A guide to the theoretical and computational toolkits for the modern study of molecular kinetics in condensed phases

Molecular Kinetics in Condensed Phases: Theory, Simulation and Analysis puts the focus on the theory, algorithms, simulations methods and analysis of molecular kinetics in condensed phases. The authors - noted experts on the topic - offer a detailed and thorough description of modern theories and simulation methods to model molecular events. They highlight the rigorous stochastic modelling of molecular processes and the use of mathematical models to reproduce experimental observations, such as rate coefficients, mean first passage times and transition path times.

The book's exploration of simulations examines atomically detailed modelling of molecules in action and the connections of these simulations to theory and experiment. The authors also explore the applications that range from simple intuitive examples of one- and two-dimensional systems to complex solvated macromolecules. This important book:

* Offers an introduction to the topic that combines theory, simulation and analysis

* Presents a guide written by authors that are well-known and highly regarded leaders in their fields

* Contains detailed examples and explanation of how to conduct computer simulations of kinetics. A detailed study of a two-dimensional system and of a solvated peptide are discussed.

* Discusses modern developments in the field and explains their connection to the more traditional concepts in chemical dynamics

Written for students and academic researchers in the fields of chemical kinetics, chemistry, computational statistical mechanics, biophysics and computational biology, Molecular Kinetics in Condensed Phases is the authoritative guide to the theoretical and computational toolkits for the study of molecular kinetics in condensed phases.

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.

Inhalt

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 OrnsteinUhlenbeck 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 FokkerPlanck Equation 29

2.1 The ChapmanKolmogorov Equation 29

2.2 The Overdamped Case 30

2.2.1 Derivation of the Smoluchowski (FokkerPlanck) Equation using the ChapmanKolmogorov Equation 30

2.2.2 Alternative Derivation of the Smoluchowski (FokkerPlanck) Equation 33

2.2.3 The Adjoint (or Reverse or Backward) FokkerPlanck 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 102

7.3.3 Exact Diagonalization 104

7.3.4 Cumulant Expansion 105

References 111

Appendix A: Gaussian Variables 111

Appendix B 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 Traject...