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Risken H. — The Fokker-Planck equation: methods of solution and applications
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Название: The Fokker-Planck equation: methods of solution and applicationsАвтор: Risken H. Аннотация: 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: Рубрика: Математика /Математическая Физика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Издание: второеГод издания: 1989Количество страниц: 472Добавлена в каталог: 25.04.2005Операции: Положить на полку |
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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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