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| Mackey M.C. — Time's arrow: the origins of thermodynamic behavior |
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| Предметный указатель |
 -space 5
Abstract ergodic theorem 35
Additivity 6 14
Adjoint operators 43 145
Anosov diffeomorphism 59
Anosov flow 60
Asymptotic periodicity 72
Asymptotic periodicity, and correlations 86
Asymptotic periodicity, and entropy 80
Asymptotic periodicity, and ergodicity 74
Asymptotic periodicity, and exactness 96
Asymptotic periodicity, and Keener map 126
Asymptotic periodicity, and perturbations 118 122 126
Asymptotic periodicity, and quadratic map 78
Asymptotic periodicity, and smoothing 71
Asymptotic periodicity, and stationary densities 72
Asymptotic periodicity, and tent map 75 82
Asymptotic periodicity, and Weak Form of Second Law 80
Asymptotic periodicity, conditions for 73 75 84
Asymptotic periodicity, in loosely coupled systems 118
Asymptotic periodicity, in strongly coupled systems 122 123
Asymptotic periodicity, period of 72 125
Average 67
Baker transformation 60
Baker transformation, and additive perturbations 123
Baker transformation, and dyadic transformation 112
Baker transformation, and entropy 64 66 106
Baker transformation, and invertibility 60 113
Baker transformation, and K-systems 67 112
Baker transformation, and Lebesgue measure 60
Baker transformation, and Liouville equation 114
Baker transformation, and microcanonical ensemble 65
Baker transformation, and mixing 60 63
Baker transformation, and trace 112 123
Baker transformation, Frobenius — Perron operator for 64 113
Baker transformation, quadratic 112
Baker transformation, stationary density for 65
Birkhoff ergodic theorem 44
Birkhoff ergodic theorem, extended to ergodic dynamics 53
Blackbody radiation 18
Boltzmann 6 31—33
Boltzmann  -space 5
Boltzmann and Loschmidt 32 45
Boltzmann and Zermelo 45
Boltzmann constant 16
Boltzmann equation 114
Boltzmann statistics 16
Boltzmann — Gibbs entropy 5
Boltzmann — Gibbs entropy additivity 6 14
Boltzmann — Gibbs entropy and baker transformation 64 106
Boltzmann — Gibbs entropy and canonical ensemble 11
Boltzmann — Gibbs entropy and conditional entropy 28
Boltzmann — Gibbs entropy and correlations 86
Boltzmann — Gibbs entropy and exactness 100
Boltzmann — Gibbs entropy and grand canonical ensemble 19
Boltzmann — Gibbs entropy and microcanonical ensemble 9
Boltzmann — Gibbs entropy and mixing 64
Boltzmann — Gibbs entropy and quadratic map 28
Boltzmann — Gibbs entropy and tent map 100
Boltzmann — Gibbs entropy and thermodynamic entropy 13
Boltzmann — Gibbs entropy maximal 9 19
Boltzmann — Gibbs entropy sampled 107
Boltzmann — Gibbs entropy uniqueness of 6
Boltzmann — Gibbs entropy, coarse grained 104
canonical ensemble 14
Canonical ensemble and entropy 11 101
Canonical ensemble and exactness 92 94 98 101
Canonical ensemble and mixing 63
Canonical ensemble and stationary density 64
Cauchy — Holder inequality 93
Cesaro convergence 56
Chebyshev inequality 118
Clausius 33
Coarse graining 103
Coarse graining and baker transformation 106
Coarse graining and dynamics 106
Coarse graining and errors 105
Coarse graining and Gibbs 103 106 109
Coarse graining and invertibility 107
Coarse graining and Strong Form of Second Law 107
Coarse graining and trace 113
Coarse graining of densities 104
Coarse graining of entropy 104 105 112
Coarse graining of mixing systems 106
Coarse graining of phase space 103
Coarse graining temporal 107
Conditional entropy 27
Conditional entropy and asymptotic periodicity 80
Conditional entropy and baker transformation 66
Conditional entropy and Boltzmann — Gibbs entropy 28 29
Conditional entropy and correlations 87
Conditional entropy and exactness 98
Conditional entropy and invertible dynamics 31 66
Conditional entropy and Keener map 126
Conditional entropy and Liouville equation 32
Conditional entropy and mixing 64 66
Conditional entropy and ordinary differential equations 32
Conditional entropy and perturbations 119 126 131 149 150
Conditional entropy and quadratic map 78 100
Conditional entropy and tent map 82 95
Conditional entropy and thermodynamic entropy difference 29
Conditional entropy coarse grained 105
Conditional entropy dependence on initial density 80 126
Conditional entropy properties of 29
Conditional entropy sampled 107
Convergence Cesaro 56
Convergence strong 35 92
Convergence weak 35 63
Correlation 67
Correlation and asymptotic periodicity 86
Correlation and entropy 86
Correlation and Frobenius — Perron operator 68
Correlation and Koopman operator 68
Correlation and mixing 68
Covariance 67
Covariance and mixing 68
Covariance normalized 67
Delay differential equation and exactness 91
Delay differential equation and laws of physics 101
density 4
Density and thermodynamic states 4
Density coarse grained 104
Density evolution 43
Density Gaussian 141 151
Density illustrated 4
Density of measure 4
Density stationary 23 38
Diffusion term 145 148
Drift term 145
Dyadic transformation 94
Dyadic transformation and baker transformation 112
Dyadic transformation and exactness 94
Dyadic transformation and Liouville equation 114
Dyadic transformation and microcanonical ensemble 94
Dyadic transformation and noninvertibility 114
Dyadic transformation and perturbations 123
Dyadic transformation and trace 112
Dyadic transformation as a factor 112 114 123
Dyadic transformation Frobenius — Perron operator for 94
Dyadic transformation stationary density for 95
Dynamical system 3
Dynamics and coarse grained entropy 106
Dynamics and Frobenius — Perron operator 39
Dynamics and Koopman operator 43
Dynamics and strong perturbations 120
Dynamics asymptotically periodic 72
Dynamics constructing exact 95
Dynamics ergodic 48
Dynamics exact 89
Dynamics Hamiltonian 33 45 48 101
| Dynamics invertible 2
Dynamics irreversible 3
Dynamics loosely coupled to perturbations 117
Dynamics measurable 39
Dynamics measure preserving 41
Dynamics mixing 58
Dynamics noninvertible 3
Dynamics nonsingular 39
Dynamics reversible 2
Dynamics time reversal invariant 2
Dynamics trajectory of 2 110
Dynamics versus perturbations 120
Energy 10 12
Energy and Maxwell — Boltzmann distribution 17
Energy continuous spectrum 17
Energy discrete spectrum 18
Energy Helmholtz 16
Ensemble 5
Entropy and correlations 86
Entropy and dynamics 30
Entropy and invertibility 31
Entropy and mixing 66
Entropy and trace 111
Entropy Boltzmann — Gibbs 5
Entropy coarse grained 104 105 107 114
Entropy conditional 28
Entropy dependence on preparation 80 125
Entropy for baker transformation 64
Entropy for K-systems 67
Entropy for quadratic map 28 78 83 100
Entropy for tent map 82 95
Ergodic theorem abstract 35
Ergodic theorem Birkhoff 44
Ergodic theorem for space and trajectory averages 53
Ergodicity 48
Ergodicity and asymptotic periodicity 74
Ergodicity and Birkhoff theorem 53
Ergodicity and canonical ensemble 54
Ergodicity and Cesaro convergence 56
Ergodicity and Frobenius — Perron operator 52 56
Ergodicity and Hamiltonian dynamics 51
Ergodicity and harmonic oscillator 49
Ergodicity and Koopman operator 51
Ergodicity and Markov operator 57
Ergodicity and microcanonical ensemble 54
Ergodicity and mixing 63
Ergodicity and perturbations 121
Ergodicity and sampled systems 109
Ergodicity and space and time averages 54
Ergodicity and stationary density 52
Ergodicity and tent map 75
Ergodicity and thermodynamic equilibrium 52 54
Ergodicity illustrated 49
Errors see “Coarse graining; Sampling”
Euler — Bernstein equations 142
Exactness 89
Exactness and asymptotic periodicity 96
Exactness and canonical ensemble 92 94 98 101
Exactness and delay differential equations 91
Exactness and dyadic transformation 94
Exactness and entropy 98 100
Exactness and factors 113
Exactness and invertibility 90
Exactness and K-systems 113
Exactness and Keener map 131
Exactness and lower bound function 97
Exactness and Markov operator 93
Exactness and microcanonical ensemble 94 100
Exactness and partial differential equations 91
Exactness and perturbations 119 122 131 135 149 152 158
Exactness and quadratic map 95 100
Exactness and sampled sequence 108
Exactness and strong convergence 92
Exactness and Strong Form of Second Law 98 100 150 152 158
Exactness and tent map 91 95 96 100
Exactness constructing 95
Exactness convergence rate 96
Exactness illustrated 90
Extension in phase 10
Factor 112
Factor and dyadic transformation 112 114 123
Factor and exactness 113
Factor and K-system 113
Factor and projection operators 113
Factor and quadratic baker transformation 112
Factor and Strong Form of Second Law 113
Factor and trace 112
Fixed point 23
Fokker — Planck equation 22 144
Fokker — Planck equation, and Liouville equation 22 148
Fokker — Planck equation, and Markov operator 22 145 147
Fokker — Planck equation, and noninvertibility 24
Fokker — Planck equation, and stationary density 23 150
Fokker — Planck equation, and thermodynamic equilibrium 150
Fokker — Planck equation, derivation 142
Fokker — Planck equation, generalized solution 146
Frobenius — Perron operator 24 39
Frobenius — Perron operator, and correlations 68
Frobenius — Perron operator, and ergodicity 52 56
Frobenius — Perron operator, and evolution of densities 44
Frobenius — Perron operator, and Koopman operator 44
Frobenius — Perron operator, and Liouville equation 47 114 148
Frobenius — Perron operator, and mixing 63
Frobenius — Perron operator, and ordinary differential equations 23 47 114 148
Frobenius — Perron operator, approximating Markov operators 136
Frobenius — Perron operator, for baker transformation 64 113
Frobenius — Perron operator, for dyadic map 94
Frobenius — Perron operator, for Hamiltonian systems 48
Frobenius — Perron operator, for invertible dynamics 41
Frobenius — Perron operator, for quadratic baker transformation 113
Frobenius — Perron operator, for quadratic map 28 78
Frobenius — Perron operator, for tent map 25 75
Frobenius — Perron operator, properties of 40
Frobenius — Perron operator, stationary density of 23 25
Gas dynamics 61
Gaussian density 141—151
Generalized solution 146
Geodesic flows 62
Gibbs  -space 5
Gibbs and coarse graining 103 106 109
Gibbs and mixing 64
Gibbs extension in phase 10
Gibbs function 16
Gibbs index of probability 5
Gibbs inequality 7
Gibbs inequality, integrated 8
Gibbs microcanonical ensemble 10
God theorem 111
grand canonical ensemble 19
Hamiltonian dynamics 33—45 48 101
Hamiltonian dynamics and ergodicity 51
Hamiltonian dynamics and Frobenius — Perron operator 48
Hamiltonian dynamics and Lebesgue measure 48 101
Hamiltonian dynamics and maximal entropy 101
harmonic oscillator 49
Hat map see “Tent map”
Helmholtz free energy 16
Index of probability 5
Initial density dependence of entropy on 80 126
Invariant measure 41
Invariant measure, and stationary density 42
Invariant measure, and thermodynamic equilibrium 43
Invariant measure, illustrated 42
Invariant set 48
Invertibility 2
Invertibility and baker transformation 60 113
Invertibility and coarse graining 107
Invertibility and entropy 31 66
Invertibility and exactness 90
Invertibility and laws of physics 101
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