Preface to the Second Edition ...................................... xiii Preface .............................................................. xv List of Symbols ..................................................... xix 1 Introduction ........................................................ 1 Part I Basics ................................... 7 2 Statistical Mechanics ............................................... 9 2.1 Entropy and Temperature ......................................... 9 2.2 Classical Statistical Mechanics ................................ 13 2.2.1 Ergodicity ............................................... 15 2.3 Questions and Exercises ........................................ 17 3 Monte Carlo Simulations ............................................ 23 3.1 The Monte Carlo Method ......................................... 23 3.1.1 Importance Sampling ...................................... 24 3.1.2 The Metropolis Method .................................... 27 3.2 A Basic Monte Carlo Algorithm .................................. 31 3.2.1 The Algorithm ............................................ 31 3.2.2 Technical Details ........................................ 32 3.2.3 Detailed Balance versus Balance .......................... 42 3.3 Trial Moves .................................................... 43 3.3.1 Translational Moves ...................................... 43 3.3.2 Orientational Moves ...................................... 48 3.4 Applications ................................................... 51 3.5 Questions and Exercises ........................................ 58 4 Molecular Dynamics Simulations ..................................... 63 4.1 Molecular Dynamics: The Idea ................................... 63 4.2 Molecular Dynamics: A Program .................................. 64 4.2.1 Initialization ........................................... 65 4.2.2 The Force Calculation .................................... 67 4.2.3 Integrating the Equations of Motion ...................... 69 4.3 Equations of Motion ............................................ 71 4.3.1 Other Algorithms ......................................... 74 4.3.2 Higher-Order Schemes ..................................... 77 4.3.3 Liouville Formulation of Time-Reversible Algorithms ...... 77 4.3.4 Lyapunov Instability ..................................... 81 4.3.5 One More Way to Look at the Verlet Algorithm ............. 82 4.4 Computer Experiments ........................................... 84 4.4.1 Diffusion ................................................ 87 4.4.2 Order-n Algorithm to Measure Correlations ................ 90 4.5 Some Applications .............................................. 97 4.6 Questions and Exercises ....................................... 105 Part II Ensembles ............................. 109 5 Monte Carlo Simulations in Various Ensembles ...................... 111 5.1 General Approach .............................................. 112 5.2 Canonical Ensemble ............................................ 112 5.2.1 Monte Carlo Simulations ................................. 113 5.2.2 Justification of the Algorithm .......................... 114 5.3 Microcanonical Monte Carlo .................................... 114 5.4 Isobaric-Isothermal Ensemble .................................. 115 5.4.1 Statistical Mechanical Basis ............................ 116 5.4.2 Monte Carlo Simulations ................................. 119 5.4.3 Applications ............................................ 122 5.5 Isotension-Isothermal Ensemble ................................ 125 5.6 Grand-Canonical Ensemble ...................................... 126 5.6.1 Statistical Mechanical Basis ............................ 127 5.6.2 Monte Carlo Simulations ................................. 130 5.6.3 Justification of the Algorithm .......................... 130 5.6.4 Applications ............................................ 133 5.7 Questions and Exercises ....................................... 135 6 Molecular Dynamics in Various Ensembles ........................... 139 6.1 Molecular Dynamics at Constant Temperature .................... 140 6.1.1 The Andersen Thermostat ................................. 141 6.1.2 Nose-Hoover Thermostat .................................. 147 6.1.3 Nose-Hoover Chains ...................................... 155 6.2 Molecular Dynamics at Constant Pressure ....................... 158 6.3 Questions and Exercises ....................................... 160 Part III Free Energies and Phase Equilibria ... 165 7 Free Energy Calculations .......................................... 167 7.1 Thermodynamic Integration ..................................... 168 7.2 Chemical Potentials ........................................... 172 7.2.1 The Particle Insertion Method ........................... 173 7.2.2 Other Ensembles ......................................... 176 7.2.3 Overlapping Distribution Method ......................... 179 7.3 Other Free Energy Methods ..................................... 183 7.3.1 Multiple Histograms ..................................... 183 7.3.2 Acceptance Ratio Method ................................. 189 7.4 Umbrella Sampling ............................................. 192 7.4.1 Nonequilibrium Free Energy Methods ...................... 196 7.5 Questions and Exercises ....................................... 199 8 The Gibbs Ensemble ................................................ 201 8.1 The Gibbs Ensemble Technique .................................. 203 8.2 The Partition Function ........................................ 204 8.3 Monte Carlo Simulations ....................................... 205 8.3.1 Particle Displacement ................................... 205 8.3.2 Volume Change ........................................... 206 8.3.3 Particle Exchange ....................................... 208 8.3.4 Implementation .......................................... 208 8.3.5 Analyzing the Results ................................... 214 8.4 Applications .................................................. 220 8.5 Questions and Exercises ....................................... 223 9 Other Methods to Study Coexistence ................................ 225 9.1 Semigrand Ensemble ............................................ 225 9.2 Tracing Coexistence Curves .................................... 233 10 Free Energies of Solids .......................................... 241 10.1 Thermodynamic Integration ................................... 242 10.2 Free Energies of Solids ..................................... 243 10.2.1 Atomic Solids with Continuous Potentials ............ 244 10.3 Free Energies of Molecular Solids ........................... 245 10.3.1 Atomic Solids with Discontinuous Potentials .......... 248 10.3.2 General Implementation Issues ........................ 249 10.4 Vacancies and Interstitials ................................. 263 10.4.1 Free Energies ........................................ 263 10.4.2 Numerical Calculations ............................... 266 11 Free Energy of Chain Molecules ................................... 269 11.1 Chemical Potential as Reversible Work ....................... 269 11.2 Rosenbluth Sampling ......................................... 271 11.2.1 Macromolecules with Discrete Conformations ........... 271 11.2.2 Extension to Continuously Deformable Molecules ....... 276 11.2.3 Overlapping Distribution Rosenbluth Method ........... 282 11.2.4 Recursive Sampling ................................... 283 11.2.5 Pruned-Enriched Rosenbluth Method .................... 285 Part IV Advanced Techniques ................... 289 12 Long-Range Interactions .......................................... 291 12.1 Ewald Sums .................................................. 292 12.1.1 Point Charges ........................................ 292 12.1.2 Dipolar Particles .................................... 300 12.1.3 Dielectric Constant .................................. 301 12.1.4 Boundary Conditions .................................. 303 12.1.5 Accuracy and Computational Complexity ................ 304 12.2 Fast Multipole Method ....................................... 306 12.3 Particle Mesh Approaches .................................... 310 12.4 Ewald Summation in a Slab Geometry .......................... 316 13 Biased Monte Carlo Schemes ....................................... 321 13.1 Biased Sampling Techniques .................................. 322 13.1.1 Beyond Metropolis .................................... 323 13.1.2 Orientational Bias ................................... 323 13.2 Chain Molecules ............................................. 331 13.2.1 Configurational-Bias Monte Carlo ..................... 331 13.2.2 Lattice Models ....................................... 332 13.2.3 Off-lattice Case ..................................... 336 13.3 Generation of Trial Orientations ............................ 341 13.3.1 Strong Intramolecular Interactions ................... 342 13.3.2 Generation of Branched Molecules ..................... 350 13.4 Fixed Endpoints ............................................. 353 13.4.1 Lattice Models ....................................... 353 13.4.2 Fully Flexible Chain ................................. 355 13.4.3 Strong Intramolecular Interactions ................... 357 13.4.4 Rebridging Monte Carlo ............................... 357 13.5 Beyond Polymers ............................................. 360 13.6 Other Ensembles ............................................. 365 13.6.1 Grand-Canonical Ensemble ............................. 365 13.6.2 Gibbs Ensemble Simulations ........................... 370 13.7 Recoil Growth ............................................... 374 13.7.1 Algorithm ............................................ 376 13.7.2 Justification of the Method .......................... 379 13.8 Questions and Exercises ..................................... 383 14 Accelerating Monte Carlo Sampling ................................ 389 14.1 Parallel Tempering .......................................... 389 14.2 Hybrid Monte Carlo .......................................... 397 14.3 Cluster Moves ............................................... 399 14.3.1 Clusters ............................................. 399 14.3.2 Early Rejection Scheme ............................... 405 15 Tackling Time-Scale Problems ..................................... 409 15.1 Constraints ................................................. 410 15.1.1 Constrained and Unconstrained Averages .............. 415 15.2 On-the-Fly Optimization: Car-Parrinello Approach ............ 421 15.3 Multiple Time Steps ......................................... 424 16 Rare Events ...................................................... 431 16.1 Theoretical Background ...................................... 432 16.2 Bennett-Chandler Approach ................................... 436 16.2.1 Computational Aspects ................................ 438 16.3 Diffusive Barrier Crossing .................................. 443 16.4 Transition Path Ensemble .................................... 450 16.4.1 Path Ensemble ........................................ 451 16.4.2 Monte Carlo Simulations .............................. 454 16.5 Searching for the Saddle Point .............................. 462 17 Dissipative Particle Dynamics .................................... 465 17.1 Description of the Technique ................................ 466 17.1.1 Justification of the Method .......................... 467 17.1.2 Implementation of the Method ......................... 469 17.1.3 DPD and Energy Conservation .......................... 473 17.2 Other Coarse-Grained Techniques ............................. 476 Part V Appendices ............................. 479 A Lagrangian and Hamiltonian ........................................ 481 A.1 Lagrangian .................................................... 483 A.2 Hamiltonian ................................................... 486 A.3 Hamilton Dynamics and Statistical Mechanics ................... 488 A.3.1 Canonical Transformation ................................ 489 A.3.2 Symplectic Condition .................................... 490 A.3.3 Statistical Mechanics ................................... 492 B Non-Hamiltonian Dynamics .......................................... 495 B.1 Theoretical Background ....................................... 495 B.2 Non-Hamiltonian Simulation of the N,V,T Ensemble ............. 497 B.2.1 The Nose-Hoover Algorithm .............................. 498 B.2.2 Nose-Hoover Chains ..................................... 502 B.3 The N,P,T Ensemble ........................................... 505 C Linear Response Theory ............................................ 509 C.1 Static Response ............................................... 509 C.2 Dynamic Response .............................................. 511 C.3 Dissipation ................................................... 513 C.3.1 Electrical Conductivity ................................. 516 C.3.2 Viscosity ............................................... 518 C.4 Elastic Constants ............................................. 519 D Statistical Errors ................................................ 525 D.1 Static Properties: System Size ................................ 525 D.2 Correlation Functions ......................................... 527 D.3 Block Averages ................................................ 529 E Integration Schemes ............................................... 533 E.1 Higher-Order Schemes .......................................... 533 E.2 Nose-Hoover Algorithms ........................................ 535 E.2.1 Canonical Ensemble ...................................... 536 E.2.2 The Isothermal-Isobaric Ensemble ........................ 540 F Saving CPU Time ................................................... 545 F.1 Verlet List ................................................... 545 F.2 Cell Lists .................................................... 550 F.3 Combining the Verlet and Cell Lists ........................... 550 F.4 Efficiency .................................................... 552 G Reference States .................................................. 559 G.1 Grand-Canonical Ensemble Simulation ........................... 559 H Statistical Mechanics of the Gibbs "Ensemble" ..................... 563 H.1 Free Energy of the Gibbs Ensemble ............................. 563 H.1.1 Basic Definitions ....................................... 563 H.1.2 Free Energy Density ..................................... 565 H.2 Chemical Potential in the Gibbs Ensemble ...................... 570 I Overlapping Distribution for Polymers ............................. 573 J Some General Purpose Algorithms ................................... 577 K Small Research Projects ........................................... 581 K.1 Adsorption in Porous Media .................................... 581 K.2 Transport Properties in Liquids ............................... 582 K.3 Diffusion in a Porous Media ................................... 583 K.4 Multiple-Time-Step Integrators ................................ 584 K.5 Thermodynamic Integration ..................................... 585 L Hints for Programming ............................................. 587 Bibliography ........................................................ 589 Author Index ........................................................ 619 Index ............................................................... 628