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

Автор: Risken H.


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.

Язык: en

Рубрика: Математика/Математическая Физика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: второе

Год издания: 1989

Количество страниц: 472

Добавлена в каталог: 25.04.2005

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Предметный указатель
Absorbing boundary      179
Absorbing wall      102f
Absorption, infrared by polar molecules      282f
Additive noise      44
Adiabatic elimination of fast variables      188ff
Adiabatic elimination of fast variables, linear process for fast variable      192ff
Adjoint operator, solutions in terms of      182
After-effect-response function      166
Amplitude correlation function for laser light      389ff
Analytic continuation for solving non-Hermitian problems      145
Anharmonic potential complex eigenvalues of Kramers equation, low friction limit      365ff
Anharmonic potential, Schrodinger equation      201f
Annihilation operator      233
Applications of tridiagonal recurrence relations      196ff
Backward Kolmogorov equation, N variables      83
Balance detailed      146
Balance for steady state      145 147
BGK collision operator      10
BGK model      10
Biorthogonal set, expansion into      137ff 155f
Birth and death process      76
Bistability between running and locked solution      328ff
Bistability of Brownian motion in periodic potentials      278
Bistable potential, diffusion over barrier      125ff
Bistable rectangular potential well      114ff
Bloch equations      377f
Bloch equations, for density operator of two-level system      224f
Boltzmann equation      9f
Boltzmann equation, with BGK and SW collision operators      420ff
Born — Oppenheimer approximation      190
Bose — Einstein distribution      411
Boson operators      108f 233
Boundary condition for first-passage time problem      180
Boundary condition for Fokker — Planck equation, one variable      102f
Boundary condition for infinite jumps      102f
Boundary condition for inverted potential      117ff
Boundary condition for Kramers equation, first-passage time problem      183
Boundary condition natural      102f
Boundary condition periodic      102f
Boundary, absorbing      179f
Brinkman's hierarchy application to periodic potential      315
Brinkman's hierarchy for Kramers equation      236
Brownian motion      1ff
Brownian motion, deterministic equation      1
Brownian motion, deterministic equation, with external force      87
Brownian motion, for free particle      240 249 254
Brownian motion, for laser model      380
Brownian motion, in a superionic conductor      280f
Brownian motion, in periodic potential      276ff
Brownian motion, in periodic potential, high friction mobility      291
Brownian motion, in periodic potential, high friction, continued fraction expansion      289ff
Brownian motion, in periodic potential, high friction, continued fraction expansion, cosine potential      297ff
Brownian motion, in periodic potential, high friction, continued fraction expansion, distribution functions      291f
Brownian motion, in periodic potential, high friction, continued fraction expansion, eigenvalues      298ff
Brownian motion, in periodic potential, high friction, continued fraction expansion, rectangular model potential      295ff
Brownian motion, in periodic potential, high friction, continued fraction expansion, saw-tooth potential      293
Brownian motion, in periodic potential, high friction, continued fraction expansion, time-dependent solution      294ff
Brownian motion, in periodic potential, high-friction limit      287ff
Brownian motion, in periodic potential, high-friction limit, stationary solution      287ff
Brownian motion, in periodic potential, inverse friction expansion      293f
Brownian motion, in periodic potential, low friction, transformation to energy      301ff
Brownian motion, in periodic potential, low-friction limit      300ff
Brownian motion, in periodic potential, normalization of equations      286f
Brownian motion, of a mathematical pendulum      280
Brownian motion, of dipoles in a constant field      282f
Brownian motion, of particles in potential      229ff
Brownian motion, of two interacting particles      88
Brownian motion, one-dimensional, in potential      87
Brownian motion, stochastic differential equation      2
Brownian motion, three-dimensional      86f
Chapman — Kolmogorov equation one variable      29
Chapman — Kolmogorov equation several variables      31
Characteristic exponent of Mathieu equation      223
Characteristic function for damped harmonic oscillator, equation of motion      427
Characteristic function for several variables      20f
Characteristic function one variable      16f
Cole — Cole plot of complex susceptibility      352
Collision operator      150
Collision operator BGK      10 420
Collision operator, Skinner & Wolynes      420
Colored noise      3 32
Colored noise, exponential, for Kubo oscillator      414ff
Colored noise, reduction to white noise      415
Completeness relation of eigenfunctions      105
Computer simulation of Langevin equations      60ff
Conditional probability      21
Conditional probability, several variables      26ff 31
Conductivity, frequency dependent      see also “Mobility” 347
Continuants      215
Continued fractions downward iteration      227
Continued fractions matrix      197
Continued fractions methods for calculating matrix      227f
Continued fractions methods for calculating ordinary      226f
Continued fractions ordinary      196ff
Continued fractions upward iteration      227 228
Continued-fraction approximants of ordinary      204
Continued-fraction asymptotic ratio of coefficients      208f
Continued-fraction evaluation for harmonic oscillator      422ff
Continued-fraction for Green's function of Kubo oscillator with colored noise      416
Continued-fraction numerical stability      208
Continued-fraction ordinary      203f
Continued-fraction solution of recurrence relations laser intensity moments      206ff
Continued-fraction solution of recurrence uniqueness      204ff
Continuity equation      230
Cooper pairs      282
Correlation coefficient      22
Correlation functions connection between different      167ff
Correlation functions, connection to spectral density      173ff
Correlation functions, for constant diffusion tensor, detailed balance      168
Correlation functions, for rotating dipoles      349
Correlation functions, for velocity and space, low friction expansion      274
Correlation functions, Kramers equation      168f
Correlation functions, matrix continued-fraction expansion      350ff
Correlation functions, of energy      170
Correlation functions, of force      169
Correlation functions, of velocity      33 169
Correlation functions, stationary      166f
Correlation functions, stationary, for laser      389ff
Correlation functions, symmetry      174
Correlation functions, two times, Ornstein — Uhlenbeck process      41f
Covariance      23
Creation operator      233
Critical force      330f
Cross correlation      21
Cumulants connection to moments      18
Cumulants for several variables      20
Cumulants one variable      17f
Current in superionic conductors      281
Current voltage characteristic of Josephson junction      282
Cycle slip      283
Damped quantum-mechanical harmonic derivation of Fokker — Planck equation      426ff
Damped quantum-mechanical harmonic oscillator      425ff
Damping constant for Brownian motion      230
Delta-correlated random process, Gaussian      230
Delta-correlation for Langevin force      3
Density matrix for two-level system      224 377
Density operator for two-level system      224 377
Detailed balance      145ff
Detailed balance condition      134 147
Detailed balance condition, for joint distribution      148
Detailed balance condition, operator equation for      147f
Detailed balance condition, relation for Fokker — Planck operator      152
Detailed balance condition, sufficient and necessary for coefficients      151
Detailed balance, for Fokker — Planck equation      147ff
Detailed balance, for master equation      145f
Detailed balance, for stationary distribution of Kramers equation      152f
Deterministic differential equation      1
Deterministic motion in cosine potential, low friction      331ff
Detuned laser, Langevin equation      393f
Diagonalizing non-Hermitian matrices      138
Differential equations applications of recurrence relations      196ff
Differential equations Green's function for tridiagonal equations      209ff
Differential equations solution of initial value problem      209ff
Differential equations systems of coupled tridiagonal      198
Differential equations systems of coupled tridiagonal with higher derivatives      198
Differential equations tridiagonal systems, initial value problem      213
Differential equations with multiplicative harmonic time dependence      222ff
Diffusion and drift coefficients independent of some variables      183
Diffusion coefficients constant      153
Diffusion coefficients constant, determination from Langevin equation      58
Diffusion coefficients constant, for Brownian motion with force      58
Diffusion coefficients constant, rational functions      121f
Diffusion constant connection to mobility      343
Diffusion constant definition      342
Diffusion constant Einstein's result      35
Diffusion constant for finite external force      346
Diffusion matrix      84
Diffusion matrix semidefinite case      152
Diffusion matrix, singular      152
Diffusion matrix, symmetry      84
Diffusion over potential barrier      122ff
Diffusion over potential barrier, bistable and metastable      125ff
Diffusion over potential barrier, transform to Fredholm integral equation      129f
Diffusion tensor      5
Diffusion tensor contravariant form      92
Diffusion tensor transformation      91
Dipoles, rotation in a constant field      282f
Dissipation of energy for parabolic potential      178
Dissipation-fluctuation theorem      see “Fluctuation-dissipation theorem”
Distribution function      see also “Probability density” 15
Distribution function, for slow variable      191
Distribution function, for time-integrated velocity      187
Distribution function, of first-passage time      180
Distribution function, one variable      4
Distribution function, positivity, several variables      86
Drift and diffusion coefficients for laser model      381f
Drift and diffusion coefficients independent of some variables      183
Drift coefficient determination from Langevin equation      58
Drift coefficient for Brownian motion with force      58
Drift coefficient linear      153
Drift coefficient rational functions      121f
Drift coefficient reversible and irreversible      149f
Drift coefficient transformation on time reversal      149f
Drift vector      5 84
Drift vector, contravariant form      93
Drift vector, transformation      91
Dynamic structure factor      340f
Eigenfunction expansion for laser transient solution      398
Eigenfunction expansion, one variable Fokker — Planck equation      101
Eigenfunction, zeroth      143
Eigenfunctions, and eigenvalues, inverse friction expansion      266ff
Eigenfunctions, check by using sum rules      171
Eigenfunctions, completeness relation      105 143
Eigenfunctions, for bistable potentials      126
Eigenfunctions, for bistable potentials, approximation for high barriers      126
Eigenfunctions, numerical methods for determining      119f
Eigenfunctions, of inverted potentials      117ff
Eigenfunctions, of Kramers equation with linear force      241ff
Eigenfunctions, of the Hermitian part, orthonormality      143
Eigenfunctions, orthogonality      104
Eigenfunctions, orthonormality      104
Eigenvalue equation      137f
Eigenvalue problem for scalar recurrence relations      214ff
Eigenvalue problem for vector recurrence relations      220ff
Eigenvalues (complex) of Kramers equation in anharmonic potential, low friction      365ff
Eigenvalues (real) of Kramers equation in periodic potential, low friction      368ff
Eigenvalues and eigenfunctions, inverse friction expansion      266f
Eigenvalues bounds of real part      144
Eigenvalues connection to linear response mobility      345
Eigenvalues determination by variational problem      120
Eigenvalues for bistable potentials, approximation for high barriers      126
Eigenvalues for inverted parabolic potential of Kramers equation      246ff
Eigenvalues for metastable potential, asymmetric      128
Eigenvalues for non-Hermitian operators      144
Eigenvalues numerical methods for determining      119ff
Eigenvalues of inverted potentials      117ff
Eigenvalues of Kramers equation      255ff
Eigenvalues of Kramers equation, in periodic potential      359ff
Eigenvalues of Kramers equation, in periodic potential, external force      363ff
Eigenvalues of Kramers equation, in periodic potential, external force, low friction      365ff
Eigenvalues of Kramers equation, with linear force      241ff
Eigenvalues of laser amplitude fluctuations      390
Eigenvalues of laser Fokker — Planck equation      396ff
Eigenvalues of laser intensity fluctuations      392
Eigenvalues positivity      104f
Eigenvalues positivity of real part      143f
Eigenvalues upper and lower bounds      120
Eigenvectors for vector recurrence relations      221
Einstein relation, connection to fluctuation-dissipation theorem      175
Einstein's result for the diffusion constant      35
EINSTEIN's summation convention      38 138 148 155
Elimination of fast variables adiabatic      188ff 192ff
Elimination of fast variables Fokker — Planck vs. Langevin equation      194
Elimination of variables, Nakajima — Zwanzig projector formalism      194f
Energy correlation function      170
Energy dissipation      173
Equation of motion one variable      4
Equation of motion Ornstein — Uhlenbeck process      157
Equilibrium distribution      146
Equivalence of solutions of the forward and backward Kramers — Moyal equation      69f
Escape rate improved Kramers'      124
Escape rate Kramers'      124
Even variables      147
Expansion coefficients of Kramers equation, equations of motion      236
Expansion of solutions into biorthogonal set      137ff
Expansion of solutions into complete set      121f
Expectation values, Ornstein — Uhlenbeck process, equations of motion      157
External field      164
external noise      59
Factorial moment, photoelectron counting      408f
Fast variables      188f
Fast variables, adiabatic elimination of      188ff
Fast variables, for periodic potential      279
First-passage time      179
First-passage time problems      179ff
First-passage time problems, for Kramers equation, boundary condition      183
First-passage time, boundary conditions      180
First-passage time, distribution function      180
First-passage time, equation for moments      183
First-passage time, for metastable potential      130ff
First-passage time, formal solution      181
First-passage time, mean      182
First-passage time, moments      180
Fluctuating control parameter for laser model      431ff
Fluctuation-dissipation theorem      167
Fluctuation-dissipation theorem, connection to Einstein relation      175
Fokker — Planck equation      70
Fokker — Planck equation path integral solutions      74f
Fokker — Planck equation, adiabatic elimination of fast variables      188f
Fokker — Planck equation, alternative derivation      429f
Fokker — Planck equation, analytic solutions      7
Fokker — Planck equation, and Kramers — Moyal expansion      72
Fokker — Planck equation, boundary conditions, one variable      102f
Fokker — Planck equation, covariant form      91ff
Fokker — Planck equation, decomposition into Hermitian and anti-Hermitian part      140f
Fokker — Planck equation, derivation      63ff 429f
Fokker — Planck equation, eigenfunction expansion      339
Fokker — Planck equation, eigenvalues, reduction to Schrodinger equation      107
Fokker — Planck equation, exact solution, Ornstein — Uhlenbeck process      153ff
Fokker — Planck equation, expansion into complete set      121f
Fokker — Planck equation, expansion into Hermite functions      234f
Fokker — Planck equation, finite jump      112f
Fokker — Planck equation, for arbitrary friction, stationary distribution      314ff
Fokker — Planck equation, for bistable rectangular potential well      114ff
Fokker — Planck equation, for Brownian motion in periodic potentials      276ff
Fokker — Planck equation, for Brownian motion three-dimensional      86f
Fokker — Planck equation, for Brownian motion two interacting particles      88
Fokker — Planck equation, for damped quantum-mechanical harmonic oscillator, derivation      426ff
Fokker — Planck equation, for infinite square well Schrodinger potential      110f
Fokker — Planck equation, for Kubo oscillator with colored noise      415
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