Understanding Molecular Simulation (eBook)

Cover 1
Copyright Page 5
Contents 6
Preface to the Second Edition 14
Preface 16
List of Symbols 20
Chapter 1. Introduction 24
Part I: Basics 30
Chapter 2. Statistical Mechanics 32
2.1 Entropy and Temperature 32
2.2 Classical Statistical Mechanics 36
2.3 Questions and Exercises 40
Chapter 3. Monte Carlo Simulations 46
3.1 The Monte Carlo Method 46
3.2 A Basic Monte Carlo Algorithm 54
3.3 Trial Moves 66
3.4 Applications 74
3.5 Questions and Exercises 81
Chapter 4. Molecular Dynamics Simulations 86
4.1 Molecular Dynamics: The Idea 86
4.2 Molecular Dynamics: A Program 87
4.3 Equations of Motion 94
4.4 Computer Experiments 107
4.5 Some Applications 120
4.6 Questions and Exercises 128
Part II: Ensembles 132
Chapter 5. Monte Carlo Simulations in Various Ensembles 134
5.1 General Approach 135
5.2 Canonical Ensemble 135
5.3 Microcanonical Monte Carlo 137
5.4 Isobaric-Isothermal Ensemble 138
5.5 Isotension-Isothermal Ensemble 148
5.6 Grand-Canonical Ensemble 149
5.7 Questions and Exercises 158
Chapter 6. Molecular Dynamics in Various Ensembles 162
6.1 Molecular Dynamics at Constant Temperature 163
6.2 Molecular Dynamics at Constant Pressure 181
6.3 Questions and Exercises 183
Part III: Free Energies and Phase Equilibria 188
Chapter 7. Free Energy Calculations 190
7.1 Thermodynamic Integration 191
7.2 Chemical Potentials 195
7.3 Other Free Energy Methods 206
7.4 Umbrella Sampling 215
7.5 Questions and Exercises 222
Chapter 8. The Gibbs Ensemble 224
8.1 The Gibbs Ensemble Technique 226
8.2 The Partition Function 227
8.3 Monte Carlo Simulations 228
8.4 Applications 243
8.5 Questions and Exercises 246
Chapter 9. Other Methods to Study Coexistence 248
9.1 Semigrand Ensemble 248
9.2 Tracing Coexistence Curves 256
Chapter 10. Free Energies of Solids 264
10.1 Thermodynamic Integration 265
10.2 Free Energies of Solids 266
10.3 Free Energies of Molecular Solids 268
10.4 Vacancies and Interstitials 286
Chapter 11. Free Energy of Chain Molecules 292
11.1 Chemical Potential as Reversible Work 292
11.2 Rosenbluth Sampling 294
Part IV: Advanced Techniques 312
Chapter 12. Long-Range Interactions 314
12.1 Ewald Sums 315
12.2 Fast Multipole Method 329
12.3 Particle Mesh Approaches 333
12.4 Ewald Summation in a Slab Geometry 339
Chapter 13. Biased Monte Carlo Schemes 344
13.1 Biased Sampling Techniques 345
13.2 Chain Molecules 354
13.3 Generation of Trial Orientations 364
13.4 Fixed Endpoints 376
13.5 Beyond Polymers 383
13.6 Other Ensembles 388
13.7 Recoil Growth 397
13.8 Questions and Exercises 406
Chapter 14. Accelerating Monte Carlo Sampling 412
14.1 Parallel Tempering 412
14.2 Hybrid Monte Carlo 420
14.3 Cluster Moves 422
Chapter 15. Tackling Time-Scale Problems 432
15.1 Constraints 433
15.2 On-the-Fly Optimization: Car-Parrinello Approach 444
15.3 Multiple Time Steps 447
Chapter 16. Rare Events 454
16.1 Theoretical Background 455
16.2 Bennett-Chandler Approach 459
16.3 Diffusive Barrier Crossing 466
16.4 Transition Path Ensemble 473
16.5 Searching for the Saddle Point 485
Chapter 17. Dissipative Particle Dynamics 488
17.1 Description of the Technique 489
17.2 Other Coarse-Grained Techniques 499
Part V: Appendices 502
A Lagrangian and Hamiltonian 504
A.1 Lagrangian 506
A.2 Hamiltonian 509
A.3 Hamilton Dynamics and Statistical Mechanics 511
B Non-Hamiltonian Dynamics 518
B.1 Theoretical Background 518
B.2 Non-Hamiltonian Simulation of the N,V,T Ensemble 520
B.3 The N,P,T Ensemble 528
C Linear Response Theory 532
C.1 Static Response 532
C.2 Dynamic Response 534
C.3 Dissipation 536
C.4 Elastic Constants 542
D Statistical Errors 548
D.1 Static Properties: System Size 548
D.2 Correlation Functions 550
D.3 Block Averages 552
E Integration Schemes 556
E.1 Higher-Order Schemes 556
E.2 Nosé-Hoover Algorithms 558
F Saving CPU Time 568
F.1 Verlet List 568
F.2 Cell Lists 573
F.3 Combining the Verlet and Cell Lists 573
F.4 Efficiency 575
G Reference States 582
G.1 Grand-Canonical Ensemble Simulation 582
H Statistical Mechanics of the Gibbs Ensemble 586
H.1 Free Energy of the Gibbs Ensemble 586
H.2 Chemical Potential in the Gibbs Ensemble 593
I Overlapping Distribution for Polymers 596
J Some General Purpose Algorithms 600
K Small Research Projects 604
K.1 Adsorption in Porous Media 604
K.2 Transport Properties in Liquids 605
K.3 Diffusion in a Porous Media 606
K.4 Multiple-Time-Step Integrators 607
K.5 Thermodynamic Integration 608
L Hints for Programming 610
Bibliography 612
Author Index 642
Index 651

Chapter 2 Statistical Mechanics (p. 10-11)

The topic of this book is computer simulation. Computer simulation allows us to study properties of many-particle systems. However, not all properties can be directly measured in a simulation. Conversely, most of the quantities that can be measured in a simulation do not correspond to properties that aremeasured in real experiments. To give a specific example: in aMolecular Dynamics simulation of liquid water, we could measure the instantaneous positions and velocities of all molecules in the liquid. However, this kind of information cannot be compared to experimental data, because no real experiment provides us with such detailed information. Rather, a typical experiment measures an average property, averaged over a large number of particles and, usually, also averaged over the time of the measurement. If we wish to use computer simulation as the numerical counterpart of experiments, we must know what kind of averages we should aim to compute. In order to explain this, we need to introduce the language of statistical mechanics. This we shall do here. We provide the reader with a quick (and slightly dirty) derivation of the basic expressions of statistical mechanics. The aimof these derivations is only to show that there is nothing mysterious about concepts such as phase space, temperature and entropy and many of the other statistical mechanical objects that will appear time and again in the remainder of this book.

2.1 Entropy and Temperature

Most of the computer simulations that we discuss are based on the assumption that classical mechanics can be used to describe the motions of atoms and molecules. This assumption leads to a great simplification in almost all calculations, and it is therefore most fortunate that it is justified in many cases of practical interest. Surprisingly, it turns out to be easier to derive the basic laws of statistical mechanics using the language of quantum mechanics. We will follow this route of least resistance. In fact, for our derivation, we need only little quantum mechanics. Specifically, we need the fact that a quantum mechanical system can be found in different states.