Table of Contents
Preface v
1 Introduction 1
1.1 Computational statistical physics: some landmarks 1
1.1.1 Some orders of magnitude 2
1.1.2 Aims of molecular simulation 3
1.2 Microscopic description of physical systems 6
1.2.1 Interactions 6
1.2.2 Dynamics of isolated systems 13
1.2.3 Thermodynamic ensembles 20
1.3 Free energy and its numerical computation 33
1.3.1 Absolute free energy 34
1.3.2 Relative free energies 37
1.3.3 Free energy and metastability 44
1.3.4 Computational techniques 51
1.4 Summary of the mathematical tools and structure of the book 59
2 Sampling methods 61
2.1 Markov chain methods 63
2.1.1 Some background material on the theory of Markov chains 64
2.1.2 The Metropolis-Hastings algorithm 67
2.1.3 Hybrid Monte-Carlo 72
2.1.4 Generalized Metropolis-Hastings variants 74
2.2 Continuous stochastic dynamics 77
2.2.1 Mathematical background on Markovian continuous processes 78
2.2.2 Overdamped Langevin Process 86
2.2.3 Langevin process 88
2.2.4 Overdamped limit of the Langevin dynamics 97
2.3 Convergence of sampling methods 105
2.3.1 Sampling errors 105
2.3.2 Rate of convergence for stochastic processes 113
2.4 Methods for alchemical free energy differences 118
2.4.1 Free energy perturbation 119
2.4.2 Bridge sampling 132
2.5 Histogram methods 138
2.5.1 Principle of histogram methods 138
2.5.2 Extended bridge sampling 142
3 Thermodynamic integration and sampling with constraints 149
3.1 Introduction: The alchemical setting 150
3.1.1 General strategy 150
3.1.2 Numerical application 152
3.2 The reaction coordinate case: configurational space sampling 154
3.2.1 Reaction coordinate and free energy 154
3.2.2 The mean force 163
3.2.3 Sampling measures on submanifolds of Rn 168
3.2.4 Sampling measures on submanifolds of Rn: discretization 180
3.2.5 Computing the mean force 188
3.2.6 On the efficiency of constrained sampling 200
3.3 The reaction coordinate case: Phase space sampling 203
3.3.1 Constrained mechanical systems 204
3.3.2 Phase space measures for constrained systems 209
3.3.3 Hamilton and Poisson formalisms with constraints 219
3.3.4 Constrained Langevin processes 227
3.3.5 Numerical implementation 232
3.3.6 Thermodynamic integration with constrained Langevin processes 242
4 Nonequilibrium methods 259
4.1 The Jarzynski equality in the alchemical case 260
4.1.1 Markovian nonequilibrium simulations 260
4.1.2 Importance weights of nonequilibrium simulations 262
4.1.3 Practical implementation 266
4.1.4 Degeneracy of weights 269
4.1.5 Error analysis 275
4.2 Generalized Jarzynski-Crooks fluctuation identity 284
4.2.1 Derivation of the identity 285
4.2.2 Relationship with standard equalities in the physics and chemistry literature 291
4.2.3 Numerical strategies 293
4.3 Nonequilibrium stochastic methods in the reaction coordinate case 296
4.3.1 Overdamped nonequilibrium dynamics 296
4.3.2 Hamiltonian and Langevin nonequilibrium dynamics 305
4.3.3 Numerical results 323
4.4 Path sampling strategies 324
4.4.1 The path ensemble 324
4.4.2 Sampling switching paths 327
5 Adaptive methods 339
5.1 Adaptive algorithms: A general framework 340
5.1.1 Updating formulas 343
5.1.2 Extended dynamics 350
5.1.3 Discretization methods 353
5.1.4 Classical examples of adaptive methods 365
5.1.5 Numerical illustration 369
5.2 Convergence of the adaptive biasing force method 372
5.2.1 Presentation of the studied ABF dynamics 372
5.2.2 Precise statements of the convergence results 377
5.2.3 Proofs 390
6 Selection 405
6.1 Replica selection framework 407
6.1.1 Weighted replica ensembles 407
6.1.2 Resampling strategies 413
6.1.3 Discrete-time version 419
6.1.4 Numerical application 422
6.2 Selection in adaptive methods 424
6.2.1 Motivation for the selection term 424
6.2.2 Numerical application 428
Appendix A Most important notation used throughout this book 431
A.1 General notation 431
A.2 Physical spaces and energies 433
A.3 Spaces with constraints, projection operators 434
A.4 Measures 436
A.5 Free energy 438
Bibliography 441
Index 455
