Contents vii

279

9 Coarse graining from particles to ¬‚uid dynamics

9.1 Introduction 279

9.2 The macroscopic equations of ¬‚uid dynamics 281

9.3 Coarse graining in space 288

9.4 Conclusion 295

297

10 Mesoscopic continuum dynamics

10.1 Introduction 297

10.2 Connection to irreversible thermodynamics 298

10.3 The mean ¬eld approach to the chemical potential 301

305

11 Dissipative particle dynamics

11.1 Representing continuum equations by particles 307

11.2 Prescribing ¬‚uid parameters 308

11.3 Numerical solutions 309

11.4 Applications 309

313

Part II Physical and Theoretical Concepts

315

12 Fourier transforms

12.1 De¬nitions and properties 315

12.2 Convolution and autocorrelation 316

12.3 Operators 317

12.4 Uncertainty relations 318

12.5 Examples of functions and transforms 320

12.6 Discrete Fourier transforms 323

12.7 Fast Fourier transforms 324

12.8 Autocorrelation and spectral density from FFT 325

12.9 Multidimensional Fourier transforms 331

335

13 Electromagnetism

13.1 Maxwell™s equation for vacuum 335

13.2 Maxwell™s equation for polarizable matter 336

13.3 Integrated form of Maxwell™s equations 337

13.4 Potentials 337

13.5 Waves 338

13.6 Energies 339

13.7 Quasi-stationary electrostatics 340

13.8 Multipole expansion 353

13.9 Potentials and ¬elds in non-periodic systems 362

13.10 Potentials and ¬elds in periodic systems of charges 362

viii Contents

379

14 Vectors, operators and vector spaces

14.1 Introduction 379

14.2 De¬nitions 380

14.3 Hilbert spaces of wave functions 381

14.4 Operators in Hilbert space 382

14.5 Transformations of the basis set 384

14.6 Exponential operators and matrices 385

14.7 Equations of motion 390

14.8 The density matrix 392

397

15 Lagrangian and Hamiltonian mechanics

15.1 Introduction 397

15.2 Lagrangian mechanics 398

15.3 Hamiltonian mechanics 399

15.4 Cyclic coordinates 400

15.5 Coordinate transformations 401

15.6 Translation and rotation 403

15.7 Rigid body motion 405

15.8 Holonomic constraints 417

423

16 Review of thermodynamics

16.1 Introduction and history 423

16.2 De¬nitions 425

16.3 Thermodynamic equilibrium relations 429

16.4 The second law 432

16.5 Phase behavior 433

16.6 Activities and standard states 435

16.7 Reaction equilibria 437

16.8 Colligative properties 441

16.9 Tabulated thermodynamic quantities 443

16.10 Thermodynamics of irreversible processes 444

453

17 Review of statistical mechanics

17.1 Introduction 453

17.2 Ensembles and the postulates of statistical mechanics 454

17.3 Identi¬cation of thermodynamical variables 457

17.4 Other ensembles 459

17.5 Fermi“Dirac, Bose“Einstein and Boltzmann statistics 463

17.6 The classical approximation 472

17.7 Pressure and virial 479

17.8 Liouville equations in phase space 492

17.9 Canonical distribution functions 497

Contents ix

17.10 The generalized equipartition theorem 502

505

18 Linear response theory

18.1 Introduction 505

18.2 Linear response relations 506

18.3 Relation to time correlation functions 511

18.4 The Einstein relation 518

18.5 Non-equilibrium molecular dynamics 519

523

19 Splines for everything

19.1 Introduction 523

19.2 Cubic splines through points 526

19.3 Fitting splines 530

19.4 Fitting distribution functions 536

19.5 Splines for tabulation 539

19.6 Algorithms for spline interpolation 542

19.7 B-splines 548

References 557

Index 587

Preface

This book was conceived as a result of many years research with students

and postdocs in molecular simulation, and shaped over several courses on

the subject given at the University of Groningen, the Eidgen¨ssische Tech-

o

nische Hochschule (ETH) in Z¨rich, the University of Cambridge, UK, the

u

University of Rome (La Sapienza), and the University of North Carolina

at Chapel Hill, NC, USA. The leading theme has been the truly interdisci-

plinary character of molecular simulation: its gamma of methods and models

encompasses the sciences ranging from advanced theoretical physics to very

applied (bio)technology, and it attracts chemists and biologists with limited

mathematical training as well as physicists, computer scientists and mathe-

maticians. There is a clear hierarchy in models used for simulations, ranging

from detailed (relativistic) quantum dynamics of particles, via a cascade of

approximations, to the macroscopic behavior of complex systems. As the

human brain cannot hold all the specialisms involved, many practical simu-

lators specialize in their niche of interest, adopt “ often unquestioned “ the

methods that are commonplace in their niche, read the literature selectively,

and too often turn a blind eye on the limitations of their approaches.

This book tries to connect the various disciplines and expand the horizon

for each ¬eld of application. The basic approach is a physical one, and an

attempt is made to rationalize each necessary approximation in the light

of the underlying physics. The necessary mathematics is not avoided, but

hopefully remains accessible to a wide audience. It is at a level of abstrac-

tion that allows compact notation and concise reasoning, without the bur-

den of excessive symbolism. The book consists of two parts: Part I follows

the hierarchy of models for simulation from relativistic quantum mechanics

to macroscopic ¬‚uid dynamics; Part II reviews the necessary mathematical,

physical and chemical concepts, which are meant to provide a common back-

ground of knowledge and notation. Some of these topics may be super¬‚uous

xi

xii Preface