The Langevin equation: with applications to stochastic problems in physics, chemistry and electrical engineering
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| Format: | Book |
| Language: | English |
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Singapore
World Scientific
2012
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| Edition: | 3. ed. |
| Series: | World scientific series in contemporary chemical physics
27 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXII, 827 S. graph. Darst. |
| ISBN: | 9789814355667 |
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| 245 | 1 | 0 | |a The Langevin equation |b with applications to stochastic problems in physics, chemistry and electrical engineering |c William T. Coffey ; Yuri P. Kalmykov |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a Singapore |b World Scientific |c 2012 | |
| 300 | |a XXII, 827 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a World scientific series in contemporary chemical physics |v 27 | |
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| _version_ | 1804148574470012928 |
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| adam_text | Titel: The Langevin equation
Autor: Coffey, William
Jahr: 2012
CONTENTS
Preface to the Third Edition...........................................................................................vii
Contents............................................................................................................................xv
Chapter 1 Historical Background and Introductory Concepts.................................1
1.1. Brownian motion..................................................................................................1
1.2. Einstein s explanation of Brownian movement...................................................5
1.3. The Langevin equation.......................................................................................10
1.3.1. Calculation of Avogadro s number.......................................................14
1.4. Einstein s Method..............................................................................................16
1.5. Essential concepts in Statistical Mechanics.......................................................21
1.5.1. Ensemble of systems............................................................................ 23
1.5.2. Phase space...........................................................................................23
1.5.3. Representative point..............................................................................24
1.5.4. Ergodic hypothesis................................................................................24
1.5.5. Calculation of averages.........................................................................25
1.5.6. Liouville equation..................................................................................27
1.5.7. Reduction of the Liouville equation.................................................. 28
1.5.8. Langevin equation for a system with one degree of freedom...............29
1.5.9. Intuitive derivation of the Klein-Kramers equation.............................30
1.5.10. Conditions under which a Maxwellian distribution in the
velocities may be deemed to be attained...............................................32
1.5.11. Very-high-damping (VHD) regime......................................................33
1.5.12. Very-low-damping (VLD) regime........................................................38
1.6. Probability theory...............................................................................................42
1.6.1. Random variables and probability distributions....................................43
1.6.2. The Gaussian distribution......................................................................45
1.6.3. Moment-generating functions...............................................................48
1.6.4. Central limit theorem.............................................................................52
1.6.5. Random processes.................................................................................53
1.6.6. Wiener-Khinchin theorem....................................................................55
1.7. Application to the Langevin equation................................................................56
1.8. Wiener process...................................................................................................58
1.8.1. Variance of the Wiener process.............................................................59
1.8.2. Wiener integrals....................................................................................61
1.9. The Fokker-Planck equation.............................................................................63
1.10. Drift and diffusion coefficients..........................................................................69
1.11. Solution of the one-dimensional Fokker-Planck equation................................72
1.12. The Smoluchowski equation..............................................................................75
1.13. Escape of particles over potential barriers: Kramers theory.............................77
1.13.1. Escape rate in the IHD limit..................................................................83
1.13.2. Kramers calculation of the escape rate in the VLD limit.....................87
1.13.3. Range of validity of the IHD and VLD formulas..................................90
1.13.4. Extension of Kramers theory to many dimensions in the
IHD limit...............................................................................................92
1.13.5. Langer s treatment of the IHD limit......................................................94
1.13.6. Kramers formula as a special case of Langer s formula......................99
1.13.7. Kramers turnover problem.................................................................101
1.14. Applications of the theory of Brownian movement in a potential...................112
1.15. Rotational Brownian motion: application to dielectric relaxation...................114
1.15.1. Breakdown of the Debye theory at high frequencies..........................118
1.16. Superparamagnetism: magnetic after-effect....................................................121
1.17. Brown s treatment of Neel relaxation..............................................................128
1.18. Asymptotic expressions for the Neel relaxation time......................................132
1.18.1. Magnetization reversal time in a uniaxial superparamagnet:
application of Kramers method..........................................................132
1.18.2. Escape rate formulas for superparamagnets........................................136
1.19. Ferrofluids........................................................................................................141
1.20. Depletion effect in a biased bistable potential.................................................143
1.21. Stochastic resonance........................................................................................149
1.22. Anomalous diffusion........................................................................................151
1.22.1. Empirical formulas for the complex dielectric permittivity................156
1.22.2. Theoretical justification for anomalous relaxation behavior...............157
1.22.3. Anomalous dielectric relaxation of an assembly of dipolar
molecules.............................................................................................160
References.................................................................................................................163
Chapter 2 Langevin Equations and Methods of Solution.......................................171
2.1. Criticisms of the Langevin equation................................................................171
2.2. Doob s interpretation of the Langevin equation..............................................172
2.3. Nonlinear Langevin equation with a multiplicative noise term: Ito and
Stratonovich rules............................................................................................174
2.4. Derivation of differential-recurrence relations from the one-dimensional
Langevin equation............................................................................................178
2.5. Nonlinear Langevin equation in several dimensions.......................................180
2.6. Average of the multiplicative noise term in the Langevin equation............... 183
2.6.1. Multiplicative noise term for rotation in space...................................185
2.6.2. Multiplicative noise terms in the non-inertial limit.............................186
2.6.3. Explicit average of the noise-induced terms for a planar rotator........188
2.7. Methods of solution of differential-recurrence relations arising from the
nonlinear Langevin equation........................................................................... 190
2.7.1. Matrix method.................................................................................... 191
2.7.2. Initial conditions..................................................................................193
2.7.3. Matrix continued-fraction solution of recurrence equations...............195
2.8. Linear response theory....................................................................................201
2.9. Integral relaxation time....................................................................................206
2.10. Linear response theory for systems with dynamics governed by single-
variable Fokker-Planck equations....................................................................208
2.11. Smallest non-vanishing eigenvalue: continued-fraction approach..................212
2.11.1 Evaluation of k from a scalar three-term recurrence relation............213
2.11.2 Evaluation of k from a matrix three-term recurrence relation.......... 216
2.12. Effective relaxation time..................................................................................218
2.13. Evaluation of the dynamic susceptibility using rmX, ref, and A]...................220
2.14. Nonlinear transient response of a Brownian particle......................................222
References.................................................................................................................225
Chapter 3 Brownian Motion of a Free Particle and a Harmonic Oscillator........229
3.1. Introduction......................................................................................................229
3.2. Ornstein-Uhlenbeck theory of Brownian motion............................................230
3.3. Stationary solution of the Langevin equation: the Wiener-Khinchin
theorem............................................................................................................231
3.4. Application to phase diffusion in MRI............................................................235
3.5. Brownian motion of a harmonic oscillator......................................................241
3.6. Rotational Brownian motion of a fixed-axis rotator........................................243
3.7. Torsional oscillator model: example of the use of the Wiener integral...........246
References................................................................................................................250
Chapter 4 Rotational Brownian Motion About a Fixed Axis in iV-Fold
Cosine Potentials......................................................................................253
4.1. Introduction......................................................................................................253
4.2. Langevin equation for rotation about a fixed axis...........................................254
4.3. Longitudinal and transverse effective relaxation times...................................257
4.4. Polarizabilities and relaxation times of a fixed-axis rotator with two
equivalent sites.................................................................................................261
4.4.1. Matrix solution for the longitudinal response.....................................263
4.4.2. Continued-fraction solution for the longitudinal response..................265
4.4.3. Comparison of the longitudinal relaxation time with the Kramers
theory...................................................................................................271
4.4.4. Continued-fraction solution for the transverse response.....................272
4.5. Effect of a d.c. bias field on the orientational relaxation of a fixed-axis
rotator with two equivalent sites......................................................................277
4.5.1. Longitudinal response.........................................................................277
4.5.2. Transverse response............................................................................280
4.5.3. Relaxation times..................................................................................281
References.................................................................................................................283
Chapter 5 Brownian Motion in a Tilted Periodic Potential: Application
to the Josephson Tunnelling Junction...................................................285
5.1. Introduction......................................................................................................285
5.2. Langevin equations..........................................................................................286
5.3. Josephson junction: dynamic model................................................................287
5.4. Reduction of the averaged Langevin equation for the junction to a set of
differential-recurrence relations......................................................................290
5.5. Current-voltage characteristics.......................................................................293
5.6. Linear response to an applied alternating current............................................298
5.7. Effective eigenvalues for the Josephson junction............................................300
5.8. Linear impedance.............................................................................................303
5.9. Spectrum of the Josephson radiation...............................................................307
5.10. Nonlinear effects in d.c. and a.c. current-voltage characteristics....................310
5.11. Concluding remarks.........................................................................................317
References.................................................................................................................318
Chapter 6 Translational Brownian Motion in a Double-Well Potential..............321
6.1. Introduction......................................................................................................321
6.2. Characteristic times of the position correlation function.................................323
6.3. Converging continued fractions for the correlation functions.........................332
6.4. Two-mode approximation................................................................................336
6.5. Stochastic resonance........................................................................................338
6.6. Concluding remarks.........................................................................................340
References.................................................................................................................340
Chapter 7 Non-inertial Rotational Diffusion in Axially Symmetric
External Potentials: Applications to Orientational Relaxation
of Molecules in Fluids and Liquid Crystals..........................................343
7.1. Introduction......................................................................................................343
7.2. Rotational diffusion in a potential: Langevin equation approach....................344
7.2.1. Averaging the non-inertial Euler-Langevin equation.........................346
7.2.2. Differential-recurrence equations for spherical harmonics.................348
7.2.3. Examples of differential-recurrence equations...................................350
7.3. Brownian rotation in a uniaxial potential........................................................352
7.3.1. Longitudinal relaxation.......................................................................355
7.3.2. Susceptibility and relaxation times: continued-fraction solution........358
7.3.3. Integral form and asymptotic expansions of the exact solution..........363
7.3.4. Transverse response: continued-fraction solution...............................365
7.3.5. Complex susceptibilities.....................................................................368
7.4. Brownian rotation in a uniform d.c. external field......................................... 372
7.4.1. Longitudinal response: continued-fraction solution............................374
7.4.2. Transverse response: continued-fraction solution...............................377
7.4.3. Comparison with experimental data....................................................380
7.5. Nonlinear transient responses in dielectric and Kerr-effect relaxation...........382
7.5.1. Nonlinear transient dielectric and Kerr-effect relaxation times..........384
7.5.2. Nonlinear step-on transient response: matrix continued-fraction
solution...............................................................................................387
7.6. Nonlinear dielectric relaxation of polar molecules in a strong a.c. electric
field: steady-state response.............................................................................392
7.7. Concluding remarks.........................................................................................397
References.................................................................................................................398
Chapter 8 Anisotropic Non-inertial Rotational Diffusion in an
External Potential: Application to Linear and Nonlinear
Dielectric Relaxation and the Dynamic Kerr Effect............................403
8.1. Introduction...................................................................................................403
8.2. Anisotropic non-inertial rotational diffusion of an asymmetric top in an
external potential..............................................................................................404
8.2.1. Euler-Langevin equation in the non-inertial limit..............................404
8.2.2. Differential-recurrence equation for Wigner s D functions................408
8.3. Application to dielectric relaxation..................................................................413
8.3.1. Linear dielectric response of an assembly of asymmetric tops...........413
8.3.2. Nonlinear response in superimposed a.c. and strong d.c. bias fields:
perturbation solution...........................................................................415
8.3.3. Rotational Brownian motion and dielectric relaxation in nematic
liquid crystals......................................................................................418
8.4. Kerr-effect relaxation......................................................................................429
8.4.1. Basic relations.....................................................................................431
8.4.2. Dynamic Kerr effect: matrix formulation...........................................432
8.4.3. Nonlinear dielectric and Kerr-effect transients in strong fields..........447
8.5. Concluding remarks........................................................................................ 453
References................................................................................................................454
Chapter 9 Brownian Motion of Classical Spins: Application to
Magnetization Relaxation in Superparamagnets.................................459
9.1. Introduction......................................................................................................459
9.2. Brown s model: Langevin equation approach.................................................462
9.2.1. Derivation of the differential-recurrence equation for the
statistical moments from the stochastic Gilbert equation...................464
9.2.2. Fokker-Planck equation approach......................................................467
9.2.3. Differential-recurrence equations for a uniaxial superparamagnet
in an external magnetic field...............................................................469
9.2.4. Comparison of the Gilbert, Landau-Lifshitz, and Kubo
kinetic models of the Brownian rotation of classical spins................ 470
9.2.5. Calculation of the observables............................................................473
9.3. Magnetization relaxation in uniaxial superparamagnets..................................477
9.3.1. Longitudinal relaxation.......................................................................479
9.3.2. Characteristic times and magnetic susceptibility................................483
9.3.3. Magnetic stochastic resonance............................................................488
9.3.4. Dynamic magnetic hysteresis..............................................................493
9.3.5. Transverse response: ferromagnetic resonance...................................503
9.4. Reversal time of the magnetization in superparamagnets with nonaxially
symmetric potentials: escape-rate theory approach.........................................509
9.4.1. Basic equations for the escape rate.....................................................512
9.4.2. Discrete orientation model..................................................................515
9.4.3. Reversal time for cubic anisotropy.....................................................520
9.4.4. Uniaxial superparamagnet in a uniform d.c. magnetic field...............523
9.4.5. Biaxial superparamagnet in a uniform d.c. magnetic field applied
along the easy axis..............................................................................530
9.4.6. Mixed anisotropy: breakdown of the paraboloid approximation.......532
9.4.7. Reversal time of a superantiferromagnetic nanoparticle.................... 542
9.4.8. Switching-field curves and surfaces....................................................546
9.5. Magnetization relaxation in superparamagnets with non-axially symmetric
anisotropy: matrix continued-fraction approach..............................................553
9.5.1. Uniaxial superparamagnet in an oblique field.....................................553
9.5.2. Cubic anisotropy.................................................................................562
9.6. Nonlinear a.c. stationary response of superparamagnets.................................574
9.7. Concluding remarks.........................................................................................581
References.................................................................................................................585
Chapter 10 Inertial Effects in Rotational and Translational Brownian Motion
for a Single Degree of Freedom............................................................. 595
10.1. Introduction......................................................................................................595
10.2. Inertial effects in nonlinear dielectric response...............................................601
10.2.1. Linear response for non-inertial rotation about a fixed axis...............603
10.2.2. Differential-recurrence equations for nonlinear transients..................605
10.2.3. Matrix continued-fraction solution.....................................................608
10.2.4. Analytic equations for the integral relaxation time.............................610
10.2.5. Inertial effects.....................................................................................612
10.3. Brownian motion of a fixed-axis rotator in a double-well potential...............616
10.3.1. Matrix continued-fraction solution.....................................................617
10.3.2. Turnover formula for the escape rate..................................................619
10.3.3. Analytic formulas for the longitudinal correlation time.....................625
10.4. Brownian motion of a fixed-axis rotator in an asymmetric double-well
potential........................................................................................................... 629
10.4.1. Basic equations................................................................................... 630
10.4.2. Matrix continued-fraction solution.....................................................633
10.4.3. Turnover formula for y1_1 and VHD and VLD asymptotes for rµ.
Complex susceptibility....................................................................... 634
10.5. Brownian motion in a tilted periodic potential................................................642
10.5.1. Matrix continued-fraction solution.....................................................644
10.5.2. Turnover equation for the decay rate..................................................648
10.6. Translational Brownian motion in a double-well potential.............................652
10.6.1. Matrix continued-fraction solution.....................................................654
10.6.2. Longest and integral relaxation times. Dynamic susceptibility..........657
10.7. Concluding remarks.........................................................................................663
References.................................................................................................................664
Chapter 11 Inertial Effects in Rotational Diffusion in Space: Application to
Orientational Relaxation in Molecular Liquids and Ferrofluids....... 669
11.1. Introduction......................................................................................................669
11.2. Inertial rotational Brownian motion of a thin rod in space..............................671
11.2.1. Recurrence equations for the correlation functions.............................671
11.2.2. First-rank correlation function............................................................676
11.2.3. Second-rank correlation function........................................................678
11.2.4. Orientational correlation function of an arbitrary rank L.....................679
11.3. Rotational Brownian motion of a symmetrical top..........................................682
11.3.1. Recurrence equations for the correlation functions............................682
11.3.2. First- and second-rank correlation functions......................................684
11.4. Inertial rotational Brownian motion of a rigid dipolar rotator in a uniaxial
biased potential................................................................................................691
11.4.1. Basic relations.....................................................................................691
11.4.2. Asymptotic formulas for the longest relaxation time......................... 698
11.5. Itinerant oscillator model of rotational motion in liquids................................703
11.5.1. Generalization of the Onsager model: relation to the cage model......704
11.5.2. Dipole correlation function.................................................................710
11.5.3. Matrix continued-fraction solution for the complex susceptibility.... 712
11.5.4. Comparison with experimental data................................................... 715
11.6. Application of the cage model to ferrofluids.............................................718
Appendix A: Statistical averages of the Hermite polynomials of the components
of the angular velocity for linear molecules........................................729
Appendix B: Averages of the components of the angular velocities........................730
Appendix C: Sack s continued-fraction solution for the sphere...............................733
References.................................................................................................................734
Chapter 12 Anomalous Diffusion and Relaxation....................................................739
12.1. Discrete-and continuous-time random walks.................................................739
12.2. Fractional diffusion equation for the continuous-time random
walk model...................................................................................................... 745
12.3. Solution of fractional diffusion equations.......................................................754
12.3.1. Anomalous diffusion of a fixed-axis rotator in a potential..................759
12.3.2. Fractional diffusion of a particle in a double-well potential...............762
12.4. Characteristic times of anomalous diffusion...................................................766
12.4.1. Mean first-passage time for normal diffusion.....................................766
12.4.2. First-passage-time distribution for anomalous diffusion....................770
12.4.3. Spectral definition of the characteristic times.....................................774
12.5. Inertial effects in anomalous relaxation...........................................................777
12.5.1. Slow transport process governed by trapping.....................................777
12.5.2. Calculation of the complex susceptibility...........................................779
12.5.3. Comment on the use of the telegraph equation...................................784
12.6. Barkai and Silbey s fractional kinetic equation...............................................786
12.6.1. Complex susceptibility........................................................................789
12.6.2. Fractional kinetic equation for the needle model................................792
12.7. Anomalous diffusion in a periodic potential...................................................797
12.8. Fractional Langevin equation..........................................................................803
12.8.1. Application to phase diffusion in MRI................................................806
12.9. Concluding remarks.........................................................................................811
Appendix: Fractal dimension, anomalous exponents, and random walks................814
References.................................................................................................................816
Index...............................................................................................................................821
|
| any_adam_object | 1 |
| author | Coffey, William T. |
| author_facet | Coffey, William T. |
| author_role | aut |
| author_sort | Coffey, William T. |
| author_variant | w t c wt wtc |
| building | Verbundindex |
| bvnumber | BV039701142 |
| classification_tum | PHY 064f PHY 015f PHY 013f |
| ctrlnum | (OCoLC)724665109 (DE-599)BVBBV039701142 |
| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| edition | 3. ed. |
| format | Book |
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| id | DE-604.BV039701142 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:09:18Z |
| institution | BVB |
| isbn | 9789814355667 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024549692 |
| oclc_num | 724665109 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-29T DE-11 DE-20 |
| owner_facet | DE-91G DE-BY-TUM DE-29T DE-11 DE-20 |
| physical | XXII, 827 S. graph. Darst. |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | World Scientific |
| record_format | marc |
| series | World scientific series in contemporary chemical physics |
| series2 | World scientific series in contemporary chemical physics |
| spelling | Coffey, William T. Verfasser aut The Langevin equation with applications to stochastic problems in physics, chemistry and electrical engineering William T. Coffey ; Yuri P. Kalmykov 3. ed. Singapore World Scientific 2012 XXII, 827 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier World scientific series in contemporary chemical physics 27 Langevin-Gleichung (DE-588)4166699-9 gnd rswk-swf Brownsche Bewegung (DE-588)4128328-4 gnd rswk-swf Langevin-Gleichung (DE-588)4166699-9 s DE-604 Brownsche Bewegung (DE-588)4128328-4 s 1\p DE-604 Kalmykov, Jurij P. Sonstige oth World scientific series in contemporary chemical physics 27 (DE-604)BV009659296 27 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024549692&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Coffey, William T. The Langevin equation with applications to stochastic problems in physics, chemistry and electrical engineering World scientific series in contemporary chemical physics Langevin-Gleichung (DE-588)4166699-9 gnd Brownsche Bewegung (DE-588)4128328-4 gnd |
| subject_GND | (DE-588)4166699-9 (DE-588)4128328-4 |
| title | The Langevin equation with applications to stochastic problems in physics, chemistry and electrical engineering |
| title_auth | The Langevin equation with applications to stochastic problems in physics, chemistry and electrical engineering |
| title_exact_search | The Langevin equation with applications to stochastic problems in physics, chemistry and electrical engineering |
| title_full | The Langevin equation with applications to stochastic problems in physics, chemistry and electrical engineering William T. Coffey ; Yuri P. Kalmykov |
| title_fullStr | The Langevin equation with applications to stochastic problems in physics, chemistry and electrical engineering William T. Coffey ; Yuri P. Kalmykov |
| title_full_unstemmed | The Langevin equation with applications to stochastic problems in physics, chemistry and electrical engineering William T. Coffey ; Yuri P. Kalmykov |
| title_short | The Langevin equation |
| title_sort | the langevin equation with applications to stochastic problems in physics chemistry and electrical engineering |
| title_sub | with applications to stochastic problems in physics, chemistry and electrical engineering |
| topic | Langevin-Gleichung (DE-588)4166699-9 gnd Brownsche Bewegung (DE-588)4128328-4 gnd |
| topic_facet | Langevin-Gleichung Brownsche Bewegung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024549692&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009659296 |
| work_keys_str_mv | AT coffeywilliamt thelangevinequationwithapplicationstostochasticproblemsinphysicschemistryandelectricalengineering AT kalmykovjurijp thelangevinequationwithapplicationstostochasticproblemsinphysicschemistryandelectricalengineering |