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Risken H. — The Fokker-Planck equation: methods of solution and applications
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Íàçâàíèå: The Fokker-Planck equation: methods of solution and applicationsÀâòîð: Risken H. Àííîòàöèÿ: This book deals with the derivation of the Fokker-Planck equation, methods of solving it and some of its applications. Various methods such as the simulation method, the eigenfunction expansion, numerical integration, the variational method, and the matrix continued-fraction method are discussed. This is the first time that this last method, which is very effective in dealing with simple Fokker-Planck equations having two variables, appears in a textbook. The methods of solution are applied to the statistics of a simple laser model and to Brownian motion in potentials. It is shown that the solution of the equation for Brownian motion in a variety of potentials can be expressed in terms suitable for evaluation on a computer. A supplement is included, containing a short review of new material together with some recent references. The book should be very useful to graduate students in physics, chemical physics, and electrical engineering, and also to research workers in these fields.
ßçûê: Ðóáðèêà: Ìàòåìàòèêà /Ìàòåìàòè÷åñêàÿ Ôèçèêà /Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Èçäàíèå: âòîðîåÃîä èçäàíèÿ: 1989Êîëè÷åñòâî ñòðàíèö: 472Äîáàâëåíà â êàòàëîã: 25.04.2005Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
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Fokker — Planck equation, for laser see also “Laser Fokker — Planck equation” 382ff Fokker — Planck equation, for laser, quantum mechanical derivation 376 Fokker — Planck equation, for low friction mobility 312ff Fokker — Planck equation, for low friction, stationary distribution 304ff Fokker — Planck equation, for low friction, stationary distribution, x-dependent 307ff Fokker — Planck equation, for low friction, stationary mobility 335ff Fokker — Planck equation, for metastable rectangular potential well 119 Fokker — Planck equation, for motion in periodic potential, normalization 286f Fokker — Planck equation, for parabolic potential 109 Fokker — Planck equation, for stochastic process with colored noise 416ff Fokker — Planck equation, for stochastic process with colored noise, generalizations 419f Fokker — Planck equation, for V-shaped potential 111f Fokker — Planck equation, generalizations 8f Fokker — Planck equation, generalized 9 Fokker — Planck equation, generalized potential 141 Fokker — Planck equation, heuristic derivation 6 Fokker — Planck equation, infinite jump 113 Fokker — Planck equation, instationary solutions 337 Fokker — Planck equation, jump conditions for the eigenfunctions 113f Fokker — Planck equation, jumps of the potential 112ff Fokker — Planck equation, lowest eigenvalue 158 Fokker — Planck equation, N variables 5 Fokker — Planck equation, N variables, eigenfunction expansions 139ff Fokker — Planck equation, N variables, methods of solution 133ff Fokker — Planck equation, nonperiodic solutions 338 Fokker — Planck equation, numerical integration method 120f Fokker — Planck equation, numerical methods for eigenfunctions 119ff Fokker — Planck equation, numerical methods for eigenvalues 119ff Fokker — Planck equation, one dimension 87 Fokker — Planck equation, one variable 4 72 87 Fokker — Planck equation, one variable, eigenfunction expansion 101 Fokker — Planck equation, one variable, methods of solution 96ff Fokker — Planck equation, one variable, normalized form 97 Fokker — Planck equation, one variable, Schrodinger form 107 Fokker — Planck equation, one variable, stationary solution 98 Fokker — Planck equation, one variable, Sturm — Liouville form 106 Fokker — Planck equation, rigorous derivation 6 Fokker — Planck equation, Schrodinger potential 142 Fokker — Planck equation, several variables, examples 86ff Fokker — Planck equation, solution by expansion into a complete set 159f Fokker — Planck equation, solution by Fourier transform 154 Fokker — Planck equation, solution by matrix continued-fraction method 160f Fokker — Planck equation, solution by numerical integration 159 Fokker — Planck equation, solution by WKB method 162 Fokker — Planck equation, solution for inverted parabolic potential 109f Fokker — Planck equation, solution for Ornstein — Uhlenbeck process 100f Fokker — Planck equation, solution for parabolic potential 108f Fokker — Planck equation, solution for small times N variables 85f Fokker — Planck equation, solution for small times one variable 73f Fokker — Planck equation, solution for Wiener process 99 Fokker — Planck equation, solution methods reduction to Hermitian problem 159 Fokker — Planck equation, solution methods transformation of variables 158 Fokker — Planck equation, stationary distribution 314ff Fokker — Planck equation, stationary distribution, matrix continued-fraction method 317ff Fokker — Planck equation, stationary distribution, mobility in periodic potential 334 Fokker — Planck equation, stationary distribution, numerical calculation 320ff Fokker — Planck equation, stationary drift velocity for periodic potential 318f Fokker — Planck equation, three-dimensional 87 Fokker — Planck equation, transformation of variables 88f Fokker — Planck equation, transformation to energy variable 301ff Fokker — Planck equation, transformation to Hermitian form, one variable 103f Fokker — Planck equation, variational method 158 Fokker — Planck equation, variational method, for solving 120 Fokker — Planck operator anti-Hermitian part 141 Fokker — Planck operator approach of solutions to limit solution 134ff Fokker — Planck operator decomposition in reversible and irreversible part 150 Fokker — Planck operator for parabolic potential 109 Fokker — Planck operator Hermitian part 141 Fokker — Planck operator inverse friction expansion 259ff Fokker — Planck operator N variables, Hermiticity and potential conditions 134 Fokker — Planck operator relation for detailed balance 152 Formal solution 66f 69 83 Forward Kolmogorov equation 70 Fredholm integral equation 130 Friction constant for Brownian motion 230 Friction, position dependent, for Kramers equation 275 Gaussian distribution 23 Gaussian distribution instationary 100 Gaussian distribution, for two variables 238 Gaussian distribution, general 24 Gaussian distribution, moments 24 Gaussian Langevin force, delta-correlated 44 Generation and recombination process see “Birth and death process” Generation process 76 Glauber's P representation 427 Green — Kubo expressions 163 Green's function for Brownian motion in parabolic potential 177 Green's function for Kramers equation inverse friction expansion 261 268ff Green's function for Ornstein — Uhlenbeck process 38f Green's function for systems of tridiagonal equations, Laplace-transform 210 Green's function for tridiagonal differential equations 209ff Green's function for Wiener process 99 Green's function matrix 217 Green's function matrix, Laplace transform 217 Green's function of Kramers equation 251ff Green's function of Kramers equation, matrix elements 251 Green's function, 272ff H-Theorem 135 Haken's slaving principle 189 Harmonic mixing in cosine potential 226 Harmonic oscillator damped quantum-mechanical 425ff Harmonic oscillator evaluation of a matrix continued fraction 422ff Harmonic oscillator Green's function 422ff Harmonic time dependence, differential equations with multiplicative 222f Hermitian form of irreversible operator, Kramers equation 233 High-friction limit for motion in periodic potential 287ff Independence of drift and diffusion coefficients of some variable 183 Independence of drift and diffusion examples 184 Inertial effects in Brownian motion 278 Inertial effects in Brownian motion, of dipoles 282f Initial value problem for tridiagonal systems of differential equations 213 Integral equation for diffusion over barrier, transform to Fredholm equation 129f Intensity correlation function for laser light 389ff Internal noise 59 Inverse friction expansion eigenvalues and eigenfunctions 266ff Inverse friction expansion for Brownian motion in periodic potential 293f Inverse friction expansion for Fokker — Planck operator 259ff Inverse friction expansion for Green's function 417ff Inverse friction expansion for Kramers equation 257ff Inverse friction expansion for parabolic potential 265f Inverted parabolic potential 245f Inverted potential boundary condition 117ff Inverted potential eigenvalues and eigenfunctions 117ff Irrelevant variables 189 Irreversible drift coefficients 149f Irreversible operator 150 231f Irreversible probability current 150 Ito's definition of stochastic integrals 50ff Joint distribution see also “Joint probability” Joint distribution for Kramers equation, stationary 253 Joint probability change on time-reversal 149 Joint probability detailed balance condition 148 Joint probability expansion into eigenmodes for laser 388ff Joint probability for Markov processes, several variables 85 Joint probability in terms of eigenfunctions 105 143 Joint probability one dimension, Ornstein — Uhlenbeck process 101 Joint probability Ornstein — Uhlenbeck process 156 Joint probability stationary state, several variables 85 Josephson tunneling junction 281f Jump conditions for Fokker — Planck equation 112ff Klein — Kramers equation 7 Kolmogorov equation, forward 70 Kramers equation 7 Kramers equation, boundary condition for first-passage time problem 183 Kramers equation, detailed balance for stationary distribution 152f Kramers equation, eigenvalue problem 255ff Kramers equation, eigenvalues for inverted parabolic potential 246ff Kramers equation, eigenvectors 255ff Kramers equation, expansion into complete set 250 Kramers equation, for Brownian motion in periodic potentials 276ff Kramers equation, for Brownian motion in periodic potentials, eigenfunctions 359ff Kramers equation, for Brownian motion in periodic potentials, eigenvalues 359ff Kramers equation, for linear force eigenvalues and eigenfunctions 241ff Kramers equation, for linear force stationary distribution 240 Kramers equation, for linear force transition probability 238 244 Kramers equation, for linear force variance matrix 239 Kramers equation, for rotating dipoles 348 Kramers equation, forms 229f Kramers equation, free Brownian motion 254 Kramers equation, Green's function 251ff Kramers equation, initial value problem 251 Kramers equation, inverse friction expansion 257ff Kramers equation, matrix continued-fraction solutions 249ff Kramers equation, memory kernel 251 Kramers equation, normalization of eigenfunctions 256 Kramers equation, normalization of variables 230f Kramers equation, position dependent friction 275 Kramers equation, response and correlation function 168f Kramers equation, solution, reversible part 232 Kramers equation, solutions 229ff Kramers equation, solutions, for inverted parabolic potential 245ff Kramers equation, solutions, for parabolic potential 237ff Kramers equation, stationary joint distribution 253 Kramers equation, stationary solution 137 Kramers equation, symmetry relations for eigenfunctions 256 Kramers equation, three dimensions 230 Kramers equation, transition probability 252 Kramers equation, velocity correlation function 253 Kramers equation, with linear force, solutions 237ff Kramers — Kronig relations 172f Kramers — Moyal backward expansion formal solution 69 Kramers — Moyal backward expansion N variables 82f Kramers — Moyal backward expansion one variable 67ff Kramers — Moyal coefficients one variable 48ff Kramers — Moyal coefficients several variables 54ff Kramers — Moyal coefficients van Kampen's expansion 77 Kramers — Moyal equation, equivalence of forward and backward expansion 69 Kramers — Moyal expansion for birth and death process 76 Kramers — Moyal expansion truncated 71 77 Kramers — Moyal forward expansion formal solution 66f Kramers — Moyal forward expansion N variables 82 Kramers — Moyal forward expansion one variable 8 63ff Kramers' escape rate theory 122ff Kubo oscillator 45 4l4ff Langevin equation determination from drift and diffusion coefficients 56 Langevin equation for Brownian motion in periodic potentials 277 Langevin equation for Brownian motion one dimension 32 Langevin equation for Brownian motion three dimensions 36 Langevin equation for detuned laser 393f Langevin equation for motion in periodic potential, normalization 286f Langevin equation for single mode laser 379ff Langevin equation nonlinear see “Nonlinear Langevin equation uniqueness” 56 Langevin force 2f Laplace-transform of Green's function for systems of tridiagonal equations 210 Laser drift and diffusion coefficients 381f Laser equation of motion for density operator 377f Laser equations, semiclassical, one mode, homogeneously broadened 377ff Laser fluctuating control parameter 382 Laser Fokker — Planck equation 382ff Laser Fokker — Planck equation, eigenfunction expansion 398 Laser Fokker — Planck equation, eigenvalues by matrix continued fraction method 396ff Laser Fokker — Planck equation, eigenvalues for fluctuating control parameter 433ff Laser Fokker — Planck equation, expansion into complete set 494ff Laser Fokker — Planck equation, expansion of transition probability into eigenmodes 387ff Laser Fokker — Planck equation, fluctuating control parameter 432ff Laser Fokker — Planck equation, for detuned laser 394 Laser Fokker — Planck equation, intensity distribution far above threshold 404ff Laser Fokker — Planck equation, normalization of variables 383 Laser Fokker — Planck equation, solution by matrix continued fraction method 394ff Laser Fokker — Planck equation, stationary cumulants 386f Laser Fokker — Planck equation, stationary distribution 432f Laser Fokker — Planck equation, stationary expectation values 384ff Laser Fokker — Planck equation, stationary moments 384f Laser Fokker — Planck equation, stationary solution 384ff Laser Fokker — Planck equation, transformation to additive noise 433 Laser Fokker — Planck equation, transformation to Morse potential 433ff Laser Fokker — Planck equation, transient moments far above threshold 406 Laser Fokker — Planck equation, transient solution for amplitude 402f Laser Fokker — Planck equation, transient solution for intensity 398ff Laser intensity moments 215f Laser intensity, moment equation 199 384 Laser Langevin equations 379ff Laser Langevin equations fluctuating control parameter 431ff Laser Langevin equations linearization of 381 Laser light, statistical properties 374ff Laser nonlinear Langevin equation 375f Laser semiclassical treatment 375 Laser threshold condition 379 Laser transient solution without noise 375 Light, statistical properties of laser 374ff Linear process for fast variable, adiabatic elimination 192ff Linear response 163ff 276 Linear response functions 164ff Linear response, connection to lowest eigenvalue 345 Linear response, for Kramers equation 168f Linear response, for Ornstein — Uhlenbeck process 171 Linear response, of velocity for parabolic potential 177 Linear response, to energy 170 Linear response, to temperature 169f Linewidth of laser light amplitude, stationary 390f Linewidth of laser light intensity, stationary 391ff Ljapunov function 135 Locked solution 278 328 Lowest eigenvalue of Fokker — Planck equation 158 Markov approximation in projector formalism 195 Markov process 9 27f Markov property 59f Markovian variables, properties of time-integrated 184ff Markovian variables, time integrals of 184ff Master equation 11 Master equation, for birth and death process 76 Master equation, for continuous variables 146f Master equation, for generation and recombination process 76 Master equation, for Poisson process 78 Master equation, generalized 11 Master equation, with nearest neighbor coupling 198 Master equation, with two nearest-neighbor coupling 201 Mathieu equation 222f Mathieu equation, generalizations 223f Matrix continued-fraction method 121f 218f Matrix continued-fraction method application to partial differential equations 160f Matrix continued-fraction solutions of Kramers equations 249ff Maxwell distribution, one-dimensional 16 73 Mean first-passage time see also “First passage time” 182 Mean-squared deviation 19 Mean-squared displacement 34ff Memory effects 9 Memory function 213 Memory function, for Kramers equation, inverse friction expansion 261 Memory kernel 213 Memory kernel, for Kramers equation 251 Memory matrix-kernel solution for vector recurrence relations 219 Metastable potential , diffusion over barrier 125ff Metastable potential asymmetric 128f Metastable rectangular potential well 119 Metric tensor 94f Mobility 276 Mobility, connection to diffusion constant 343 Mobility, linear and nonlinear 276 Moments, connection to cumulants 18 Moments, for laser intensity, equation of motion 199 384 Moments, for stationary intensity of laser 215f Moments, of first-passage time 180 Moments, of first-passage time, equation 183 Morse potential, application to fluctuating control parameter 433ff Motion in periodic potential, without noise 329ff Multiplicative noise 44 Nakajima — Zwanzig projection formalism, connection to adiabatic elimination 194f Natural boundary condition 102f Noise additive 44
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