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Simulating.the.Physical.World.Hierarchical.Modeling.from.Quantum.Mechanics.to.
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Preface page xi Symbols, units and constants xv Part I A Modeling Hierarchy for Simulations 1 1 Introduction 3 1.1 What is this book about? 3 1.2 A modeling hierarchy 9 1.3 Trajectories and distributions 13 1.4 Further reading 14 2 Quantum mechanics: principles and relativistic effects 19 2.1 The wave character of particles 19 2.2 Non-relativistic single free particle 23 2.3 Relativistic energy relations for a free particle 25 2.4 Electrodynamic interactions 31 2.5 Fermions, bosons and the parity rule 36 3 From quantum to classical mechanics: when and how 39 3.1 Introduction 39 3.2 From quantum to classical dynamics 42 3.3 Path integral quantum mechanics 44 3.4 Quantum hydrodynamics 64 3.5 Quantum corrections to classical behavior 70 4 Quantum chemistry: solving the time-independent Schr¨odinger equation 77 4.1 Introduction 77 4.2 Stationary solutions of the TDSE 78 4.3 The few-particle problem 79 4.4 The Born–Oppenheimer approximation 97 v vi Contents 4.5 The many-electron problem of quantum chemistry 98 4.6 Hartree–Fock methods 99 4.7 Density functional theory 102 4.8 Excited-state quantum mechanics 105 4.9 Approximate quantum methods 106 4.10 Nuclear quantum states 107 5 Dynamics of mixed quantum/classical systems 109 5.1 Introduction 109 5.2 Quantum dynamics in a non-stationary potential 114 5.3 Embedding in a classical environment 129 6 Molecular dynamics 139 6.1 Introduction 139 6.2 Boundary conditions of the system 140 6.3 Force field descriptions 149 6.4 Solving the equations of motion 189 6.5 Controlling the system 194 6.6 Replica exchange method 204 6.7 Applications of molecular dynamics 207 7 Free energy, entropy and potential of mean force 211 7.1 Introduction 211 7.2 Free energy determination by spatial integration 213 7.3 Thermodynamic potentials and particle insertion 218 7.4 Free energy by perturbation and integration 221 7.5 Free energy and potentials of mean force 227 7.6 Reconstruction of free energy from PMF 231 7.7 Methods to derive the potential of mean force 234 7.8 Free energy from non-equilibrium processes 239 8 Stochastic dynamics: reducing degrees of freedom 249 8.1 Distinguishing relevant degrees of freedom 249 8.2 The generalized Langevin equation 251 8.3 The potential of mean force 255 8.4 Superatom approach 256 8.5 The fluctuation–dissipation theorem 257 8.6 Langevin dynamics 263 8.7 Brownian dynamics 268 8.8 Probability distributions and Fokker–Planck equations 269 8.9 Smart Monte Carlo methods 272 8.10 How to obtain the friction tensor 274 Contents vii 9 Coarse graining from particles to fluid dynamics 279 9.1 Introduction 279 9.2 The macroscopic equations of fluid dynamics 281 9.3 Coarse graining in space 288 9.4 Conclusion 295 10 Mesoscopic continuum dynamics 297 10.1 Introduction 297 10.2 Connection to irreversible thermodynamics 298 10.3 The mean field approach to the chemical potential 301 11 Dissipative particle dynamics 305 11.1 Representing continuum equations by particles 307 11.2 Prescribing fluid parameters 308 11.3 Numerical solutions 309 11.4 Applications 309 Part II Physical and Theoretical Concepts 313 12 Fourier transforms 315 12.1 Definitions 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 13 Electromagnetism 335 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 fields in non-periodic systems 362 13.10 Potentials and fields in periodic systems of charges 362 viii Contents 14 Vectors, operators and vector spaces 379 14.1 Introduction 379 14.2 Definitions 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 15 Lagrangian and Hamiltonian mechanics 397 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 16 Review of thermodynamics 423 16.1 Introduction and history 423 16.2 Definitions 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 17 Review of statistical mechanics 453 17.1 Introduction 453 17.2 Ensembles and the postulates of statistical mechanics 454 17.3 Identification 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 18 Linear response theory 505 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 19 Splines for everything 523 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 |
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