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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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| Предметный указатель |
Metastability 342
Microscopic reversibility 159
moment 30
Moment generating function 32
Moments and characteristic function 32
Multitime averages in quantum systems 394
Multivariate master equation 303
Multivariate master equation for diffusing system 307 308
Natural boundary for FPE 122
Noise 11
Nonanticipating function, defined 86
Nonlinear absorber 415
Nyquist’s formula 163
Nyquist’s theorem 18 20
Omstein — Uhlenbeck process 74—77
Omstein — Uhlenbeck process and Onsager relations 160
Omstein — Uhlenbeck process and quantum harmonic oscillator 396
Omstein — Uhlenbeck process in small noise expansion of SDE 181
Omstein — Uhlenbeck process, complex, defined 410
Omstein — Uhlenbeck process, eigenfunctions 134
Omstein — Uhlenbeck process, existence theorems for 411—415
Omstein — Uhlenbeck process, exit time from a circle 171
Omstein — Uhlenbeck process, FPE from 396 415
Omstein — Uhlenbeck process, generalised 408 419
Omstein — Uhlenbeck process, generalised, defined 409
Omstein — Uhlenbeck process, multivariate 109 111
Omstein — Uhlenbeck process, Oscillator, noisy 105
Omstein — Uhlenbeck process, spectrum matrix 111
Omstein — Uhlenbeck process, stationary distribution 126
Omstein — Uhlenbeck process, stationary variance 110
Omstein — Uhlenbeck process, stationary variance in two dimensions 110
Omstein — Uhlenbeck process, time correlation function 76
Omstein — Uhlenbeck process, time correlation matrix in stationary state 111
Omstein — Uhlenbeck process, time dependent 115
Onsager relations 160—165
Operator correspondences in Glauber — Sudarshan P-representation 383
Operator identities for complex P-representation 414
Operator identities for positive P-representation 414—415
Ornstein — Uhlenbeck process (SDE) 106—107
Pauli matrices, defined 399
Phase space master equation 331 341
Phase transition, second order in reaction diffusion system 324
Phenomenological flux 162
Phenomenological force 161
Poisson distribution 13 14 38—39
Poisson distribution and coherent states 379
Poisson distribution relaxation in reaction diffusion system 316—318
Poisson process 13 72—75
Poisson representation 235 277— 301
Poisson representation and Boltzmann master equation 338 341
Poisson representation and time correlation functions 289—294
Poisson representation and unimolecular reactions 279
Poisson representation and unimolecular reactions in reaction diffusion system 314—315
Poisson representation and unimolecular reactions positive, defined 285—289
Poisson representation and unimolecular reactions real, defined 282
Poisson representation and unimolecular reactions, complex, defined 282
Poisson representation and unimolecular reactions, relation to generalised P-representation 413
Poissonian time dependent solutions of certain master equations 241
Positive P-representation, defined 410
Positive P-representation, FPE from 416—418
Positive P-representation, operator identities for 414—415
Potential barrier, escape over 138— 141
Potential conditions for FPE in complex P-representation 418—419
Potential conditions for many-variable FPE 146
Potential solution of one-variable FPE 124
Prey-predator system 8
Prey-predator system, chemical FPE for 268 273
probability 22
Probability a priori 23
Probability axioms 22
Probability current; defined 119
probability density 28
Probability, conditional 25
Probability, joint 24
Projection operator in adiabatic elimination 198
q-representation 409
Quantum characteristic function 386—388
Quantum Liouville equation 382
Quantum Markov process, derivation of 388—394
Quantum Markov process, time correlation functions in 402 408
Quantum master equation defined 394
Quantum master for harmonic oscillator 395 399
Quantum master, derivation of 390—394
Quantum mechanics of harmonic oscillator 373
Quantum regression theorem 404 405
Quantum regression theorem, P-representation 409
Random telegraph process 77 79
Random telegraph process and driven atom 402
Random variable 24
Random variable, Gaussian 36
Random variables independent 27
Random walk 70—73
Random walk, approximation by FPE 247
Rayleigh process 144 147
Rayleigh process, eigenfunctions 135
Reaction diffusion chemical equation 305
Reaction diffusion master equation; defined 313
| Reaction diffusion system 303
Reaction diffusion system in Poisson representation 314
Reaction diffusion system, fluctuating partial differential equations 313
Reaction diffusion systems, divergence problems 327
Reflecting boundary condition for backward FPE 129
Regression theorem 65
Regression theorem, quantum 404 405
Relaxation time, dependence on peak populations 352
Sample path, continuous 5 45—47
Scaling assumption in approximation of master equation 246
Schrodinger’s equation 374
SDE (see “Stochastic differential equation”)
Separatrix 361 372
Set of events 21
Sets of probability zero 29
Shot noise 12
Sigma algebra 22
Slave noisy 226
Slave noisy, silent 225
Slaving principle 197
Small noise expansion for SDE 180—187
Small noise expansion of FPE 187—194
Smoluchowski equation 197
Smoluchowski equation and escape problem 369
Smoluchowski equation for diffusion in a double well 349
Smoluchowski equation, boundary conditions 205
Smoluchowski equation, corrected 209
Smoluchowski equation, corrections to 206—210
Smoluchowski equation, defined 197 203
Spatial correlations 315—328
Spectrum 16
Spectrum matrix and eigenfunctions 169
Spectrum matrix with detailed balance 150
Spectrum of Omstein — Uhlenbeck process 111
Spectrum, flat 19
Splitting probability in double well potential 345
Standard deviation 30
Stationary 18
Stationary distribution of FPE, asymptotic method for 194
Stationary Markov process, defined 56
Stationary process, approach to 61 63
Stationary solutions, many-variable FPE 146—148
Stationary systems 17
Statistical mechanics 6
Stochastic differential equation 5 7 14 15 80—115
Stochastic differential equation and positive Poisson re 287 289
Stochastic differential equation as limit of nonwhite process 210—218
Stochastic differential equation, connection with FPE 96
Stochastic differential equation, definition and properties of 92—102
Stochastic differential equation, dependence of solutions on initial conditions 101
Stochastic differential equation, dependence of solutions on parameters 101
Stochastic differential equation, partial 305
Stochastic differential equation, small noise expansion for 180—187
Stochastic integral, defined 83
Stochastic partial differential equation 305
Stochastic process, defined 42
Stochastic process, separable 42
Stratonovich 86
Stratonovich SDE as limit of nonwhite process 210—218
Stratonovich stochastic integral 86
System size 235
System size expansion and critical fluctuations 255
System size expansion and critical fluctuations and Kramers — Moyal expansion 251
System size expansion and critical fluctuations and Kurtz’s theorem 254
System size expansion and critical fluctuations in birth-death master equation 266
System size expansion and critical fluctuations in Boltzmann master equation 337 340
System size expansion and critical fluctuations of master equation 250—257
Third-order noise 295
Third-order noise, defined 299—300
Time correlation function (see “Autocorrelation function”)
Time correlation functions and P-representation 405 408
Time correlation functions and quantum harmonic oscillator 405 408
Time correlation functions for two-level atom 408
Time correlation functions in quantum Markov processes 402 408
Time reversal invariance 150
Transition probability 10
Trimolecular reaction as a limit of bimolecular reaction 294 299
Two-level atom driven 399 402
Two-level atom driven, time correlations 408
Unimolecular reactions in Poisson representation 279
Unstable state, decay from 347
Van Kampen’s system size expansion 250
Variance 30
Variational principle for eigenfunctions of FPE 168
viscosity 6
Volterra-Lotka system (see also “Prey-predator system”)
von Neumann’s equation 382
White noise 19—20 80—83
White noise as limit of nonwhite process 210—218
Wiener process 46 47 67—70
Wiener process as limit of random walk 72
Wiener process, autocorrelation function 70
Wiener process, eigenfunctions 132
Wiener process, independence of increments 69
Wiener process, irregularity of sample paths 68
Wiener process, nondifferentiability of sample paths 68
Wiener — Khinchin theorem 17
Wigner representation 409
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