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BK

First published

Hoboken : Wiley, 2020

xix, 261 stran, 8 nečíslovaných stran obrazových příloh : ilustrace (některé barevné) ; 24 cm

ISBN 978-1-119-17677-0 (vázáno)

Obsahuje bibliografické odkazy a rejstřík

002000517

Acknowledgments xiii // Introduction: Historical Background and Recent Developments that Motivate this Book xv // 1 The Langevin Equation and Stochastic Processes 1 // 1.1 General Framework 1 // 1.2 The Ornstein-Uhlenbeck (OU) Process 5 // 1.3 The Overdamped Limit 8 // 1.4 The Overdamped Harmonic Oscillator: An Ornstein-Uhlenbeck process 11 // 1.5 Differential Form and Discretization 12 // 1.5.1 Euler-Maruyama Discretization (EMD) and Itô Processes 15 // 1.5.2 Stratonovich Discretization (SD) 17 // 1.6 Relation Between Itô and Stratonovich Integrals 19 // 1.7 Space Varying Diffusion Constant 21 // 1.8 Itô vs Stratonovich 23 // 1.9 Detailed Balance 23 // 1.10 Memory Kernel 25 // 1.11 The Many Particle Case 26 References 26 // 2 The Fokker-Planck Equation 29 // 2.1 The Chapman-Kolmogorov Equation 29 // 2.2 The Overdamped Case 30 // 2.2.1 Derivation of the Smoluchowski (Fokker-Planck) Equation using the // Chapman-Kolmogorov Equation 30 // 2.2.2 Alternative Derivation of the Smoluchowski (Fokker-Planck) Equation 33 // 2.2.3 The Adjoint (or Reverse or Backward) Fokker-Planck Equation 34 // 2.3 The Underdamped Case 34 // 2.4 The Free Case 35 // 2.4.1 Overdamped Case 35 // 2.4.2 Underdamped Case 36 // 2.5 Averages and Observables 37 // References 39 // 3 The Schrödinger Representation 41 // 3.1 The Schrödinger Equation 41 // 3.2 Spectral Representation 43 // 3.3 Ground State and Convergence to the Boltzmann Distribution 44 References 47 // 4 Discrete Systems: The Master Equation and Kinetic Monte Carlo 49 // 4.1 The Master Equation 49 // 4.1.1 Discrete-Time Markov Chains 49 // 4.1.2 Continuous-Time Markov Chains, Markov Processes 51 // 4.2 Detailed Balance 53 // 4.2.1 Final State Only 54 // 4.2.2 Initial State Only 54 // 4.2.3 Initial and Final State 55 // 4.2.4 Metropolis Scheme 55 // 4.2.5 Symmetrization 55 // 4.3 Kinetic Monte Carlo (KMC) 58 // References 61 //

5 Path Integrals 63 // 5.1 The Itô Path Integral 63 // 5.2 The Stratonovich Path Integral 66 // References 67 // 6 Barrier Crossing 69 // 6.1 First Passage Time and Transition Rate 69 // 6.1.1 Average Mean First Passage Time 71 // 6.1.2 Distribution of First Passage Time 73 // 6.1.3 The Free Particle Case 74 // 6.1.4 Conservative Force 75 // 6.2 Kramers Transition Time: Average and Distribution 77 // 6.2.1 Kramers Derivation 78 // 6.2.2 Mean First Passage Time Derivation 80 // 6.3 Transition Path Time: Average and Distribution 81 // 6.3.1 Transition Path Time Distribution 82 // 6.3.2 Mean Transition Path Time 84 References 86 // 7 Sampling Transition Paths 89 // 7.1 Dominant Paths and Instantons 92 // 7.1.1 Saddle-Point Method 92 // 7.1.2 The Euler-Lagrange Equation: Dominant Paths 92 // 7.1.3 Steepest Descent Method 96 // 7.1.4 Gradient Descent Method 97 // 7.2 Path Sampling 98 // 7.2.1 Metropolis Scheme 98 // 7.2.2 Langevin Scheme 99 // 7.3 Bridge and Conditioning 99 // 7.3.1 Free Particle 102 // 7.3.2 The Ornstein-Uhlenbeck Bridge І02 // 7.3.3 Exact Diagonalization 104 // 7.3.4 Cumulant Expansion 105 References 111 // Appendix A: Gaussian Variables 111 Appendix В 113 // 8 The Rate of Conformational Change: Definition and Computation 117 // 8.1 First-order Chemical Kinetics 117 // 8.2 Rate Coefficients from Microscopic Dynamics 119 // 8.2.1 Validity of First Order Kinetics 120 // 8.2.2 Mapping Continuous Trajectories onto Discrete Kinetics and Computing Exact Rates 123 // 8.2.3 Computing the Rate More Efficiently 126 // 8.2.4 Transmission Coefficient and Variational Transition State Theory 128 // 8.2.5 Harmonic Transition-State Theory 129 References 131 // 9 Zwanzig-Caldeiga-Leggett Model for Low-Dimensional Dynamics 133 // 9.1 Low-Dimensional Models of Reaction Dynamics From a Microscopic Hamiltonian 133 //

9.2 Statistical Properties of the Noise and the Fluctuation-dissipation Theorem 137 // 9.2.1 Ensemble Approach 138 // 9.2.2 Single-Trajectory Approach 139 // 9.3 Time-Reversibility of the Langevin Equation 142 // References 145 // 10 Escape from a Potential Well in the Case of Dynamics Obeying the Generalized Langevin Equation: General Solution Based on the Zwanzig-Caldeira-Leggett Hamiltonian 147 // 10.1 Derivation of the Escape Rate 147 // 10.2 The Limit of Kramers Theory ISO // 10.3 Significance of Memory Effects 152 // 10.4 Applications of the Kramers Theory to Chemical Kinetics in Condensed Phases, Particularly in Biomolecular Systems 153 // 10.5 A Comment on the Use of the Term “Free Energy" in Application to Chemical Kinetics and Equilibrium 155 // References 156 // 11 Diffusive Dynamics on a Multidimensional Energy Landscape 157 // 11.1 Generalized Langevin Equation with Exponential Memory can be Derived from a 2D Markov Model 157 // 11.2 Theory of Multidimensional Barrier Crossing 161 // 11.3 Breakdown of the Langer Theory in the Case of Anisotropic Diffusion: the Berezhkovskii-Zitserman Case 167 References 171 // 12 Quantum Effects in Chemical Kinetics 173 // 12.1 When is a Quantum Mechanical Description Necessary? 173 // 12.2 How Do the Laws of Quantum Mechanics Affect the Observed Transition Rates? 174 // 12.3 Semiclassical Approximation and the Deep Tunneling Regime 177 // 12.4 Path Integrals, Ring-Polymer Quantum Transition-State Theory, Instantons and Centroids 184 // References 191 // 13 Computer Simulations of Molecular Kinetics: // Foundation 193 // 13.1 Computer Simulations: Statement of Goals 193 // 13.2 The Empirical Energy 195 // 13.3 Molecular States 197 // 13.4 Mean First Passage Time 199 // 13.5 Coarse Variables 199 // 13.6 Equilibrium, Stable, and Metastable States 200 // References 202 //

14 The Master Equation as a Model for Transitions Between Macrostates 203 // References 211 // 15 Direct Calculation of Rate Coefficients with Computer Simulations 213 // 15.1 Computer Simulations of Trajectories 213 // 15.2 Calculating Rate with Trajectories 219 References 221 // 16 A Simple Numerical Example of Rate Calculations 223 // References 231 // 17 Rare Events and Reaction Coordinates 233 // References 240 // 18 Celling 241 // References 252 // 19 An Example of the Use of Cells: Alanine Dipeptide 255 // References 257 // Index 259