Transport Processes in Macroscopically Disordered Media (eBook)

Professor Andrew Snarskii obtained his physics undergraduate and Master Science degrees from Chernivtsi State University in 1972. In 1976 he received PhD also from Chernivtsi State University. He received degree of doctor of science (habilitation degree) from Kiev Institute of Physics in 1991. His fields of research include thermoelectricity, physical processes in percolation structures, deterministic chaos, fractals, theory of complex networks. Now he is a full tenured Professor of Kiev Polytechnic University. Dr. Igor V. Bezsudnov graduated from Moscow Institute of Electronics and Mathematics in 1985. Since then he always worked in research and development departments of different companies. In 2012 he received Ph.D in physics from Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine. His fields of research include the behaviour of inhomogeneous media near the percolation threshold, phenomenon of self-organized criticality, thermoelectric properties of disordered media, computer numerical modelling of complex media. Now his affiliation is vice director of NPP Nauka-Service, Moscow, chef of R&D. Mr. Vladimir A. Sevryukov graduated from Bauman Moscow State Technical University in 1983. His work has always been connected with the development and application of advanced technologies and scientific achievements. Fields of research and interest includes percolation systems and their transport properties, computer modelling of highly disordered media. Currently he is the director of NPP Nauka-Service,Moscow. Dr. Alexander Morozovskiy graduated from Kiev Polytechnic University in 1982. He worked as researcher in Kiev Institute of Metal Physics. He received his PhD from Kiev Institute of Metal Physics in 1988. His area of research includes theory of percolation, superconductivity, market microstructure, credit risk, econophysics. Currently he is working at Citibank. Professor Joseph Malinsky obtained his physics undergraduate and (advanced) Master of Science degrees from Kiev State University in 1973. In 1985 he has received Ph.D in physics from the Graduate Center of CUNY under the supervision of Professor Joseph L.Birman. His fields of research include areas of Condensed Matter Physics, Biophysics, Mathematical Biology etc. His affiliations include CCNY, BCC, Graduate Center of City University of NY (physics program), Mount Sinai Medical School (Departments of Biophysics and Biomathematics). Now he is a full tenured Professor.

Preface 6
Contents 9
Methods 13
1 Introduction 14
1.1 Types of Macroscopically Disordered Media 14
1.2 Classification of Physical Properties. Physical Analogies 16
References 17
2 The Methods of Description of Random Media 18
2.1 Effective Kinetic Coefficients, or What Do We Measure 18
2.2 Correlation Length and Self-averaging 22
References 23
3 Effective Conductivity of Macroscopically Disordered Media 25
3.1 Double-Sided Estimates of the Effective Kinetic Coefficients 25
3.2 Approximations of Maxwell, Garnett, and Bruggeman 28
3.3 Periodically Located Inclusions 38
3.4 Plain-Layered Systems 43
References 48
4 Elements of Geometrical Theory of Percolation 50
4.1 Percolation Problem 50
4.2 Basic Concepts of Geometric Percolation 52
References 54
5 Effective Conductivity of Percolation Media 55
5.1 Analogy with the Phenomenological Theory of Second-Order Phase Transitions. Scaling and Critical Exponents 55
5.2 Effective Conductivity as an Order Parameter. Phenomenological Description 59
5.3 Calculation of Critical Indices 64
5.4 Hierarchical Model of Percolation Structure 71
5.5 Examples of Applications of Percolation Theory 80
References 81
6 Self-dual Media 84
6.1 Locally Isotropic Media 84
6.2 Locally Anisotropic Media 93
References 100
7 Continual Percolation Problem 102
7.1 Types of Continual Percolation Problems 102
7.2 Swiss Cheese Media 104
References 108
8 Media with Exponentially Broad Spectrum of Local Properties 110
8.1 Formulation of the Problem and Approximate Calculation of the Effective Conductivity 110
8.2 Correlation Length and Pre-exponential Factor 112
References 117
9 Finite Scaling 119
9.1 Properties of Percolation Systems with Dimensions Lesser Than Their Correlation Length 119
9.2 Finite-Size Scaling for Self-dual Media 125
References 128
10 Conductivity of Percolation Layer 129
10.1 Effective Conductivity of the Percolation Systems in the Cases with Some Sizes Are Lesser and the Other Greater Than Percolation Length. Definition of the Problem 129
10.2 Solution Technique 131
References 134
Processes 135
11 AC Conductivity 136
11.1 EMT-Approximation 136
11.2 The Method of Percolation Theory 138
References 144
12 Galvanomagnetic Properties of Macroscopically Disordered Media 145
12.1 Introduction 145
12.2 Layered Media in the Magnetic Field 148
12.3 Dual Media in the Magnetic Field 149
12.4 Strongly Inhomogeneous Media in the Vicinity of the Percolation Threshold, Two-Dimensional Case 152
12.5 Strong Disorder, Three-Dimensional Case 158
References 162
13 Flicker-Noise (1/f-Noise) 165
13.1 Flicker-Noise in Inhomogeneous Media 165
13.2 Flicker-Noise in Inhomogeneous Media—EMT-Approximation 168
13.3 Flicker-Noise in Percolation Systems 169
13.4 Abnormally High Rate of Flicker-Noise in Self-dual Media 174
13.5 Flicker-Noise in the Systems with Exponentially Broad Spectrum of the Resistances 176
13.6 Flicker-Noise for Fluctuation of Phase Concentration 181
References 182
14 Higher Current Moments 185
14.1 Definitions 185
14.2 Critical Exponents of the Higher Current Moments 186
References 190
15 Thermoelectric Properties 192
15.1 EMT-Approximation 192
15.2 Thermoelectric Properties of the Self-dual Media 195
15.3 Critical Region of Concentration—Behavior of /alpha_{{/rm e}} in the Vicinity of Percolation 198
15.4 Isomorphism 201
References 207
16 Effective Elastic Properties 209
16.1 Basic Concepts of Elasticity Theory 209
16.2 Effective Module in the Vicinity of Percolation Threshold 211
References 218
17 Nonlinear Properties of Composites 220
17.1 Types of Nonlinearity 220
17.2 The Case of Weak Nonlinearity 221
17.3 The Case of Strong Nonlinearity 227
References 237
18 Effective Properties of Ferromagnetic Composites 239
18.1 Nonlinearity and Hysteresis in Ferromagnets 239
18.2 Hysteresis-Less Case 240
18.3 Ferromagnetic Composites with a Nonzero Hysteresis Loop 242
References 245
19 Temperature Coefficient of Resistance and Third Harmonic Generation Close to Percolation Threshold 246
19.1 Temperature Coefficient of Resistance 246
19.2 Third Harmonic Generation 247
References 250
20 Instability and Chaos in the Macroscopically Inhomogeneous Media with Weak Dissipation 251
20.1 Dual Media 251
20.2 Ladder Filter 257
References 261
21 Percolation-Similar Description of Abrikosov Vortex 262
21.1 The Pinning of the Abrikosov Vortexes 263
21.2 The Case of the Wide Pinning Force Distribution 264
References 269
22 Anderson Localization in the Percolation Structure 272
22.1 Anderson Localization 272
22.2 Anderson Metal–Insulator Transition in Percolation Structure 273
References 275
23 Conclusion 276
References 277