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Risken H. — The Fokker-Planck equation: methods of solution and applications
Risken H. — The Fokker-Planck equation: methods of solution and applications



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Íàçâàíèå: The Fokker-Planck equation: methods of solution and applications

Àâòîð: 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: en

Ðóáðèêà: Ìàòåìàòèêà/Ìàòåìàòè÷åñêàÿ Ôèçèêà/

Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö

ed2k: ed2k stats

Èçäàíèå: âòîðîå

Ãîä èçäàíèÿ: 1989

Êîëè÷åñòâî ñòðàíèö: 472

Äîáàâëåíà â êàòàëîã: 25.04.2005

Îïåðàöèè: Ïîëîæèòü íà ïîëêó | Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
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Ïðåäìåòíûé óêàçàòåëü
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, $\chi$-representation      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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