Frenkel D. Understanding molecular simulation: from algorithms to applications
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Frenkel D. Understanding molecular simulation: from algorithms to applications / Frenkel D., Smit B. - San Diego: Academic Press, 2002. - 638 p. - (Computational science series; vol.1). - ISBN 0-12-267351-4.
 
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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


Вверх Frenkel D. Understanding molecular simulation: from algorithms to applications / Frenkel D., Smit B. - San Diego: Academic Press, 2002. - 638 p. - (Computational science series; vol.1). - ISBN 0-12-267351-4.

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