bibliography 265