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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.

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: 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

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Noise colored      3 32 416ff
Noise colored, for Kubo oscillator      414ff
Noise multiplicative      44
Noise non-Markovian, reduction to Markovian      60
Noise of spontaneous emission      375
Noise white      3 32
Noise-induced drift      see “Spurious drift”
Noiseless motion in cosine potential      329ff
Noiseless motion in cosine potential, low friction      331ff
Non — Hermitian problems, reduction to Hermitian ones by analytic continuation      145
Non — Markovian noise, reduction to Markovian      60
Non — Markovian process      9
Nonlinear Langevin equation one variable      44
Nonlinear Langevin equation several variables      54
Nonlinear response      276
Normalization for motion in periodic potential      286f
Normalization of eigenvectors of Kramers equation      256
Normalization of the diffusion coefficient      96f
Normalization of the probability distribution      72
Normalization of the probability distribution, several variables      85
Normalization of variables for Kramers equation      230f
Numerical integration method for solving the Fokker — Planck equation      120f
Odd variables      147
Onsager — Machlup function      75f
Operator equation for detailed balance condition      147 148
Ornstein — Uhlenbeck process      153ff
Ornstein — Uhlenbeck process, for damped harmonic oscillator      428
Ornstein — Uhlenbeck process, for laser      381
Ornstein — Uhlenbeck process, for one variable, joint distribution      109
Ornstein — Uhlenbeck process, joint probability      156
Ornstein — Uhlenbeck process, potential condition      156f
Ornstein — Uhlenbeck process, several variables      38ff
Ornstein — Uhlenbeck process, solution by Fourier transform      42ff
Ornstein — Uhlenbeck process, solution for one variable      100f
Ornstein — Uhlenbeck process, spectral density matrix      43
Ornstein — Uhlenbeck process, stationary distribution      156
Ornstein — Uhlenbeck process, transition probability      156
Ornstein — Uhlenbeck process, two-dimensional      238ff
Ornstein — Uhlenbeck process, with external field      171
Orthogonality of eigenfunctions      104
Oscillator, self sustained      374
Pade approximants      214
Parabolic potential Green's function for Brownian motion      177
Parabolic potential inverse friction expansion      265f
Parabolic potential Kramers equation for inverted      245ff
Parabolic potential solution of Fokker — Planck equation for      108f
Partial differential equations application of recurrence relations      196ff
Partial differential equations reduction to vector recurrence relations      202
Partial differential equations with multiplicative harmonic time dependence      225f
Path integral solutions of the Fokker — Planck equation      74f
Pawula theorem      70f
Pendulum, Brownian motion of mathematical      280
Periodic boundary condition      102f
Periodicity of stationary solution for periodic potential      315ff
Phase between superconductors      282
Phase locked loop      283ff
Phase locked loop, order of      285
Photoelectron counting distribution connection to intensity distribution      408ff
Photoelectron counting distribution for laser light (measured), near threshold      411f
Photoelectron counting distribution for laser light, short time intervals      409
Photoelectron counting factorial moment      408f
Photoelectron counting for arbitrary time intervals      412f
Photon counting      see “Photoelectron counting”
Poisson distribution      78
Poisson process      78ff
Poisson process, special birth and death process      76
Positivity of distribution functions, one variable      75
Positivity of distribution functions, several variables      86
Positivity of eigenvalues      104f
Positivity of real part of eigenvalues      143f
Potential conditions      141
Potential conditions Ornstein — Uhlenbeck process      156f
Potential conditions, simplified form      133f
Potential effective, for low friction motion      328
Potential of Fokker — Planck equation, calculation from stationary solution      108
Potential of the one-variable Schrodinger equation      107
Potential parabolic      108f
Potential parabolic, inverted      109f
Potential periodic      see “Brownian motion”
Probability current      72
Probability current contravariant form      93
Probability current, for Kramers equation      230
Probability current, for N variables      84
Probability current, reversible and irreversible      150
probability density      14
Probability density, for a time-dependent variable      25
Probability density, for discrete variables      15
Probability density, for several time-dependent variables      30
Probability density, for several variables      19
Probability density, negative values      80
Projection operator formalism for elimination of variables      194f
Pulse-response function      165
Purely random process      27
Random process general      28
Random process purely      27
Random process several, time-dependent      30f
Random process transformation, one variable      16
Random variables      13
Ray method      see “WKB method”
Rayleigh equation      374
Recombination process      76
Recurrence relations finite tridiagonal      198
Recurrence relations for ratios      203
Recurrence relations for vectors      161
Recurrence relations for vectors eigenvalue problem      220ff
Recurrence relations for vectors initial value problem      217ff
Recurrence relations for vectors memory matrix-kernel solution      219
Recurrence relations for vectors solutions      216
Recurrence relations for vectors Taylor expansion method      220
Recurrence relations for vectors tridiagonal forms with higher derivatives      222
Recurrence relations for vectors uniqueness of solutions by continued fractions      216
Recurrence relations M nearest-neighbor coupling      200
Recurrence relations M nearest-neighbor coupling, reduction to vector      200
Recurrence relations one-sided      198 212
Recurrence relations pentadiagonal      200
Recurrence relations scalar      197ff
Recurrence relations tridiagonal applications      196ff
Recurrence relations tridiagonal solutions      196ff
Recurrence relations tridiagonal vector      199ff
Recurrence relations truncation of      204
Recurrence relations two-sided      212
Recurrence relations upiteration of      205 320f
Recursion relations for inhomogeneous tridiagonal systems      213
Reduction of number of variables      179
Reflecting wall      102f
Relaxation function      166
Relevant variables      189
Relevant variables, distribution function      191
Relevant variables, for low friction      301
Response      see also “Linear response”
Response functions after-effect-response      166
Response functions for Brownian particles in periodic potential      347ff
Response functions for constant diffusion tensor, detailed balance      168
Response functions for linear response      164ff
Response functions pulse-response      165
Response functions step-response      165f
Response linear and nonlinear      276
Response linear systems      163
Response of dipoles in external field      347
Reversible drift coefficients      149f
Reversible operators      150 231f
Reversible part of Kramers equation, solution      232
Reversible probability current      150
Rice's method      43
Riemann's curvature tensor      95
Rotating wave approximation      374f
Rotation of dipoles in a constant field      282f
Running solution      278 328
Schrodinger equation for anharmonic potential      201f
Schrodinger equation for discrete variable      198
Schrodinger equation transform of one-variable Fokker — Planck equation      107
SchrOdinger potential infinite square well      110f
SchrOdinger potential of Fokker — Planck equation      142
Semi-invariants      see “Cumulants”
Sine — Gordon equation, connection to equation of motion in periodic potential      285
Slaving principle      189
Slip of the slow variable      274
Slow variables      188f
Slow variables, distribution function      191
Slow variables, for periodic potential      279
Smoluchowski equation      see also “Brownian motion in periodic potential high 7f 96 257
Smoluchowski equation, correction terms      293f
Smoluchowski equation, derivation from Kramers equation      257
Smoluchowski equation, stationary solution      98
Spectral decomposition of matrix      155
Spectral density      30
Spectral density matrix, Ornstein — Uhlenbeck process      43
Spectral density, connection to correlation function      173f
Spectral density, connection to susceptibility      174
Spontaneous emission noise      375
Spurious drift      45 47 50
Spurious drift, several variables      55
Stationary distribution for Kramers equation      240
Stationary distribution Ornstein — Uhlenbeck process      156
Stationary process      26
Stationary solution      144
Stationary solution, of one-variable Fokker — Planck equation      98
Stationary solution, uniqueness      134ff
Statistics of laser light      374ff
Step-response function      165f
Stochastic differential equation for Brownian motion      2 7
Stochastic differential equation for Brownian with colored Gaussian noise      414ff
Stochastic force      2f
Stochastic process, classification      26
Stratonovich's definition of stochastic integrals      50ff
Streaming operator      150
Sturm — Liouville equation form of Fokker — Planck equation      106
Sum rules      171 176
Sum rules for susceptibility      176
Summation convention      see “Einstein's summation convention”
Superconductor in Josephson junction      281f
Superionic conductor model      280f
susceptibility      277
Susceptibility complex      172
Susceptibility complex, Cole — Cole plot      352
Susceptibility complex, for Brownian motion in parabolic potential      177
Susceptibility complex, for Brownian motion in periodic potential      351ff
Susceptibility complex, for Brownian motion in periodic potential, zero friction limit      355ff
Susceptibility complex, for Brownian particles in periodic potential      347ff
Susceptibility complex, for even and odd variables      175f
Susceptibility complex, for rotating dipoles      349
Symmetry relations for eigenfunctions of Kramers equation      256
Taylor expansion for solving tridiagonal recurrence relations      214
Telegrapher's equation      258
Thermal velocity      231
Time integral of Markovian variables      184ff
Time scales, slow and fast      188
Time-integrated Markovian variables, distribution      184ff
Time-integrated velocity for Brownian particle      187f
Trajectories for motion in cosine potential      329f
Transformation of variables drift and diffusion coefficients      58 91
Transformation of variables Fokker — Planck equation      88f
Transformation of variables Langevin equation      57
Transient of amplitude for laser      402ff
Transient of cumulants for laser      403 407
Transient of distribution for laser      399
Transient of intensity for laser      400 403 407
Transient of variance for laser      401
Transition probability expansion into eigenmodes for laser      387ff
Transition probability for Kramers equation      252
Transition probability for Kramers equation, with linear force      238 244
Transition probability for large times      343ff
Transition probability for large times, connection to stationary distribution      345
Transition probability for Ornstein — Uhlenbeck process      100 156
Transition probability for small times      73f
Transition probability for small times N variables      85
Transition probability for Wiener process      99
Transition probability in terms of biorthogonal set      139
Transition probability in terms of eigenfunctions      105 143
Tridiagonal recurrence relations, solution by Taylor expansion      214
Tridiagonal vector recurrence relation      see “Recurrence relations”
Truncation of recurrence relation      204
Tunneling junction, Josephson      281f
Upiteration of recurrence relation      205
Upiteration of recurrence relation for periodic potentials      320f
van der Pol equation      374
Van der Pol oscillator, including white noise      375f
Variables, reduction of the number      179
Variance      23
Variational problem for one-variable Fokker — Planck equation      120
Vector recurrence relation      see “Recurrence relation”
Velocity correlation function      33
Velocity correlation function for Brownian motion in periodic potential      350ff
Velocity correlation function for harmonic potential      177
Velocity correlation function for Kramers equation      253
Velocity correlation function for Kramers equation with linear force      245
Velocity correlation function inverse friction expansion for Kramers equation      274
Velocity distribution, stationary      36
Velocity, thermal      231
Volume element, transformation      89
White noise      3 32
Wiener process      40
Wiener process for phase of laser amplitude      381
Wiener process solution for one variable      99
Wiener — Khintchine theorem      30
WKB method, solution of Fokker — Planck equation by      162
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