Ron Elber is Professor of Chemistry at the University of Texas at Austin and W. A. "Tex" Moncrief, Jr. Endowed Chair in Computational Life Sciences and Biology in the Oden Institute for Computational Engineering and Sciences. Dmitrii E. Makarov is Professor of Chemistry at the University of Texas at Austin. His research is in the field of computational and theoretical chemical physics. Henri Orland is Directeur de Recherches at the Institut de Physique Théorique, the French Alternative Energies and Atomic Energy Commission, CEA, France.

• Acknowledgments xiiiIntroduction: Historical Background and Recent Developments that Motivate this Book xv1 The Langevin Equation and Stochastic Processes 11.1 General Framework 11.2 The Ornstein-Uhlenbeck (OU) Process 51.3 The Overdamped Limit 81.4 The Overdamped Harmonic Oscillator: An Ornstein–Uhlenbeck process 111.5 Differential Form and Discretization 121.5.1 Euler-Maruyama Discretization (EMD) and Itô Processes 151.5.2 Stratonovich Discretization (SD) 171.6 Relation Between Itô and Stratonovich Integrals 191.7 Space Varying Diffusion Constant 211.8 Itô vs Stratonovich 231.9 Detailed Balance 231.10 Memory Kernel 251.11 The Many Particle Case 26References 262 The Fokker–Planck Equation 292.1 The Chapman–Kolmogorov Equation 292.2 The Overdamped Case 302.2.1 Derivation of the Smoluchowski (Fokker–Planck) Equation using the Chapman–Kolmogorov Equation 302.2.2 Alternative Derivation of the Smoluchowski (Fokker–Planck) Equation 332.2.3 The Adjoint (or Reverse or Backward) Fokker–Planck Equation 342.3 The Underdamped Case 342.4 The Free Case 352.4.1 Overdamped Case 352.4.2 Underdamped Case 362.5 Averages and Observables 37References 393 The Schrödinger Representation 413.1 The Schrödinger Equation 413.2 Spectral Representation 433.3 Ground State and Convergence to the Boltzmann Distribution 44References 474 Discrete Systems: The Master Equation and Kinetic Monte Carlo 494.1 The Master Equation 494.1.1 Discrete-Time Markov Chains 494.1.2 Continuous-Time Markov Chains, Markov Processes 514.2 Detailed Balance 534.2.1 Final State Only 544.2.2 Initial State Only 544.2.3 Initial and Final State 554.2.4 Metropolis Scheme 554.2.5 Symmetrization 554.3 Kinetic Monte Carlo (KMC) 58References 615 Path Integrals 635.1 The Itô Path Integral 635.2 The Stratonovich Path Integral 66References 676 Barrier Crossing 696.1 First Passage Time and Transition Rate 696.1.1 Average Mean First Passage Time 716.1.2 Distribution of First Passage Time 736.1.3 The Free Particle Case 746.1.4 Conservative Force 756.2 Kramers Transition Time: Average and Distribution 776.2.1 Kramers Derivation 786.2.2 Mean First Passage Time Derivation 806.3 Transition Path Time: Average and Distribution 816.3.1 Transition Path Time Distribution 826.3.2 Mean Transition Path Time 84References 867 Sampling Transition Paths 897.1 Dominant Paths and Instantons 927.1.1 Saddle-Point Method 927.1.2 The Euler-Lagrange Equation: Dominant Paths 927.1.3 Steepest Descent Method 967.1.4 Gradient Descent Method 977.2 Path Sampling 987.2.1 Metropolis Scheme 987.2.2 Langevin Scheme 997.3 Bridge and Conditioning 997.3.1 Free Particle 1027.3.2 The Ornstein-Uhlenbeck Bridge 1027.3.3 Exact Diagonalization 1047.3.4 Cumulant Expansion 105References 111Appendix A: Gaussian Variables 111Appendix B 1138 The Rate of Conformational Change: Definition and Computation 1178.1 First-order Chemical Kinetics 1178.2 Rate Coefficients from Microscopic Dynamics 1198.2.1 Validity of First Order Kinetics 1208.2.2 Mapping Continuous Trajectories onto Discrete Kinetics and Computing Exact Rates 1238.2.3 Computing the Rate More Efficiently 1268.2.4 Transmission Coefficient and Variational Transition State Theory 1288.2.5 Harmonic Transition-State Theory 129References 1319 Zwanzig-Caldeiga-Leggett Model for Low-Dimensional Dynamics 1339.1 Low-Dimensional Models of Reaction Dynamics From a Microscopic Hamiltonian 1339.2 Statistical Properties of the Noise and the Fluctuation-dissipation Theorem 1379.2.1 Ensemble Approach 1389.2.2 Single-Trajectory Approach 1399.3 Time-Reversibility of the Langevin Equation 142References 14510 Escape from a Potential Well in the Case of Dynamics Obeying the Generalized Langevin Equation: General Solution Based on the Zwanzig-Caldeira-Leggett Hamiltonian 14710.1 Derivation of the Escape Rate 14710.2 The Limit of Kramers Theory 15010.3 Significance of Memory Effects 15210.4 Applications of the Kramers Theory to Chemical Kinetics in Condensed Phases, Particularly in Biomolecular Systems 15310.5 A Comment on the Use of the Term “Free Energy” in Application to Chemical Kinetics and Equilibrium 155References 15611 Diffusive Dynamics on a Multidimensional Energy Landscape 15711.1 Generalized Langevin Equation with Exponential Memory can be Derived from a 2D Markov Model 15711.2 Theory of Multidimensional Barrier Crossing 16111.3 Breakdown of the Langer Theory in the Case of Anisotropic Diffusion: the Berezhkovskii-Zitserman Case 167References 17112 Quantum Effects in Chemical Kinetics 17312.1 When is a Quantum Mechanical Description Necessary? 17312.2 How Do the Laws of Quantum Mechanics Affect the Observed Transition Rates? 17412.3 Semiclassical Approximation and the Deep Tunneling Regime 17712.4 Path Integrals, Ring-Polymer Quantum Transition-State Theory, Instantons and Centroids 184References 19113 Computer Simulations of Molecular Kinetics: Foundation 19313.1 Computer Simulations: Statement of Goals 19313.2 The Empirical Energy 19513.3 Molecular States 19713.4 Mean First Passage Time 19913.5 Coarse Variables 19913.6 Equilibrium, Stable, and Metastable States 200References 20214 The Master Equation as a Model for Transitions Between Macrostates 203References 21115 Direct Calculation of Rate Coefficients with Computer Simulations 21315.1 Computer Simulations of Trajectories 21315.2 Calculating Rate with Trajectories 219References 22116 A Simple Numerical Example of Rate Calculations 223References 23117 Rare Events and Reaction Coordinates 233References 24018 Celling 241References 25219 An Example of the Use of Cells: Alanine Dipeptide 255References 257Index 259

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Ron Elber is Professor of Chemistry at the University of Texas at Austin and W. A. "Tex" Moncrief, Jr. Endowed Chair in Computational Life Sciences and Biology in the Oden Institute for Computational Engineering and Sciences. Dmitrii E. Makarov is Professor of Chemistry at the University of Texas at Austin. His research is in the field of computational and theoretical chemical physics. Henri Orland is Directeur de Recherches at the Institut de Physique Théorique, the French Alternative Energies and Atomic Energy Commission, CEA, France.

• Acknowledgments xiiiIntroduction: Historical Background and Recent Developments that Motivate this Book xv1 The Langevin Equation and Stochastic Processes 11.1 General Framework 11.2 The Ornstein-Uhlenbeck (OU) Process 51.3 The Overdamped Limit 81.4 The Overdamped Harmonic Oscillator: An Ornstein–Uhlenbeck process 111.5 Differential Form and Discretization 121.5.1 Euler-Maruyama Discretization (EMD) and Itô Processes 151.5.2 Stratonovich Discretization (SD) 171.6 Relation Between Itô and Stratonovich Integrals 191.7 Space Varying Diffusion Constant 211.8 Itô vs Stratonovich 231.9 Detailed Balance 231.10 Memory Kernel 251.11 The Many Particle Case 26References 262 The Fokker–Planck Equation 292.1 The Chapman–Kolmogorov Equation 292.2 The Overdamped Case 302.2.1 Derivation of the Smoluchowski (Fokker–Planck) Equation using the Chapman–Kolmogorov Equation 302.2.2 Alternative Derivation of the Smoluchowski (Fokker–Planck) Equation 332.2.3 The Adjoint (or Reverse or Backward) Fokker–Planck Equation 342.3 The Underdamped Case 342.4 The Free Case 352.4.1 Overdamped Case 352.4.2 Underdamped Case 362.5 Averages and Observables 37References 393 The Schrödinger Representation 413.1 The Schrödinger Equation 413.2 Spectral Representation 433.3 Ground State and Convergence to the Boltzmann Distribution 44References 474 Discrete Systems: The Master Equation and Kinetic Monte Carlo 494.1 The Master Equation 494.1.1 Discrete-Time Markov Chains 494.1.2 Continuous-Time Markov Chains, Markov Processes 514.2 Detailed Balance 534.2.1 Final State Only 544.2.2 Initial State Only 544.2.3 Initial and Final State 554.2.4 Metropolis Scheme 554.2.5 Symmetrization 554.3 Kinetic Monte Carlo (KMC) 58References 615 Path Integrals 635.1 The Itô Path Integral 635.2 The Stratonovich Path Integral 66References 676 Barrier Crossing 696.1 First Passage Time and Transition Rate 696.1.1 Average Mean First Passage Time 716.1.2 Distribution of First Passage Time 736.1.3 The Free Particle Case 746.1.4 Conservative Force 756.2 Kramers Transition Time: Average and Distribution 776.2.1 Kramers Derivation 786.2.2 Mean First Passage Time Derivation 806.3 Transition Path Time: Average and Distribution 816.3.1 Transition Path Time Distribution 826.3.2 Mean Transition Path Time 84References 867 Sampling Transition Paths 897.1 Dominant Paths and Instantons 927.1.1 Saddle-Point Method 927.1.2 The Euler-Lagrange Equation: Dominant Paths 927.1.3 Steepest Descent Method 967.1.4 Gradient Descent Method 977.2 Path Sampling 987.2.1 Metropolis Scheme 987.2.2 Langevin Scheme 997.3 Bridge and Conditioning 997.3.1 Free Particle 1027.3.2 The Ornstein-Uhlenbeck Bridge 1027.3.3 Exact Diagonalization 1047.3.4 Cumulant Expansion 105References 111Appendix A: Gaussian Variables 111Appendix B 1138 The Rate of Conformational Change: Definition and Computation 1178.1 First-order Chemical Kinetics 1178.2 Rate Coefficients from Microscopic Dynamics 1198.2.1 Validity of First Order Kinetics 1208.2.2 Mapping Continuous Trajectories onto Discrete Kinetics and Computing Exact Rates 1238.2.3 Computing the Rate More Efficiently 1268.2.4 Transmission Coefficient and Variational Transition State Theory 1288.2.5 Harmonic Transition-State Theory 129References 1319 Zwanzig-Caldeiga-Leggett Model for Low-Dimensional Dynamics 1339.1 Low-Dimensional Models of Reaction Dynamics From a Microscopic Hamiltonian 1339.2 Statistical Properties of the Noise and the Fluctuation-dissipation Theorem 1379.2.1 Ensemble Approach 1389.2.2 Single-Trajectory Approach 1399.3 Time-Reversibility of the Langevin Equation 142References 14510 Escape from a Potential Well in the Case of Dynamics Obeying the Generalized Langevin Equation: General Solution Based on the Zwanzig-Caldeira-Leggett Hamiltonian 14710.1 Derivation of the Escape Rate 14710.2 The Limit of Kramers Theory 15010.3 Significance of Memory Effects 15210.4 Applications of the Kramers Theory to Chemical Kinetics in Condensed Phases, Particularly in Biomolecular Systems 15310.5 A Comment on the Use of the Term “Free Energy” in Application to Chemical Kinetics and Equilibrium 155References 15611 Diffusive Dynamics on a Multidimensional Energy Landscape 15711.1 Generalized Langevin Equation with Exponential Memory can be Derived from a 2D Markov Model 15711.2 Theory of Multidimensional Barrier Crossing 16111.3 Breakdown of the Langer Theory in the Case of Anisotropic Diffusion: the Berezhkovskii-Zitserman Case 167References 17112 Quantum Effects in Chemical Kinetics 17312.1 When is a Quantum Mechanical Description Necessary? 17312.2 How Do the Laws of Quantum Mechanics Affect the Observed Transition Rates? 17412.3 Semiclassical Approximation and the Deep Tunneling Regime 17712.4 Path Integrals, Ring-Polymer Quantum Transition-State Theory, Instantons and Centroids 184References 19113 Computer Simulations of Molecular Kinetics: Foundation 19313.1 Computer Simulations: Statement of Goals 19313.2 The Empirical Energy 19513.3 Molecular States 19713.4 Mean First Passage Time 19913.5 Coarse Variables 19913.6 Equilibrium, Stable, and Metastable States 200References 20214 The Master Equation as a Model for Transitions Between Macrostates 203References 21115 Direct Calculation of Rate Coefficients with Computer Simulations 21315.1 Computer Simulations of Trajectories 21315.2 Calculating Rate with Trajectories 219References 22116 A Simple Numerical Example of Rate Calculations 223References 23117 Rare Events and Reaction Coordinates 233References 24018 Celling 241References 25219 An Example of the Use of Cells: Alanine Dipeptide 255References 257Index 259

Select Guide Rating

Ron Elber is Professor of Chemistry at the University of Texas at Austin and W. A. "Tex" Moncrief, Jr. Endowed Chair in Computational Life Sciences and Biology in the Oden Institute for Computational Engineering and Sciences. Dmitrii E. Makarov is Professor of Chemistry at the University of Texas at Austin. His research is in the field of computational and theoretical chemical physics. Henri Orland is Directeur de Recherches at the Institut de Physique Théorique, the French Alternative Energies and Atomic Energy Commission, CEA, France.

• Acknowledgments xiiiIntroduction: Historical Background and Recent Developments that Motivate this Book xv1 The Langevin Equation and Stochastic Processes 11.1 General Framework 11.2 The Ornstein-Uhlenbeck (OU) Process 51.3 The Overdamped Limit 81.4 The Overdamped Harmonic Oscillator: An Ornstein–Uhlenbeck process 111.5 Differential Form and Discretization 121.5.1 Euler-Maruyama Discretization (EMD) and Itô Processes 151.5.2 Stratonovich Discretization (SD) 171.6 Relation Between Itô and Stratonovich Integrals 191.7 Space Varying Diffusion Constant 211.8 Itô vs Stratonovich 231.9 Detailed Balance 231.10 Memory Kernel 251.11 The Many Particle Case 26References 262 The Fokker–Planck Equation 292.1 The Chapman–Kolmogorov Equation 292.2 The Overdamped Case 302.2.1 Derivation of the Smoluchowski (Fokker–Planck) Equation using the Chapman–Kolmogorov Equation 302.2.2 Alternative Derivation of the Smoluchowski (Fokker–Planck) Equation 332.2.3 The Adjoint (or Reverse or Backward) Fokker–Planck Equation 342.3 The Underdamped Case 342.4 The Free Case 352.4.1 Overdamped Case 352.4.2 Underdamped Case 362.5 Averages and Observables 37References 393 The Schrödinger Representation 413.1 The Schrödinger Equation 413.2 Spectral Representation 433.3 Ground State and Convergence to the Boltzmann Distribution 44References 474 Discrete Systems: The Master Equation and Kinetic Monte Carlo 494.1 The Master Equation 494.1.1 Discrete-Time Markov Chains 494.1.2 Continuous-Time Markov Chains, Markov Processes 514.2 Detailed Balance 534.2.1 Final State Only 544.2.2 Initial State Only 544.2.3 Initial and Final State 554.2.4 Metropolis Scheme 554.2.5 Symmetrization 554.3 Kinetic Monte Carlo (KMC) 58References 615 Path Integrals 635.1 The Itô Path Integral 635.2 The Stratonovich Path Integral 66References 676 Barrier Crossing 696.1 First Passage Time and Transition Rate 696.1.1 Average Mean First Passage Time 716.1.2 Distribution of First Passage Time 736.1.3 The Free Particle Case 746.1.4 Conservative Force 756.2 Kramers Transition Time: Average and Distribution 776.2.1 Kramers Derivation 786.2.2 Mean First Passage Time Derivation 806.3 Transition Path Time: Average and Distribution 816.3.1 Transition Path Time Distribution 826.3.2 Mean Transition Path Time 84References 867 Sampling Transition Paths 897.1 Dominant Paths and Instantons 927.1.1 Saddle-Point Method 927.1.2 The Euler-Lagrange Equation: Dominant Paths 927.1.3 Steepest Descent Method 967.1.4 Gradient Descent Method 977.2 Path Sampling 987.2.1 Metropolis Scheme 987.2.2 Langevin Scheme 997.3 Bridge and Conditioning 997.3.1 Free Particle 1027.3.2 The Ornstein-Uhlenbeck Bridge 1027.3.3 Exact Diagonalization 1047.3.4 Cumulant Expansion 105References 111Appendix A: Gaussian Variables 111Appendix B 1138 The Rate of Conformational Change: Definition and Computation 1178.1 First-order Chemical Kinetics 1178.2 Rate Coefficients from Microscopic Dynamics 1198.2.1 Validity of First Order Kinetics 1208.2.2 Mapping Continuous Trajectories onto Discrete Kinetics and Computing Exact Rates 1238.2.3 Computing the Rate More Efficiently 1268.2.4 Transmission Coefficient and Variational Transition State Theory 1288.2.5 Harmonic Transition-State Theory 129References 1319 Zwanzig-Caldeiga-Leggett Model for Low-Dimensional Dynamics 1339.1 Low-Dimensional Models of Reaction Dynamics From a Microscopic Hamiltonian 1339.2 Statistical Properties of the Noise and the Fluctuation-dissipation Theorem 1379.2.1 Ensemble Approach 1389.2.2 Single-Trajectory Approach 1399.3 Time-Reversibility of the Langevin Equation 142References 14510 Escape from a Potential Well in the Case of Dynamics Obeying the Generalized Langevin Equation: General Solution Based on the Zwanzig-Caldeira-Leggett Hamiltonian 14710.1 Derivation of the Escape Rate 14710.2 The Limit of Kramers Theory 15010.3 Significance of Memory Effects 15210.4 Applications of the Kramers Theory to Chemical Kinetics in Condensed Phases, Particularly in Biomolecular Systems 15310.5 A Comment on the Use of the Term “Free Energy” in Application to Chemical Kinetics and Equilibrium 155References 15611 Diffusive Dynamics on a Multidimensional Energy Landscape 15711.1 Generalized Langevin Equation with Exponential Memory can be Derived from a 2D Markov Model 15711.2 Theory of Multidimensional Barrier Crossing 16111.3 Breakdown of the Langer Theory in the Case of Anisotropic Diffusion: the Berezhkovskii-Zitserman Case 167References 17112 Quantum Effects in Chemical Kinetics 17312.1 When is a Quantum Mechanical Description Necessary? 17312.2 How Do the Laws of Quantum Mechanics Affect the Observed Transition Rates? 17412.3 Semiclassical Approximation and the Deep Tunneling Regime 17712.4 Path Integrals, Ring-Polymer Quantum Transition-State Theory, Instantons and Centroids 184References 19113 Computer Simulations of Molecular Kinetics: Foundation 19313.1 Computer Simulations: Statement of Goals 19313.2 The Empirical Energy 19513.3 Molecular States 19713.4 Mean First Passage Time 19913.5 Coarse Variables 19913.6 Equilibrium, Stable, and Metastable States 200References 20214 The Master Equation as a Model for Transitions Between Macrostates 203References 21115 Direct Calculation of Rate Coefficients with Computer Simulations 21315.1 Computer Simulations of Trajectories 21315.2 Calculating Rate with Trajectories 219References 22116 A Simple Numerical Example of Rate Calculations 223References 23117 Rare Events and Reaction Coordinates 233References 24018 Celling 241References 25219 An Example of the Use of Cells: Alanine Dipeptide 255References 257Index 259

Select Guide Rating