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| Àâòîðèçàöèÿ |
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| Ïîèñê ïî óêàçàòåëÿì |
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| Zwanzig R. — Nonequilibrium statistical mechanics |
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| Ïðåäìåòíûé óêàçàòåëü |
Adjoint operators, Fokker — Planck equations 42—43
BGK equation, approximation to Boltzmann equation 93
BGK equation, defining local equilibrium distribution 94
BGK equation, H-theorem derivation 95—96
BGK equation, hydrodynamics 96—100
BGK equation, ratio of thermal conductivity to viscosity 100
Bloch equations, approximate equation for average spin matrices 118
Bloch equations, average magnetization 117
Bloch equations, density matrix 118
Bloch equations, derivation 115—121
Bloch equations, energy absorption 120—121
Bloch equations, memory kernel 118—119
Bloch equations, time correlation function 118—119
Brownian motion, harmonic oscillator heat bath 21—24
Brownian motion, heavy mass in harmonic lattice 24—28
Brownian motion, kinetics of first-order isomerization reaction 14—15
Brownian motion, Langevin equation for harmonic oscillator 17—18
Brownian motion, mean first passage time 77—78
Brownian motion, molecular dipole in periodic potential 18
Brownian motion, nonlinear equations for slow variables 183—184
Brownian particle, force by interaction with medium 4
Brownian particle, friction coefficient by Stokes' law 4
Brownian particle, mean squared displacement 11—12
Brownian particle, self-diffusion coefficient 12
Brownian particle, velocity correlation function 10—11
Brownian rotator, orientational time correlation function 45—47
Chemical kinetics, bimolecular reaction 64
Chemical kinetics, concentration variables 65
Chemical kinetics, Fokker — Planck equations 65
Chemical kinetics, rate equation 66
Chemical kinetics, use of master equations 64—66
Chemical reaction rates, transition state theory 67—72
Collisions, hydrodynamics 97
Collisions, rate of occurrence for BGK equation 95
Collisions, rotational diffusion 85
Continued fractions, common ways to represent 205
Continued fractions, evaluating by truncation 88
Continued fractions, expansion 91
Continued fractions, recursion relation 206—207
Convolution, Laplace transforms 204
Correlation function, spectral density 139
Cumulants, definition 189
Cumulants, equation for cumulant generating function 190
Cumulants, perturbation expansion for equation of motion for 190—192
Delta function, occurrence in Golden Rule formula 52
Delta function, use in derivation of identities 139—141
Density matrix, equilibrium 103
Density matrix, partitioning into diagonal and off-diagonal parts 125
Density matrix, quantum Liouville equation 106 133
Density matrix, two-level system in heat bath 110—111 116—117
Dephasing, Hamiltonian as elementary model 110
Differential equations, first-order linear 198—200
Differential equations, second-order 199
Diffusion see also "Energy diffusion"
Diffusion, one-dimensional 8—9
Diffusion, solution of ordinary equation in angle space 88
Diffusion, three-dimensional 10
Diffusion, transport equation 207
Diffusion, velocity correlation function 8—10
Dipole-dipole correlation function, absorption coefficient proportional to spectral density of 12
Dipole-dipole correlation function, Langevin equation for rotational Brownian motion 13
Dipole-dipole correlation function, orientational time correlation function 13—14
Dipole-dipole correlation function, two-level system in heat bath 111
Dynamic linear response, classical mechanics 130—132
Dynamic linear response, density matrix 133
Dynamic linear response, frequency dependent response 135—136
Dynamic linear response, Liouville equation 130—131 133—134
Dynamic linear response, phase space distribution function 133
Dynamic linear response, Poisson bracket 130—131
Dynamic linear response, quantum commutator 133
Dynamic linear response, quantum mechanics 133—135
Dynamic linear response, time-dependent analog of static susceptibility 132
Dynamic linear response, total response a sum of individual responses 132
Electron transfer kinetics, heat bath Hamiltonian 107—108
Electron transfer kinetics, Marcus's reorganization energy 110
Electron transfer kinetics, polarization of environment 106—107
Electron transfer kinetics, quantum analog of Kramers problem 106
Electron transfer kinetics, rate of energy dissipation 139
Electron transfer kinetics, rate of transition between states by Golden Rule formula 108
Electron transfer kinetics, spin-boson Hamiltonian 108
Energy absorption rate, optical absorption coefficient 53—55
Energy diffusion, energy diffusion coefficient 80
Energy diffusion, energy diffusion equation 81
Energy diffusion, escape rate as function of friction 82f
Energy diffusion, mean first passage time 82
Energy diffusion, rate of escape from potential well over barrier 81
First passage times, adjoint equation 75—76
First passage times, derivation 74—76
First passage times, distribution 74
First passage times, Kramers problem 76—78
First passage times, mean 75
First-order linear differential equations 198—200
Fluctuation-dissipation theorem, analog of, for Langevin equation 16—17
Fluctuation-dissipation theorem, arbitrary distinction between systematic behavior and noise 24
Fluctuation-dissipation theorem, balance between friction and noise 6—7
Fluctuation-dissipation theorem, correlation function of noise 161
Fluctuation-dissipation theorem, harmonic oscillator example 23
Fluctuation-dissipation theorem, non-Markovian version 20
Fluctuation-dissipation theorem, non-Markovian version in matrix form 21
Fluctuation-dissipation theorem, requiring symmetry and antisymmetry 17
Fluctuation-dissipation theorem, steady-state solution of Fokker — Planck equation 39
Fluctuation-dissipation theorem, two-variable Brownian motion of particle moving in potential 39
Fluctuation-renormalization bare transport coefficient 192
Fluctuation-renormalization bare transport coefficient, change in memory functions 160—161
Fluctuation-renormalization bare transport coefficient, nonlinear Langevin equations 188
Fokker — Planck equations, averages and adjoint operators 42—43
Fokker — Planck equations, choice of initial conditions 197
Fokker — Planck equations, derivation 36—39
Fokker — Planck equations, derivation of nonlinear 177—180
Fokker — Planck equations, Green's function in linear case 43—44 187—188
Fokker — Planck equations, Heisenberg approach 42—43
Fokker — Planck equations, long time steady-state solution of arbitrary 39
Fokker — Planck equations, noise-averaged distribution function 38 175
Fokker — Planck equations, nonlinear 180—181
Fokker — Planck equations, properties 41—42
Fokker — Planck equations, Schrodinger approach 42
Fokker — Planck equations, slow variables 183 187
Fokker — Planck equations, Smoluchowski equation 40—41
Fokker — Planck equations, substitution of Smoluchowski equation leading to Schrodinger-like equation 41
Fokker — Planck equations, two-variable Brownian motion of particle moving in potential 39—40
Frequency dependent magnetic susceptibility, linear response theory 137—138
Frequency dependent response, linear response in quantum mechanics 135—136
Gaussian random variables, application to Gaussian white noise 202—203
Gaussian random variables, distribution function 201
Gaussian random variables, linear combination of 202
Gaussian random variables, mean values and mean squared fluctuations 201
Gaussian random variables, moment generating function 200—201
Gaussian random variables, noise 23
Gaussian random variables, probability distribution 200
Gaussian random variables, properties of multivariate 201—202
Golden Rule, definition 48
Golden Rule, derivation 48—51
Golden Rule, energy absorption equation 142
Golden Rule, flaws of standard treatment 52—53
Golden Rule, short time behavior 53
Golden Rule, transition states in Pauli master equation 124
Golden Rule, uses in electron transfer kinetics 108—109
Gram — Schmidt process, orthonormalizing vectors in Hilbert space 146
Harmonic lattice, instructive model of Brownian motion 24
Harmonic lattice, Laplace transforms for solving equations of motion 25
Harmonic lattice, recurrence paradox 195—196
Harmonic lattice, reversibility paradox 195
Harmonic lattice, transform of normalized velocity correlation function 26
Harmonic lattice, velocity correlation function 24—25 27—28
Harmonic oscillator, Brownian motion in, heat bath 21—24
Harmonic oscillator, heat bath master equation 60—61
Harmonic oscillator, Langevin equation 22
Harmonic oscillator, Langevin equation for Brownian motion 17—18
Harmonic oscillator, memory function 22—23
Harmonic oscillator, noise 23 24
| Heat bath see also "Two-level system"
Heat bath master equation, application 59—61
Heat bath master equation, derivation 126
Heat bath master equation, Golden Rule transition rates 58
Heat bath master equation, harmonic oscillator 60—61
Heat bath master equation, microcanonical character 59
Heat bath master equation, principle of detailed balance 59
Heat bath, Bloch equations, two-level system 115—121
Heat bath, Brownian motion in harmonic oscillator 21—24
Heat bath, dephasing, two-level system 110—115
Heisenberg equations of motion, electron transfer kinetics 107
Hilbert space, basis of mode-coupling theory 170
Hilbert space, expansion of velocity correlation function 171
Hilbert space, matrix form of Liouville equation 144—146
Hilbert space, projection operators 143—149
Hilbert space, quantum mechanics 101
Hilbert space, subspace of relevant variables 148—149
Inverting Laplace transforms 204
Ion mobility, linear response theory 136—137
Isomerization reaction, kinetics of first-order 14—15
Kinetic models, BGK equation and H-theorem 93—96
Kinetic models, BGK equation and hydrodynamics 96—100
Kinetic models, Boltzmann equation 83—84
Kinetic models, collision integral 84
Kinetic models, orientational time correlation function 89—92
Kinetic models, rotational diffusion 84—88
Kinetic models, rotational relaxation 89—92
Kramers problem, Brownian particle escape from potential well 73 76
Kramers problem, electron transfer reactions as quantum analog of 106
Kramers problem, energy diffusion concept 78—82
Kramers problem, mean first passage time 77—78
Kramers problem, relation to transition state theory rate 78
Kramers — Kronig relation, frequency dependent response 136
Kubo transform, linear response in quantum mechanics 135
Kubo transform, noise in generalized Langevin equation 155
Langevin equations, Brownian motion of harmonic oscillator 17—18
Langevin equations, choice of initial conditions 197
Langevin equations, derivation for Brownian motion of system with harmonic oscillator heat bath 21—24
Langevin equations, derivation of nonlinear 177—180
Langevin equations, general treatment 15—16
Langevin equations, generalized derivation 149—151
Langevin equations, generalized derivation, deriving Fokker — Planck equation 177
Langevin equations, generalized derivation, eliminating projection operators 158—159
Langevin equations, generalized derivation, identities 157—160
Langevin equations, generalized derivation, initial nonequilibrium states 151—155
Langevin equations, generalized derivation, linear, for slow variables 165—168
Langevin equations, generalized derivation, Mori's procedure 161—162
Langevin equations, generalized derivation, noise 151—157
Langevin equations, generalized derivation, non-Markovian fluctuation-dissipation theorem 157
Langevin equations, generalized derivation, nonlinear to linear example 160—165
Langevin equations, generalized derivation, procedure constructing initial nonequilibrium distribution 154—155
Langevin equations, Markovian and non-Markovian 19—21
Langevin equations, nonlinear 18—19 180
Laplace transforms, continued fractions 206
Laplace transforms, convolution 204
Laplace transforms, definition 203
Laplace transforms, functions 204
Laplace transforms, handling time derivatives 87
Laplace transforms, integral 203—204
Laplace transforms, inverting 204
Laplace transforms, orientational time correlation function, numerical inversion 92f
Laplace transforms, solving equations of motion of harmonic lattice 25
Linear differential equations, first-order 198—200
Linear response theory, applications 136—139
Linear response theory, initial ensemble 197
Linear response theory, streaming velocity 186
Linear response, applications of theory 136—139
Linear response, determining equilibrium 127—129
Linear response, dynamic 130—136
Linear response, dynamic in classical mechanics 130—132
Linear response, dynamic in quantum mechanics 133—135
Linear response, energy absorption 141—142
Linear response, energy absorption in electric field 138—139
Linear response, frequency dependent magnetic susceptibility 137—138
Linear response, frequency dependent response 135—136
Linear response, identities 139—142
Linear response, mobility of ion 136—137
Linear response, quantum mechanical version 129—130
Linear response, quantum perturbation theory 130
Linear response, static 127—130
Liouville equation, dynamical variables 33—35
Liouville equation, evolution of dynamical variable 34
Liouville equation, formal operator solution 32
Liouville equation, Liouville operator 32—33
Liouville equation, matrix form 144—146
Liouville equation, partitioning 146—147
Long time tails, deriving, stress correlation function 173
Long time tails, mode-coupling theory 169—170
Magnetic susceptibility, linear response theory 137—138
Master equations, abstract 61—63
Master equations, chemical kinetics 64—66
Master equations, derivation of quantum mechanical or Pauli 121—124
Master equations, Golden Rule transition rates of heat bath 58
Master equations, harmonic oscillator illustration of heat bath 59—61
Master equations, heat bath 57—59
Master equations, matrix or operator equation 62
Master equations, Pauli 56—57
Master equations, random walks 63—64
Master equations, use of operator methods for averages 62
Memory kernel, derivation of nonlinear equations 179—180
Memory kernel, derivation of Pauli master equation 123 126
Memory kernel, eliminating projection operators 158—159
Memory kernel, linear Langevin equations for slow variables 165
Memory kernel, non-Markovian fluctuation-dissipation theorem 157
Memory kernel, two-level system in heat bath 118—119
Mobility of ion, linear response theory 136—137
Mode-coupling theory, deriving long time tail of stress correlation function 173
Mode-coupling theory, Hilbert space picture of dynamics 170
Mode-coupling theory, long time tails 169—170
Mode-coupling theory, product of two slow variables 171
Mode-coupling theory, self-diffusion example 170—173
Mori Langevin equation, differences between, and exact Langevin 160
Mori Langevin equation, memory function 164—165
Mori Langevin equation, noise 165
Mori Langevin equation, nonlinear to linear example 160—165
Mori Langevin equation, slow variables 166
Nernst — Planck equation, treatment of electrolyte solutions 160
Noise, application of Gaussian random variables to white 202—203
Noise, arbitrary distinction from systematic behavior 24
Noise, averaged, in Langevin equation 155—156
Noise, generalized Langevin equations 151—157
Noise, nonlinear Langevin equation 184—185
Nonlinear Langevin and Fokker — Planck equations, derivation 177—181
Nonlinear Langevin and Fokker — Planck equations, illustration of nonlinear system interacting with harmonic oscillator heat bath 183—184
Nonlinear Langevin and Fokker — Planck equations, memory kernel 179—180
Nonlinear Langevin and Fokker — Planck equations, noise and initial states 184—185
Nonlinear Langevin and Fokker — Planck equations, reduced distribution functions 175—177
Nonlinear Langevin and Fokker — Planck equations, slow variables 181—183
Optical absorption coefficient, classical time correlation function of total electric dipole moment 56
Optical absorption coefficient, focus on rate of energy absorption 53—55
Optical absorption coefficient, frequency dependence 12
Optical absorption coefficient, frequency dependence by time correlation function 53
Optical absorption coefficient, theory of optical absorption 55—56
Orientational time correlation function, approaching ideal rotator limit 91
Orientational time correlation function, deriving exact expression 89—92
Orientational time correlation function, exponential decay 92
Orientational time correlation function, Laplace transform of correlation function 90—91
Orientational time correlation function, results of numerical inversion of Laplace transform 92f
Pauli master equation, derivation 121—124
Pauli master equation, Golden Rule 124
Pauli master equation, memory kernels 123 126
Pauli master equation, microcanonical character 57
Pauli master equation, projection operator method 124—126
Probability distribution, Gaussian random variables 200
Projection operator method, deriving quantum mechanical master equation 124—126
Projection operator method, use in deriving Langevin equations 143—144
Projection operators, derivation of generalized Langevin equations 149—151
Projection operators, deriving Fokker — Planck equation 177
Projection operators, deriving Langevin equations 178
Projection operators, eliminating projection 158—159
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