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West B.J., Bologna M., Grigolini P. Ч Physics of Fractal Operators
West B.J., Bologna M., Grigolini P. Ч Physics of Fractal Operators

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Ќазвание: Physics of Fractal Operators

јвторы: West B.J., Bologna M., Grigolini P.

јннотаци€:

This text describes how fractal phenomena, both deterministic and random, change over time, using the fractional calculus. The intent is to identify those characteristics of complex physical phenomena that require fractional derivatives or fractional integrals to describe how the process changes over time. The discussion emphasizes the properties of physical phenomena whose evolution is best described using the fractional calculus, such as systems with long-range spatial interactions or long-time memory. In many cases, classic analytic function theory cannot serve for modeling complex phenomena; "Physics of Fractal Operators" shows how classes of less familiar functions, such as fractals, can serve as useful models in such cases. Because fractal functions, such as the Weierstrass function (long known not to have a derivative), do in fact have fractional derivatives, they can be cast as solutions to fractional differential equations. The traditional techniques for solving differential equations, including Fourier and Laplace transforms as well as Green's functions, can be generalized to fractional derivatives. Physics of Fractal Operators addresses a general strategy for understanding wave propagation through random media, the nonlinear response of complex materials, and the fluctuations of various forms of transport in heterogeneous materials. This strategy builds on traditional approaches and explains why the historical techniques fail as phenomena become more and more complicated.


язык: en

–убрика: ‘изика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 2003

 оличество страниц: 354

ƒобавлена в каталог: 13.12.2005

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ѕредметный указатель
action      12
Activation energy      278
Aggregation      307
Anagram      235
Anisotropy      173
Anomalous diffusion      301Ч304
Anomalous diffusion, stochastic oscillator      319
Ant      301
Antipersistent process      302
Applied strain      241
Applied stress      241
Arrhenius form      278
Atomists      75
Autocorrelation function      289Ч295
Average      196
Average, of displacement      197 317
Basin of attraction      286
Bernoulli sequence binomial, theorem      201
Binomial, coefficient      201
biology      184
Biomedical phenomena      265
Boltzmann      3 38 75 214 244 292
Boltzmann, distribution      323
Boson      315
Boundary value problem      313
Box dimension      210 211
Branching systems      265
Brownian motion      5 15 39 213Ч215 260
Brownian motion, fractional      71 203Ч207 269
Canonical distributions      18 19
Cantor sets      38Ч47
Cardiovascular system      323
Central limit theorem      5 226
Central limit theorem, generalization of      226 306Ч309
Chain condition      215Ч219 257
Chaos      2 184 207 302
Chaos, weak and strong      324
Characteristic creep time      243
Characteristic function      196 204 209
Characteristic function, Levy      219Ч202 308
Chemical kinetics      184
Classical mechanics      7Ч14 37
Closure      123
Clustering      224 226 304 309
Coarse-graining      186 305
Colored noise      208Ч210
Complex, moduli      238Ч240
Complex, phenomena      115 183 212 269 301
Complex, system      184
Composition rule      57
Constant functions      79
Constitutive equations      234
Constitutive equations, fractional      253
Constraints      9
Continuity      75
Continuous time random walk, CTRW      276
Convolution      82 138
Convolution, generalized      150
Convolution, of generalized exponentials      174
Correlation function      27 53 189 277
Correlation function, coefficient      53 212
Correlation function, inverse power law      1908 208 284
Correlation, length      304
Correlation, power law      211Ч214
creep      235 244
Creep, compliance      238 245 246
Dash pots      233 252 173
Deformation      234Ч240
density      125Ч127
Density matrix      275 322
Difference equations      200Ч203 207
Diffusion, anomalous      200 218
Diffusion, coefficient      189
Diffusion, Einstein      132
Diffusion, equation      151 175 217 260
Diffusion, ordinary and anomalous      186 190 223
Diffusive packet      276
Dimension, similarity      41
Dimension, similarity, complex      49
Dimension, similarity, fractal      42Ч47
Dimension, similarity, of trail      52
Dirac delta function      123 142Ч146 216 277 323
Dirac delta function, derivative of      145
Dirac delta function, fractional derivative of      145
Dirac delta function, impulse function      142
Dirac delta function, integral representation      144
Dirac delta function, series representation      143
Disordered media      301
Displacement      188
Dissipation      286
DNA sequences      46 191 269
Dynamical, systems, nonlinear      185
Dynamical, systems, nonlinear, variables      223
Dynamics      76 175 169 183
Ecology      198
Economics      184
Eigenfunction expansion      132
Eigenfunction expansion, fractional      312Ч315
Eigenvalues      176
Eigenvectors      176
Einstein process      310
Elastic, rod      129
Energy, mechanical      9
Energy, mechanical, barrier      278
Energy, mechanical, elastic      125
Enhanced diffusion      302
Equation of motion, fractional      271
Equilibrium      186
Equilibrium, statistical mechanics      315
Euclid      3
Euler relations      245
Euler relations, generalized      107
Euler Ч Lagrange equations      8Ч12
Exiton transport      276
Expansion coefficients      314
Exponential, generalized      88Ч108
Exponential, generalized, fractional      89
Exponential, generalized, generalized complex      97Ч99
Exponential, generalized, waves      129
Extremal path      8
extreme values      212
Feynman path integral      262
First-passage time      303
First-passage time, distribution      309Ч312
Fixed point      307Ч309
Fluctuations      200 207
Fluctuations, in magnetization      305
Fluctuations, in time series      211
Fokker Ч Planck equation      186 215Ч218 223 269 273 283
Fokker Ч Planck equation, fractional      312Ч319
force      237
Forces      9
Fourier amplitudes, fractional derivatives      136
Fourier series      119Ч124
Fourier series, coefficients      121Ч123
Fourier series, generalized      119
Fourier transforms      119Ч124 239
Fourier transforms, derivatives of function      134
Fourier transforms, fractional      119Ч154
Fourier transforms, generalized      135 147Ч150
Fourier transforms, inverse      148
Fourier Ч Laplace transforms      195Ч200
Fox, transforms      249 254 288
Fox, transforms, functions      321
Fractal, data set      212
Fractal, dimensions      40 212Ч214 227
Fractal, functions      3 4
Fractal, measure of      191
Fractal, media      119
Fractal, models      39
Fractal, statistics      23
Fractal, stochastic process      203
Fractal, trees      46
Fractal, wave propagation      303
Fractional      183Ч232
Fractional, Brownian motion      203Ч207 210
Fractional, calculus      31 32 200 233
Fractional, derivatives      77Ч88
Fractional, differential equations      31 176Ч179 247 253Ч257
Fractional, diffusion      303
Fractional, diffusion equation      274 319Ч323
Fractional, diffusion equation (FDE)      303 304
Fractional, exponential      243
Fractional, Laplace transforms      155Ч182
Fractional, Levy motion      321
Fractional, memory      244Ч246
Fractional, operators      54Ч64 264
Fractional, relaxation      246Ч257 278Ч282
Fractional, rheology      233Ч268
Fractional, series representation      54Ч56
Functionals      257
Functions, generalized      115
Functions, generalized, analytic      156
Galileo      1
Gauss, distribution      186 203 227 258Ч261 273 310
Gauss, statistics      204 269
Generalized Langevin equation      18
Generalized Weierstrass function (GWF)      4 47Ч54
Generalized Weierstrass function (GWF), fracal dimension      69 70
graphs      208
GreenТs function, for harmonic oscillator      158
GreenТs function, for relaxation      243 248
GreenТs function, fractional      175Ч179 288
GreenТs function, free-space      130
Hamilton Ч Jacobi theory      8
Hamiltonian dynamics      7 12Ч20 75 185 305
HamiltonТs principle      10 12Ч14
Harmonic oscillator, dissipative      277 278
Harmonic oscillator, driven      157
Harmonic oscillator, inhomogeneous      102Ч105
Harmonic oscillator, wave field      129
Heat bath, oscillators      16Ч20
Heat bath, oscillators, fractional      315Ч319
Heat equation      132
Helmholtz equation      130
Heredity      235
Hermite polynomial      316
Heterogeneity      173 212 224
Homogeneity      224
HookeТs law      126 235Ч248 244
Human lung      49
Hurst exponent      192
HuygenТs principle      142
Hydrodynamics      233
Hyperbolic functions, generalized      108
Hyperbolic partial differential equation      282
Hyperbolic random variables      227
Index of refraction      131
Infinitely divisible      262Ч266
Infinitesimal generator      111
Initial conditions, in fractional equation      247
Initial conditions, sensitive dependence on      185
Integral equation      241Ч244
Intermittency      203
Intervals of the GFW (IGFW)      65Ч70
Inverse power-law, memory      26Ч28 186 278
Inverse power-law, relaxation      76 246 251
Inverse power-law, wave equation      129
Ising model      304
Islands of stability      324
Ito calculus      21
Jump rate      278
Kelvin, model      137 238
Kernel, inverse power law      206
Koch snowflake      45
Kohlrausch Ч Williams Ч Watts Law      250
Kolmogorov Ч Arnold Ч Moser, KAM theory      14 38 185
Labyrinth      301
Lagrange      1
Langevin equation      5 14Ч22
Langevin equation, fractional      269Ч275 283Ч295
Langevin equation, with Levy fluctuations      221 222
Laplace transform      155
Laplace transform, inverse      155
Laplace transform, of fractional derivatives      160Ч163
Laplace transform, of generalized functions      163Ч166
Laplace transform, solution to fractional differential equations      247Ч257 253Ч257
Laplace transform, solutions to differential equations      166Ч168
Laplace Ч Mellin transform      321
Leibniz rule      83
Leibniz rule, generalized      83Ч66
Levy path integral      262Ч264
Levy, statistics      5 304Ч309 319
Levy, statistics, distribution      186 310Ч312 323
Levy, statistics, flights      312
Levy, statistics, fluctuations      221 222
Levy, statistics, one-sided distribution      199 311
Linear physics      1
Logarithm, generalized      92 93
Logarithm, generalized, derivative of      108
Long-range coupling      323
Long-term memory      323
Longitudianal mode      129
Macromolecules      257
magnetization      304
Mandelbrot      3 37
Markov, approximation      24
Markov, process      221 273 283
Master equation      192Ч194 275
Master equation, continuous      283Ч286
Master equation, generalized      275
Maxwell, model      236Ч239
Mean-square displacement, fractional      292Ч294
Mellin transforms      249 254
Memory, inverse power law      186
Memory, kernels      19 20 275Ч282
Memory, long-time      200 269
Metallurgy      233
Mittag Ч Leffler, function      29 30 177 243 250 252 269 287
Mittag Ч Leffler, series solution      255 312Ч315
Mode amplitude      128
Molecular chain      277
Moment generating function      196
Moments      322
Myths      183
Nerve growth      269
Neumann series      242
Newton      2 235 244
Noise      184
Non Ч Debye relaxation      246
Non Ч Markovian statistics      273
Non Ч Markovian statistics, process      275
Non-analytic functions      2
Non-differentiability      54
Non-differentiability, phenomena      200
Non-integrable systems      185
Nonlinear, dynamics      184
Nonlinear, phenomena      284
Nonlinear, physics      2
NuttingТs Law      251
Order      81
Ornstein Ч Uhlenbeck process      20 269 287 282 317
Oscillator, stochastic      315Ч319
Parametric derivatives, fractional      109Ч114
ParsevalТs theorem      150
Path integrals      257Ч264
Pausing time probability      193
Persistent process      302
Phase space equations      221
Phase transitions      76 306
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