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Zwanzig R. Ч Nonequilibrium statistical mechanics
Zwanzig R. Ч Nonequilibrium statistical mechanics

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Ќазвание: Nonequilibrium statistical mechanics

јвтор: Zwanzig R.


This is a presentation of the main ideas and methods of modern nonequilibrium statistical mechanics. It is the perfect introduction for anyone in chemistry or physics who needs an update or background in this time-dependent field. Topics covered include fluctuation-dissipation theorem; linear response theory; time correlation functions, and projection operators. Theoretical models are illustrated by real-world examples and numerous applications such as chemical reaction rates and spectral line shapes are covered. The mathematical treatments are detailed and easily understandable and the appendices include useful mathematical methods like the Laplace transforms, Gaussian random variables and phenomenological transport equations.

язык: en

–убрика: ‘изика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 2001

 оличество страниц: 233

ƒобавлена в каталог: 31.10.2010

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
ѕредметный указатель
Adjoint operators, Fokker Ч Planck equations      42Ч43
BGK equation, approximation to Boltzmann equation      93
BGK equation, defining local equilibrium distribution      94
BGK equation, H-theorem derivation      95Ч96
BGK equation, hydrodynamics      96Ч100
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      115Ч121
Bloch equations, energy absorption      120Ч121
Bloch equations, memory kernel      118Ч119
Bloch equations, time correlation function      118Ч119
Brownian motion, harmonic oscillator heat bath      21Ч24
Brownian motion, heavy mass in harmonic lattice      24Ч28
Brownian motion, kinetics of first-order isomerization reaction      14Ч15
Brownian motion, Langevin equation for harmonic oscillator      17Ч18
Brownian motion, mean first passage time      77Ч78
Brownian motion, molecular dipole in periodic potential      18
Brownian motion, nonlinear equations for slow variables      183Ч184
Brownian particle, force by interaction with medium      4
Brownian particle, friction coefficient by Stokes' law      4
Brownian particle, mean squared displacement      11Ч12
Brownian particle, self-diffusion coefficient      12
Brownian particle, velocity correlation function      10Ч11
Brownian rotator, orientational time correlation function      45Ч47
Chemical kinetics, bimolecular reaction      64
Chemical kinetics, concentration variables      65
Chemical kinetics, Fokker Ч Planck equations      65
Chemical kinetics, rate equation      66
Chemical kinetics, use of master equations      64Ч66
Chemical reaction rates, transition state theory      67Ч72
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      206Ч207
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      190Ч192
Delta function, occurrence in Golden Rule formula      52
Delta function, use in derivation of identities      139Ч141
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      110Ч111 116Ч117
Dephasing, Hamiltonian as elementary model      110
Differential equations, first-order linear      198Ч200
Differential equations, second-order      199
Diffusion      see also "Energy diffusion"
Diffusion, one-dimensional      8Ч9
Diffusion, solution of ordinary equation in angle space      88
Diffusion, three-dimensional      10
Diffusion, transport equation      207
Diffusion, velocity correlation function      8Ч10
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      13Ч14
Dipole-dipole correlation function, two-level system in heat bath      111
Dynamic linear response, classical mechanics      130Ч132
Dynamic linear response, density matrix      133
Dynamic linear response, frequency dependent response      135Ч136
Dynamic linear response, Liouville equation      130Ч131 133Ч134
Dynamic linear response, phase space distribution function      133
Dynamic linear response, Poisson bracket      130Ч131
Dynamic linear response, quantum commutator      133
Dynamic linear response, quantum mechanics      133Ч135
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      107Ч108
Electron transfer kinetics, Marcus's reorganization energy      110
Electron transfer kinetics, polarization of environment      106Ч107
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      53Ч55
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      75Ч76
First passage times, derivation      74Ч76
First passage times, distribution      74
First passage times, Kramers problem      76Ч78
First passage times, mean      75
First-order linear differential equations      198Ч200
Fluctuation-dissipation theorem, analog of, for Langevin equation      16Ч17
Fluctuation-dissipation theorem, arbitrary distinction between systematic behavior and noise      24
Fluctuation-dissipation theorem, balance between friction and noise      6Ч7
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 Ч 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      160Ч161
Fluctuation-renormalization bare transport coefficient, nonlinear Langevin equations      188
Fokker Ч Planck equations, averages and adjoint operators      42Ч43
Fokker Ч Planck equations, choice of initial conditions      197
Fokker Ч Planck equations, derivation      36Ч39
Fokker Ч Planck equations, derivation of nonlinear      177Ч180
Fokker Ч Planck equations, Green's function in linear case      43Ч44 187Ч188
Fokker Ч Planck equations, Heisenberg approach      42Ч43
Fokker Ч Planck equations, long time steady-state solution of arbitrary      39
Fokker Ч Planck equations, noise-averaged distribution function      38 175
Fokker Ч Planck equations, nonlinear      180Ч181
Fokker Ч Planck equations, properties      41Ч42
Fokker Ч Planck equations, Schrodinger approach      42
Fokker Ч Planck equations, slow variables      183 187
Fokker Ч Planck equations, Smoluchowski equation      40Ч41
Fokker Ч Planck equations, substitution of Smoluchowski equation leading to Schrodinger-like equation      41
Fokker Ч Planck equations, two-variable Brownian motion of particle moving in potential      39Ч40
Frequency dependent magnetic susceptibility, linear response theory      137Ч138
Frequency dependent response, linear response in quantum mechanics      135Ч136
Gaussian random variables, application to Gaussian white noise      202Ч203
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      200Ч201
Gaussian random variables, noise      23
Gaussian random variables, probability distribution      200
Gaussian random variables, properties of multivariate      201Ч202
Golden Rule, definition      48
Golden Rule, derivation      48Ч51
Golden Rule, energy absorption equation      142
Golden Rule, flaws of standard treatment      52Ч53
Golden Rule, short time behavior      53
Golden Rule, transition states in Pauli master equation      124
Golden Rule, uses in electron transfer kinetics      108Ч109
Gram Ч 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      195Ч196
Harmonic lattice, reversibility paradox      195
Harmonic lattice, transform of normalized velocity correlation function      26
Harmonic lattice, velocity correlation function      24Ч25 27Ч28
Harmonic oscillator, Brownian motion in, heat bath      21Ч24
Harmonic oscillator, heat bath master equation      60Ч61
Harmonic oscillator, Langevin equation      22
Harmonic oscillator, Langevin equation for Brownian motion      17Ч18
Harmonic oscillator, memory function      22Ч23
Harmonic oscillator, noise      23 24
Heat bath      see also "Two-level system"
Heat bath master equation, application      59Ч61
Heat bath master equation, derivation      126
Heat bath master equation, Golden Rule transition rates      58
Heat bath master equation, harmonic oscillator      60Ч61
Heat bath master equation, microcanonical character      59
Heat bath master equation, principle of detailed balance      59
Heat bath, Bloch equations, two-level system      115Ч121
Heat bath, Brownian motion in harmonic oscillator      21Ч24
Heat bath, dephasing, two-level system      110Ч115
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      144Ч146
Hilbert space, projection operators      143Ч149
Hilbert space, quantum mechanics      101
Hilbert space, subspace of relevant variables      148Ч149
Inverting Laplace transforms      204
Ion mobility, linear response theory      136Ч137
Isomerization reaction, kinetics of first-order      14Ч15
Kinetic models, BGK equation and H-theorem      93Ч96
Kinetic models, BGK equation and hydrodynamics      96Ч100
Kinetic models, Boltzmann equation      83Ч84
Kinetic models, collision integral      84
Kinetic models, orientational time correlation function      89Ч92
Kinetic models, rotational diffusion      84Ч88
Kinetic models, rotational relaxation      89Ч92
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      78Ч82
Kramers problem, mean first passage time      77Ч78
Kramers problem, relation to transition state theory rate      78
Kramers Ч 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      17Ч18
Langevin equations, choice of initial conditions      197
Langevin equations, derivation for Brownian motion of system with harmonic oscillator heat bath      21Ч24
Langevin equations, derivation of nonlinear      177Ч180
Langevin equations, general treatment      15Ч16
Langevin equations, generalized derivation      149Ч151
Langevin equations, generalized derivation, deriving Fokker Ч Planck equation      177
Langevin equations, generalized derivation, eliminating projection operators      158Ч159
Langevin equations, generalized derivation, identities      157Ч160
Langevin equations, generalized derivation, initial nonequilibrium states      151Ч155
Langevin equations, generalized derivation, linear, for slow variables      165Ч168
Langevin equations, generalized derivation, Mori's procedure      161Ч162
Langevin equations, generalized derivation, noise      151Ч157
Langevin equations, generalized derivation, non-Markovian fluctuation-dissipation theorem      157
Langevin equations, generalized derivation, nonlinear to linear example      160Ч165
Langevin equations, generalized derivation, procedure constructing initial nonequilibrium distribution      154Ч155
Langevin equations, Markovian and non-Markovian      19Ч21
Langevin equations, nonlinear      18Ч19 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      203Ч204
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      198Ч200
Linear response theory, applications      136Ч139
Linear response theory, initial ensemble      197
Linear response theory, streaming velocity      186
Linear response, applications of theory      136Ч139
Linear response, determining equilibrium      127Ч129
Linear response, dynamic      130Ч136
Linear response, dynamic in classical mechanics      130Ч132
Linear response, dynamic in quantum mechanics      133Ч135
Linear response, energy absorption      141Ч142
Linear response, energy absorption in electric field      138Ч139
Linear response, frequency dependent magnetic susceptibility      137Ч138
Linear response, frequency dependent response      135Ч136
Linear response, identities      139Ч142
Linear response, mobility of ion      136Ч137
Linear response, quantum mechanical version      129Ч130
Linear response, quantum perturbation theory      130
Linear response, static      127Ч130
Liouville equation, dynamical variables      33Ч35
Liouville equation, evolution of dynamical variable      34
Liouville equation, formal operator solution      32
Liouville equation, Liouville operator      32Ч33
Liouville equation, matrix form      144Ч146
Liouville equation, partitioning      146Ч147
Long time tails, deriving, stress correlation function      173
Long time tails, mode-coupling theory      169Ч170
Magnetic susceptibility, linear response theory      137Ч138
Master equations, abstract      61Ч63
Master equations, chemical kinetics      64Ч66
Master equations, derivation of quantum mechanical or Pauli      121Ч124
Master equations, Golden Rule transition rates of heat bath      58
Master equations, harmonic oscillator illustration of heat bath      59Ч61
Master equations, heat bath      57Ч59
Master equations, matrix or operator equation      62
Master equations, Pauli      56Ч57
Master equations, random walks      63Ч64
Master equations, use of operator methods for averages      62
Memory kernel, derivation of nonlinear equations      179Ч180
Memory kernel, derivation of Pauli master equation      123 126
Memory kernel, eliminating projection operators      158Ч159
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      118Ч119
Mobility of ion, linear response theory      136Ч137
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      169Ч170
Mode-coupling theory, product of two slow variables      171
Mode-coupling theory, self-diffusion example      170Ч173
Mori Langevin equation, differences between, and exact Langevin      160
Mori Langevin equation, memory function      164Ч165
Mori Langevin equation, noise      165
Mori Langevin equation, nonlinear to linear example      160Ч165
Mori Langevin equation, slow variables      166
Nernst Ч Planck equation, treatment of electrolyte solutions      160
Noise, application of Gaussian random variables to white      202Ч203
Noise, arbitrary distinction from systematic behavior      24
Noise, averaged, in Langevin equation      155Ч156
Noise, generalized Langevin equations      151Ч157
Noise, nonlinear Langevin equation      184Ч185
Nonlinear Langevin and Fokker Ч Planck equations, derivation      177Ч181
Nonlinear Langevin and Fokker Ч Planck equations, illustration of nonlinear system interacting with harmonic oscillator heat bath      183Ч184
Nonlinear Langevin and Fokker Ч Planck equations, memory kernel      179Ч180
Nonlinear Langevin and Fokker Ч Planck equations, noise and initial states      184Ч185
Nonlinear Langevin and Fokker Ч Planck equations, reduced distribution functions      175Ч177
Nonlinear Langevin and Fokker Ч Planck equations, slow variables      181Ч183
Optical absorption coefficient, classical time correlation function of total electric dipole moment      56
Optical absorption coefficient, focus on rate of energy absorption      53Ч55
Optical absorption coefficient, frequency dependence      12
Optical absorption coefficient, frequency dependence by time correlation function      53
Optical absorption coefficient, theory of optical absorption      55Ч56
Orientational time correlation function, approaching ideal rotator limit      91
Orientational time correlation function, deriving exact expression      89Ч92
Orientational time correlation function, exponential decay      92
Orientational time correlation function, Laplace transform of correlation function      90Ч91
Orientational time correlation function, results of numerical inversion of Laplace transform      92f
Pauli master equation, derivation      121Ч124
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      124Ч126
Probability distribution, Gaussian random variables      200
Projection operator method, deriving quantum mechanical master equation      124Ч126
Projection operator method, use in deriving Langevin equations      143Ч144
Projection operators, derivation of generalized Langevin equations      149Ч151
Projection operators, deriving Fokker Ч Planck equation      177
Projection operators, deriving Langevin equations      178
Projection operators, eliminating projection      158Ч159
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