Gardiner C.W.W., Haken H. — Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences :: Электронная библиотека попечительского совета мехмата МГУ

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Gardiner C.W.W., Haken H. — Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences

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Название: Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences

Авторы: Gardiner C.W.W., Haken H.

Аннотация:

Handbook of Stochastic Methods covers the foundations of Markov systems, stochastic differential equations, Fokker-Planck equations, approximation methods, chemical master equations, and quantum-mechanical Markov processes. From the reviews: "Extremely well written and informative... clear, complete, and fairly rigorous treatment of a larger number of very basic concepts in stochastic theory." Journal of Quantum Electronics "A first class book." Optica Acta In this second edition extra material has been added with recent progress in stochastic methods taken into account.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Издание: 2nd edition

Год издания: 1997

Количество страниц: 442

Добавлена в каталог: 12.02.2008

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Предметный указатель

Absorbing boundary condition for backward FPE      129
Additive invariants      336
Adiabatic elimination in quantum systems      391
Adiabatic elimination in quantum systems of fast variables      195—234
Adiabatic elimination in quantum systems of inhomogeneous modes in reaction diffusion systems      328—331
Almost certain      29
Arrhenius formula      141
Autocorrelation function      16 58
Autocorrelation function and Poisson rep.      289—294
Autocorrelation function for Markov process      64—66
Autocorrelation function in terms of eigenfunctions      131
Autocorrelation function with detailed balance      149
Autocorrelation matrix and eigenfunctions      168
Autocorrelation matrix, stationary      65
Backward differential Chapman — Kolmogorov equation      55
Backward FPE, boundary conditions for      128
Backward master equation and first passage times      259
Baker — Hausdorff formula      387
Bargmann states, defined      377
Bath correlation functions      389
Bernoulli distribution      72
Bernoulli trials      42
Birth-death master equation      10 13
Birth-death master equation and bistability      354 357
Birth-death master equation in quantum harmonic oscillator      395
Birth-death master equation, one-variable      236 262
Birth-death process      8
Birth-death process for flow      332—336
Birth-death processes, boundary, conditions for      257 259
Birth-death systems, many-variable      262 277
Bistability      342
Bistability in multivariable systems      357 371
Bistability with birth-death master equation      354—357
Bistable system, chemical      241 245
Boltzmann equation from Boltzmann master equation      337
Boltzmann master equation      303 336—339
Boltzmann master equation in Poisson representation      338 341
Boundary conditions for backward FPE      128
Boundary conditions for birth-death processes      257 259
Boundary conditions for FPE      118—125
Boundary conditions for FPE at a discontinuity      121
Boundary conditions for FPE at infinity      123
Boundary conditions for FPE, absorbing      121
Boundary conditions for FPE, reflecting      121
Boundary conditions for Kramers’ equation      205
Boundary conditions for many-variable FPE      146
Boundary conditions for Smoluchowski equation      205
Brownian motion      2
Brownian motion and adiabatic elimination      195
Brownian motion in double potential      365—371
Brownian motion, continuous sample path      46
Brownian motion, corrections to Smoluchowski equation      206—210
Brownian motion, Kramers’ equation for      155
Brownian particle      6
Cauchy process      46—47
Central limit theorem      37
Chaos      1
Chapman — Kolmogorov equation      5 43—44
Chapman — Kolmogorov equation, differential      47—51
Characteristic function      32
Characteristic function of a Gaussian      36
Characteristic function, quantum      386—388
Chemical FPE      266
Chemical intermediates, elimination of      218
Chemical reaction $\mathrm{A+2X \rightleftharpoons 3X, A \rightleftharpoons X}$       300—301
Chemical reaction $\mathrm{A+2X \rightleftharpoons 3X, A \rightleftharpoons X}$ , master equation      241
Chemical reaction $\mathrm{A+2X \rightleftharpoons 3X}$ , in Poisson rep.      294
Chemical reaction $\mathrm{A+X \rightarrow 2X+D, B+X \rightleftharpoons C}$ , solution of ME      274—216
Chemical reaction $\mathrm{A+X \rightleftharpoons 2X, B+X \rightleftharpoons C}$ , complex Poisson rep.      283 284
Chemical reaction $\mathrm{A+X \rightleftharpoons 2X, B+X \rightleftharpoons C}$ , Poisson rep. solution      279—281
Chemical reaction $\mathrm{A+X \rightleftharpoons 2X, B+X \rightleftharpoons C}$ , positive Poisson re      288
Chemical reaction $\mathrm{A+X \rightleftharpoons 2X, B+X \rightleftharpoons C}$ , spatially distributed      324 328
Chemical reaction $\mathrm{A+Y \rightleftharpoons X+Y, Y \rightleftharpoons 2X}$ , elimination of intermediate      295 299
Chemical reaction $\mathrm{B \rightarrow X, 2X \rightarrow A}$ and positive Poisson re      288—289
Chemical reaction $\mathrm{B \rightarrow X, 2X \rightarrow A}$ , complex Poisson re      284—285
Chemical reaction $\mathrm{B+X \rightleftharpoons C, A+X \rightarrow 2X}$ , spatially distributed      319 324
Chemical reaction $\mathrm{B\rightleftharpoons X, A+X \rightarrow 2X}$ , analogy to harmonic oscillator      395
Chemical reaction $\mathrm{X \rightleftharpoons A}$ , and system size expansion      252
Chemical reaction $\mathrm{X \rightleftharpoons A}$ , master equation      238
Chemical reaction $\mathrm{X \rightleftharpoons Y \rightleftharpoons A}$       218
Chemical reaction $\mathrm{X \rightleftharpoons Y \rightleftharpoons A}$ , chemical FPE      267—273
Chemical reaction $\mathrm{X \rightleftharpoons Y}$ as reaction diffusion system      315—318
Chemical reaction $\mathrm{X+A \rightleftharpoons 2X}$ , chemical FPE      267—273
Chemical reaction $\mathrm{X+A \rightleftharpoons 2X}$ , FPE      127
Chemical reaction $\mathrm{X_{1} \rightleftharpoons 2X_{2}}$ , reaction diffusion equations      314
Chemical reaction $\mathrm{X_{1} \rightleftharpoons 2X_{2}}$ , solution of master equation      276—277
Chemical reaction, nonlocal      328
Chemical reaction, prey-predator system, FPE      268 273
Chemical reaction, unimolecular      279
Coherent states, Boltzmann master equation for      336—339
Coherent states, defined      375
Coherent states, expansion of an operator in      378
Coherent states, expansion of arbitrary states in terms of      377
Coherent states, Poissonian number distribution of      379
Coherent states, properties of      376 380
Collisions and flow      339—341
Combinatorial kinetics; defined      262
Commutation relations      373
Complex P-representation, defined      410
Complex P-representation, FPE from      416
Complex P-representation, operator identities for      414
Conditional probability      25
Conditional probability and eigenfunctions      13
Continuity in stochastic processes      45 46
Corrected Smoluchowski equation and escape problem      369
Correlation      30
Correlation functions of heat bath      389
Correlation functions, defined      33
correlation length      320
Correlation time      20 58
Correlations behaviour at instability point      323
Correlations behaviour of smooth functions of bath operators      389
Correlations behaviour, space-time      318
Covariance      30
Covariance matrix      30
Creation operator      373
Critical fluctuations      255
Critical slowing down      257
Cubic process      185
Cumulant      33 36
Cumulant generating function      33
Cumulant generating function, factorial      39
Cumulant, factorial      39
Density matrix      380—382
Density matrix, properties of      381 382
Destruction operator      373
Detailed balance      148—165
Detailed balance in birth-death master equation      237—238
Detailed balance in diffusion      307
Detailed balance, conditions for      151
Detailed balance, defined      148
Deterministic process      53
Differential Chapman — Kolmogorov equation      47—51
Differential Chapman — Kolmogorov equation, backward      55
Diffusion and detailed balance      307
Diffusion and detailed balance in a gravitational field      126
Diffusion and detailed balance in multivariate master equation      307 308
Diffusion and detailed balance, coefficient      5
Diffusion and detailed balance, current, defined      303
Diffusion and detailed balance, equation      5
Diffusion and detailed balance, inhomogeneous anisotropic      309
Diffusion master equation continuum form of      308—313
Diffusion master equation continuum form of system size expansion of      308
Diffusion matrix, defined      52
Diffusion process      52
Diffusion process, approximation by jump process      246—249
Divergence problems in nonlinear reaction diffusion systems      327
Double well potential      140
Double well potential, diffusion in      342—348
Drift vector, defined      52

Eigenfunctions and autocorrelation matrix      168
Eigenfunctions and conditional probability      131 168
Eigenfunctions and exit time      172
Eigenfunctions and spectrum matrix      169
Eigenfunctions for FPE, variational principle for      168
Eigenfunctions for many-variable FPE      165
Eigenfunctions for one-variable FPE      129—136
Einstein equations and two-level atom      402
Ensemble average      17
Entrance boundary for FPE      123
Equilibration of populations in bistable system      348—357
Ergodic      17—18
Ergodic properties and stationary processes      57
Escape      342
Escape probability of particle in a double well      351
Escape time      141
Event      21
Existence theorems for P-representations      411 415
Exit boundary for FPE      123
Exit points, distribution of by asymptotic method      357 362
Exit time and eigenfunctions      172 (see also “First passage time”)
Exit time from a region, many variable FPE      170—171 (see also “First passage time”)
Exit time in a double well potential      345 (see also “First passage time”)
Exit time through a particular end of an interval      142 (see also “First passage time”)
Exit time, asymptotic analysis of      362 363
Factorial correlation in reaction diffusion system      315
Factorial cumulant      39
Factorial moment      38
Fast variable      197
Fick’s law      303
Fick’s law with fluctuations      304
First passage time for one dimensional FPE      136—142
First passage time for one dimensional FPE of particle in a double well      351
Flow and collisions      339 341
Flow as a birth death process      332—336
Flow in position space      334
Flow in velocity space      334
Fluctuating force      6 15
Fluctuation dissipation theorem      162
Flux, phenomenological      162
Fokker — Planck equation (FPE)      5 8 117—176
Fokker — Planck equation (FPE) from complex P-representation      416
Fokker — Planck equation (FPE) from P-representation      396 415
Fokker — Planck equation (FPE) from Poisson representation      278—285
Fokker — Planck equation (FPE) from positive P-representation      416—418
Fokker — Planck equation (FPE), backward, boundary conditions      128
Fokker — Planck equation (FPE), boundary conditions      118 125
Fokker — Planck equation (FPE), chemical      266
Fokker — Planck equation (FPE), connection with SDE      96
Fokker — Planck equation (FPE), defined      52—53
Fokker — Planck equation (FPE), functional      305
Fokker — Planck equation (FPE), many dimensional      143—170
Fokker — Planck equation (FPE), many variable, boundary conditions      146
Fokker — Planck equation (FPE), many variables, eigenfunctions      165
Fokker — Planck equation (FPE), one dimensional      117 143
Fokker — Planck equation (FPE), one variable, eigenfunctions      129—136
Fokker — Planck equation (FPE), small noise expansion for      187—194
Fokker — Planck equation (FPE), stationary solution for one variable      124
Fokker — Planck equation (FPE), Stratonovich form      100
Force, phenomenological      161
Fourier analysis of fluctuating functions      17
FPE      (see Fokker “Planck equation”)
Functional derivative, defined      305
Functional FPE, for reaction diffusion systems      305 307
Gaussian distribution      36—38
Gaussian random variable      36
Generalised P-representations      408 419
Generalised P-representations and time development equations      415—419
Generalised P-representations, defined      409
Generalised P-representations, relation to Poisson representation      413
Generating function      13
Generating function for birth-death master equation      273 274
Generating function for Poisson distribution      38
Glauber — Sudarshan      382 384 410
Glauber — Sudarshan P-representation      382—384 410
Haken’s model      223
Harmonic oscillator, defined      374
Harmonic oscillator, driven by fluctuating field      398
Harmonic oscillator, driven damped      397 398
Harmonic oscillator, driven in P-representation      384
Harmonic oscillator, interaction with external field      375
Harmonic oscillator, quantum master equation      395 399
Harmonic oscillator, quantum mechanics of      373
Heat bath, defined      388
Hermite polynomials      134
Hermite polynomials in adiabatic elimination      202
Homogeneous Markov process      60
Homogeneous Markov process, defined      56
Independence      7 27
Independent random variables      27
Independent random variables and characteristic function      32
Interaction picture      394
Interaction picture and driven atom      399
Ito stochastic integral, defined      84
Ito stochastic integral, properties of      88 92
Ito’s formula, derivation of      95
Johnson noise      18 20
Joint probability      24
Jump process      52
Kramers — Moyal expansion      5
Kramers — Moyal expansion and system size expansion      251
Kramers — Moyal expansion in birth-death master equation      266
Kramers — Moyal expansion in Boltzmann master equation      337 340
Kramers — Moyal expansion, defined      249
Kramers’ equation      155
Kramers’ equation, boundary conditions      205
Kramers’ method for escape problems      349—352
Kramers’ method for escape problems in several dimensions      363—371
Kurtz’s theorem      254
Laguerre polynomial      136
Langevin equation      11 14 80—83
Laplace transform and adiabatic elimination      200
Laser light scattering      7
Law of larse numbers      30
Limit almost certain      40
Limit almost certain in distribution      41
Limit almost certain in probability      41
Limit almost certain, mean square      40
Limit almost certain, stochastic      40
Limit of sequence of random variables      39
Lindeberg condition      37 46
Linear SDE multivariable      113
Linear SDE multivariable, single-variable      112
Liouville equation      53
Liouville equation, quantum      382
Liouville operators; defined      391
Local and global descriptions of chemical reaction, connection between      328—331
Local and global fluctuations      320
Markov assumption      13 43
Markov postulate      5 10
Markov process autocorrelation for      64—66
Markov process autocorrelation for continuous, defined      46
Markov process autocorrelation for continuous, homogeneous, defined      56
Markov process autocorrelation for continuous, quantum mechanical      373
Markov process autocorrelation for continuous, stationary, defined      56
Master equation      51 235—301
Master equation (cont.) quantum, derivation      390—394
Master equation, approximation by Fokker — Planck equation      246—257
Master equation, many-variable      262 277
Master equation, many-variable, Kramers — Moyal expansion for      266
Master equation, mean first passage times      259 262
Master equation, one-variable, stationary solution of      236—238
Master equation, phase space      331 341
Master equation, quantum; defined      394
Master equation, stationary solutions without detailed balance      266
Me      (see “Master equation”)
Mean first passage time in one-variable FPE      137
Mean first passage time in one-variable FPE for master equations      259 262
Mean square limit in definition of stochastic integral      84
Mean value      28 29

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