Non-equilibrium Statistical Mechanics and

Methods of non-equilibrium statistical mechanics play an increasingly important role in modern turbulence research, yet the range of relevant tools and methods is so wide and developing so fast that until now there has not been a single book covering the subject. As an introduction to modern methods of statistical mechanics in turbulence, this volume rectifies that situation. The book comprises three harmonised lecture courses by world class experts in statistical physics and turbulence: John Cardy introduces Field Theory and Non-Equilibrium Statistical Mechanics; Gregory Falkovich discusses Turbulence Theory as part of Statistical Physics; and Krzysztof Gawedzki examines Soluble Models of Turbulent Transport. To encourage readers to deepen their understanding of the theoretical material, each chapter contains exercises with solutions. Essential reading for students and researchers in the field of theoretical turbulence, this volume will also interest any scientist or engineer who applies knowledge of turbulence and non-equilibrium physics to their work.

Preface page	p. ix
Gregory Falkovich. Introduction to turbulence theory.	p. 1
Introduction	p. 1
Weak wave turbulence	p. 3
Strong wave turbulence	p. 8
Incompressible turbulence	p. 12
Three dimensional turbulence	p. 12
Two-dimensional Turbulence	p. 15
Passive Scalar Turbulence	p. 15
Two-dimensional enstrophy cascade	p. 18
Zero modes and anomalous scaling	p. 18
Inverse cascades	p. 24
Passive scalar in a compressible flow	p. 25
Inverse energy cascade in two dimensions	p. 27
Conclusion	p. 30
Exercises	p. 31
Problems	p. 31
Solutions	p. 32
References	p. 42
Krzysztof Gaw&ecedil;dzki. Soluble models of turbulent transport	p. 44
Introduction	p. 44
Lecture 1. Turbulent flow as a dynamical system	p. 45
Navier-Stokes equations	p. 46
Transport phenomena	p. 47
Problems	p. 53
Lecture 2. Multiplicative ergodic theory	p. 54
Natural measures	p. 54
Tangent flow	p. 57
Stretching exponents at long times	p. 58
Problems	p. 63
Lecture 3. Kraichnan model	p. 64
Lagrangian trajectories and eddy diffusion	p. 64
Tangent flow in Kraichnan velocities	p. 66
The uses of multiplicative large deviations	p. 70
Problems	p. 71
Lecture 4. Generalized flows and dissipative anomaly	p. 72
Two-particle dispersion	p. 73
Phases of the Lagrangian flow	p. 74
Scalar cascades	p. 77
Problems	p. 81
Lecture 5. Zero-mode scenario for intermittency	p. 81
Stochastic PDE for scalar	p. 81
Evolution of scalar correlation functions	p. 82
Zero modes	p. 83
Problems	p. 86
End remarks	p. 87
Solutions of problems	p. 88
Problems to Lecture 1	p. 88
Problems to Lecture 2	p. 92
Problems to Lecture 3	p. 93
Problems to Lecture 4	p. 100
Problems to Lecture 5	p. 103
Bibliography	p. 104
John Cardy. Reaction-diffusion processes	p. 108
Introduction	p. 108
Brownian motion	p. 108
The Einstein relation	p. 109
Correlation function	p. 110
Response function	p. 110
More general Langevin equations	p. 111
The response function formalism	p. 111
The master equation	p. 113
Detailed balance	p. 114
Stochastic particle systems	p. 114
Particles hopping on a lattice	p. 116
Two particle annihilation	p. 116
Averages of observables in the many-body formalism	p. 117
The Doi shift	p. 118
Path integral representation	p. 118
The expected number of particles and the expectation value of ¿	p. 121
Feynman diagrams and the renormalization group	p. 121
The critical dimension	p. 127
Other reaction-diffusion processes	p. 128
A + B → 0	p. 128
A + A &rightleftarpoons; C	p. 130
Reaction-diffusion and turbulence (Connaughton, Rajesh, Zaboronski)	p. 132
Cluster-cluster aggregation: model and continuum description	p. 133
Self-similar theory	p. 135
The conservation of mass and the counterpart of Kolmogorov 4/5-th law.	p. 138
Higher order correlation functions	p. 140
Refined self similarity	p. 142
Exercises	p. 143
Problems	p. 143
Solutions	p. 144
Bibliography	p. 160
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Non-equilibrium Statistical Mechanics and Turbulence

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