Diffusion : formalism and applications / by Sushanta Dattagupta.

Format:

Book

Author/Creator:

Dattagupta, S. (Sushanta), 1947- author.

Language:

English

Subjects (All):

Diffusion.

Physical Description:

xv, 292 pages : illustrations ; 23 cm

Place of Publication:

London : Taylor & Francis, 2014.

Summary:

Within a unifying framework, Diffusion: Formalism and Applications covers both classical and quantum domains, along with numerous applications. The author explores the more than two centuries-old history of diffusion, expertly weaving together a variety of topics from physics, mathematics, chemistry, and biology. The book examines the two distinct paradigms of diffusion-physical and stochastic-introduced by Fourier and Laplace and later unified by Einstein in his groundbreaking work on Brownian motion. The author describes the role of diffusion in probability theory and stochastic calculus and discusses topics in materials science and metallurgy, such as defect-diffusion, radiation damage, and spinodal decomposition. In addition, he addresses the impact of translational/rotational diffusion on experimental data and covers reaction-diffusion equations in biology. Focusing on diffusion in the quantum domain, the book also investigates dissipative tunneling, Landau diamagnetism, coherence-to-decoherence transition, quantum information processes, and electron localization. Features, Offers a lucid account of the history of diffusion from the last 200 years, Familiarizes you with the necessary formalism and practical applications, Covers stochastic processes, enabling you to acquire a solid mathematical foundation, Describes interdisciplinary topics from many areas of science and engineering, including nanoscience and quantum coherence, Links the theory of diffusion and spectroscopy Book jacket.

Contents:

Section I Classical Diffusion

1 Introduction to Brownian Motion 3

1.1 Introductory Remark's 3

1.2 Fourier Equation 3

1.3 Random Walk 5

1.4 Random Walk on a Lattice and Its Continuum Limit 9

1.5 Einstein on Brownian Motion 11

1.5.1 Dressed Viscosity 11

1.5.2 Synergy of Thermodynamics and Kinetics 12

1.5.3 Brownian Motion and Stochastic Diffusion 14

1.6 Concluding Remarks 16

Exercises 16

References 16

2 Markov Processes 19

2.1 Genesis of Markov Concept 19

2.2 Definition 20

2.2.1 Joint and Conditional Probabilities 21

2.3 Stationary Markov Process 22

References 25

3 Gaussian Processes 27

3.1 Introduction 27

3.1.1 Moments and Cumulants 29

3.1.2 Characteristic Function 30

3.1.3 Cumulant Generating Function 31

3.2 Gaussian Stochastic Processes 32

3.2.1 Novikov Theorem 33

3.2.2 Moment Theorem 33

3.2.3 Characteristic Functional 34

3.3 Stationary Gauss-Markov Process 34

3.3.1 Doob's Theorem 35

Exercises 37

References 38

4 Langevin Equations 39

4.1 Introduction 39

4.2 Free Particle in Momentum Space 40

4.3 Free Particle in Position Space 47

4.3.1 High-Friction Brownian Regime 51

4.3.2 Summary of Results 54

4.4 Harmonic Oscillator 55

4.5 Diffusive Cyclotron Motion 57

Exercises 60

References 60

5 Fokker-Planck Equation 61

5.1 Introduction 61

5.2 FP Equation in Velocity 62

5.3 FP Equation in Position: Stochastic Diffusion 63

5.4 FP Equation in Force Field: Kramers Equation 64

5.5 FP Equation in Force Field in High-Friction Limit: Smoluchowski Equation 66

5.6 FP Equation for Damped Harmonic Oscillator 67

5.7 FP Equation for Cyclotron Motion 68

5.8 Diffusion across Barrier 69

Exercise 73

References 73

6 Jump Diffusion 75

6.1 Introduction 75

6.2 Operator Notation 76

6.3 Two-State Jump and Telegraph Processes 79

6.4 Multi-State Jump Process 81

6.5 Kubo-Anderson Process 81

6.6 Interpolation Model 83

6.7 Kangaroo Process 84

Exercises 88

References 88

7 Random Walk and Anomalous Diffusion 89

7.1 Introduction to Continuous Time Random Walk (CTRW) 89

7.2 Non-Markovian Diffusion in CTRW Scheme 92

7.2.1 Application to Interpolation Model 93

7.2.2 Anomalous Diffusion 96

References 97

8 Spectroscopic Structure Factor 99

8.1 Introductory Remarks 99

8.2.1 Weak Collision Model: Gaussian Process 100

8.2.1.1 Very Slow Collisions 101

8.2.1.2 Very Fast Collisions 102

8.2.2 Strong Collision Model: Kubo-Anderson Process 102

8.2.2.1 No-Collision Term 103

8.2.2.2 One-Collision Term 103

8.2.2.3 Two-Collision Term 104

8.2.2.4 Very Slow Collisions 104

8.2.2.5 Very Fast Collisions 105

8.2.3 Boltzmann-Lorentz Model 106

8.2.3.1 No-Collision Term 107

8.2.3.2 One-Collision Term 107

8.3 Cyclotron Motion in Weak Collision and Boltzmann-Lorentz Models 109

8.3.1 Structure Factor in the Weak Collision Model 109

8.3.2 Structure Factor in the Boltzmann-Lorentz Model 111

8.4 Neutron Scattering from a Damped Harmonic Oscillator 114

8.5 Restricted Diffusion over Discrete Sites 115

8.5.1 Two-Site Case 115

8.5.2 Cage Diffusion 118

8.6 Unbounded Jump Diffusion in Empty Lattice 121

8.6.1 Large Jumps in Random Directions 123

8.6.2 Small Jumps in Random Directions 123

8.7 Vacancy-Assisted Correlated Diffusion in Solids 124

8.7.1 Analytical Results in a Simple Cubic (SC) Case 130

Appendix 135

References 139

9 Rotational Diffusion of Molecules 141

9.1 Introduction 141

9.2 Extended Diffusion Models 144

9.3 M Diffusion Model 146

9.3.1 No-Collision Term 147

9.3.2 One-Collision Term 148

9.4 J Diffusion Model 149

9.4.1 No-Collision Term 150

9.4.2 One-Collision Term 150

9.5 Interpolation Model 151

9.5.1 No-Collision Term 151

9.5.2 One-Collision Term 151

9.5.3 Two-Collision Term 152

9.6 Applications to Infrared and Raman Rotational Spectroscopy 155

References 159

10 Order Parameter Diffusion 161

10.1 Cahn-Hilliard Equation 161

10.2 Pattern Formation 165

10.3 Jump Diffusion in Ising Model: Relation with CH Equation 168

10.4 Reaction-Diffusion Models and Spatiotemporal Patterns 173

References 180

11 Diffusion of Rapidly Driven Systems 181

11.1 Introduction 181

11.2 Langevin Equation in Over-Damped Case 184

11.2.1 Dynamical Symmetry Breaking 185

11.2.2 Decay of Metastable State 186

11.3 Langevin Equation for Arbitrary Damping 187

11.4 Effective Diffusion in Periodic Potential 190

References 192

Section II Quantum Diffusion

12 Quantum Langevin Equations 195

12.1 Introduction 195

12.2 Derivation of the Classical Langevin Equation 197

12.2.1 Ohmic Dissipation 1 200

12.3 Quantum Generalization 202

12.4 Free Particle 203

12.5 Diffusive Quantum Cyclotron Motion 204

12.6 Diffusive Landau Diamagnetism 208

References 213

13 Path Integral Treatment of Quantum Diffusion 215

13.1 Introduction 215

13.2 Basic Model 217

13.3 Ohmic Dissipation and Classical Limit 219

13.4 Master Equation in High-Temperature Limit 221

13.5 Application to Dissipative Diamagnetism 222

References 226

14 Quantum Continuous Time Random Walk Model 227

14.1 Introduction 227

14.2 Formulation 228

14.2.1 Zero-Collision Term 229

14.2.2 One-Collision Term 229

14.2.3 Two-Collision Term 230

14.3 Applications 231

14.3.1 Quantum Harmonic Oscillator 231

14.3.2 Spin Relaxation 233

14.4 Finite Temperature Effects 234

14.4.1 Relaxation of the Harmonic Oscillator at Finite Temperatures 234

14.4.2 Phase Space Dynamics and Free Particle Limit 236

14.4.3 Spin Relaxation at Finite Temperatures: Bloch-Redfield Equations 236

References 237

15 Quantum Jump Models 239

15.1 Introduction 239

15.1.1 Spin Lattice Relaxation in Solids 240

15.1.2 Quantum Tunneling in Symmetric Double Well 240

15.1.3 Tunneling in Asymmetric Double Well 241

15.1.4 Dephasing of Qubit 241

15.2 Formalism 242

15.2.1 Cumulant Expansion 242

15.2.2 Resolvent Expansion 248

15.3 Polaronic Transformation 249

15.3.1 Asymmetric Spin Boson Model 251

15.4 Qubit 255

15.5 Neutron Structure Factor for H Diffusion in Metals 259

15.6 Spin Relaxation of Muon Diffusion in Metals 262

References 264

16 Quantum Diffusion: Decoherence and Localization 265

16.1 Introductory Comments 265

16.2 Landau to Bohr-van Leeuwen Transition of Diamagnetism 267

16.3 Harmonic Oscillator in Quantum Diffusive Regime 269

16.4 Decoherence in Spin Boson Model 271

16.5 Dissipationless Decoherence 271

16.6 Retrieving Quantum Information Despite Decoherence 275

16.7 Localization of Electronic States in Disordered Systems 279

References 282.

ISBN:

9781439895573

1439895570

OCLC:

859182551