Complex systems that bridge the traditional disciplines of physics, chemistry, biology, and materials science can be studied at an unprecedented level of detail using increasingly sophisticated theoretical methodology and high-speed computers. The aim of this book is to prepare burgeoning users and developers to become active participants in this exciting and rapidly advancing research area by uniting for the first time, in one monograph, the basic concepts of equilibrium and time-dependent statistical mechanics with the modern techniques used to solve the complex problems that arise in real-world applications. The book contains a detailed review of classical and quantum mechanics, in-depth discussions of the most commonly used simultaneously with modern computational techniques such as molecular dynamics and Monte Carlo, and important topics including free-energy calculations, linear-response theory, harmonic baths and the generalized Langevin equation, critical phenomena, and advanced conformational sampling methods. Burgeoning users and developers are thus provided firm grounding to become active participants in this exciting and rapidly advancing research area, while experienced practitioners will find the book to be a useful reference tool for the field.
Author(s): Tuckerman M.
Series: Oxford Graduate Texts
Publisher: OUP
Year: 2010
Language: English
Pages: 713
Contents......Page 12
1.2 Newton’s laws of motion......Page 18
1.3 Phase space: visualizing classical motion......Page 22
1.4 Lagrangian formulation of classical mechanics: A general framework for Newton’s laws......Page 26
1.5 Legendre transforms......Page 33
1.6 Generalized momenta and the Hamiltonian formulation of classical mechanics......Page 34
1.7 A simple classical polymer model......Page 41
1.8 The action integral......Page 45
1.9 Lagrangian mechanics and systems with constraints......Page 48
1.10 Gauss’s principle of least constraint......Page 51
1.11 Rigid body motion: Euler angles and quaterions......Page 53
1.12 Non-Hamiltonian systems......Page 63
1.13 Problems......Page 66
2.1 Overview......Page 70
2.2 The laws of thermodynamics......Page 72
2.3 The ensemble concept......Page 78
2.4 Phase space volumes and Liouville’s theorem......Page 80
2.5 The ensemble distribution function and the Liouville equation......Page 82
2.6 Equilibrium solutions of the Liouville equation......Page 86
2.7 Problems......Page 87
3.1 Brief overview......Page 91
3.2 Basic thermodynamics, Boltzmann’s relation, and the partition function of the microcanonical ensemble......Page 92
3.3 The classical virial theorem......Page 97
3.4 Conditions for thermal equilibrium......Page 100
3.5 The free particle and the ideal gas......Page 103
3.6 The harmonic oscillator and harmonic baths......Page 109
3.7 Introduction to molecular dynamics......Page 112
3.8 Integrating the equations of motion: Finite difference methods......Page 115
3.9 Systems subject to holonomic constraints......Page 120
3.10 The classical time evolution operator and numerical integrators......Page 123
3.11 Multiple time-scale integration......Page 130
3.12 Symplectic integration for quaternions......Page 134
3.13 Exactly conserved time step dependent Hamiltonians......Page 137
3.14 Illustrative examples of molecular dynamics calculations......Page 140
3.15 Problems......Page 146
4.1 Introduction: A different set of experimental conditions......Page 150
4.2 Thermodynamics of the canonical ensemble......Page 151
4.3 The canonical phase space distribution and partition function......Page 152
4.4 Energy fluctuations in the canonical ensemble......Page 157
4.5 Simple examples in the canonical ensemble......Page 159
4.6 Structure and thermodynamics in real gases and liquids from spatial distribution functions......Page 168
4.7 Perturbation theory and the van der Waals equation......Page 183
4.8 Molecular dynamics in the canonical ensemble: Hamiltonian formulation in an extended phase space......Page 194
4.9 Classical non-Hamiltonian statistical mechanics......Page 200
4.10 Nosé–Hoover chains......Page 205
4.11 Integrating the Nosé–Hoover chain equations......Page 211
4.12 The isokinetic ensemble: A simple variant of the canonical ensemble......Page 216
4.13 Applying the canonical molecular dynamics: Liquid structure......Page 221
4.14 Problems......Page 222
5.1 Why constant pressure?......Page 231
5.2 Thermodynamics of isobaric ensembles......Page 232
5.3 Isobaric phase space distributions and partition functions......Page 233
5.4 Pressure and work virial theorems......Page 239
5.5 An ideal gas in the isothermal-isobaric ensemble......Page 241
5.6 Extending of the isothermal-isobaric ensemble: Anisotropic cell fluctuations......Page 242
5.7 Derivation of the pressure tensor estimator from the canonical partition function......Page 245
5.8 Molecular dynamics in the isoenthalpic-isobaric ensemble......Page 250
5.9 Molecular dynamics in the isothermal-isobaric ensemble I: Isotropic volume fluctuations......Page 253
5.10 Molecular dynamics in the isothermal-isobaric ensemble II: Anisotropic cell fluctuations......Page 256
5.11 Atomic and molecular virials......Page 260
5.12 Integrating the MTK equations of motion......Page 262
5.13 The isothermal-isobaric ensemble with constraints: The ROLL algorithm......Page 269
5.14 Problems......Page 274
6.2 Euler’s theorem......Page 278
6.3 Thermodynamics of the grand canonical ensemble......Page 280
6.4 Grand canonical phase space and the partition function......Page 281
6.5 Illustration of the grand canonical ensemble: The ideal gas......Page 287
6.6 Particle number fluctuations in the grand canonical ensemble......Page 288
6.7 Problems......Page 291
7.1 Introduction to the Monte Carlo method......Page 294
7.2 The Central Limit theorem......Page 295
7.3 Sampling distributions......Page 299
7.4 Hybrid Monte Carlo......Page 311
7.5 Replica exchange Monte Carlo......Page 314
7.6 Wang–Landau sampling......Page 318
7.7 Transition path sampling and the transition path ensemble......Page 319
7.8 Problems......Page 326
8.1 Free energy perturbation theory......Page 329
8.2 Adiabatic switching and thermodynamic integration......Page 332
8.3 Adiabatic free energy dynamics......Page 336
8.4 Jarzynski’s equality and nonequilibrium methods......Page 339
8.5 The problem of rare events......Page 347
8.6 Reaction coordinates......Page 348
8.7 The blue moon ensemble approach......Page 350
8.8 Umbrella sampling and weighted histogram methods......Page 357
8.9 Wang–Landau sampling......Page 361
8.10 Adiabatic dynamics......Page 362
8.11 Metadynamics......Page 369
8.12 The committor distribution and the histogram test......Page 373
8.13 Problems......Page 375
9.1 Introduction: Waves and particles......Page 379
9.2 Review of the fundamental postulates of quantum mechanics......Page 381
9.3 Simple examples......Page 394
9.4 Identical particles in quantum mechanics: Spin statistics......Page 400
9.5 Problems......Page 403
10.1 The difficulty of many-body quantum mechanics......Page 408
10.2 The ensemble density matrix......Page 409
10.3 Time evolution of the density matrix......Page 412
10.4 Quantum equilibrium ensembles......Page 413
10.5 Problems......Page 418
11.2 General formulation of the quantum-mechanical ideal gas......Page 422
11.3 An ideal gas of distinguishable quantum particles......Page 426
11.4 General formulation for fermions and bosons......Page 428
11.5 The ideal fermion gas......Page 430
11.6 The ideal boson gas......Page 445
11.7 Problems......Page 455
12.1 Quantum mechanics as a sum over paths......Page 459
12.2 Derivation of path integrals for the canonical density matrix and the time evolution operator......Page 463
12.3 Thermodynamics and expectation values from the path integral......Page 470
12.4 The continuous limit: Functional integrals......Page 475
12.5 Many-body path integrals......Page 484
12.6 Numerical evaluation of path integrals......Page 488
12.7 Problems......Page 504
13.1 Ensembles of driven systems......Page 508
13.2 Driven systems and linear response theory......Page 510
13.3 Applying linear response theory: Green–Kubo relations for transport coefficients......Page 517
13.4 Calculating time correlation functions from molecular dynamics......Page 525
13.5 The nonequilibrium molecular dynamics approach......Page 530
13.6 Problems......Page 540
14.1 Time-dependent systems in quantum mechanics......Page 543
14.2 Time-dependent perturbation theory in quantum mechanics......Page 547
14.3 Time correlation functions and frequency spectra......Page 557
14.4 Examples of frequency spectra......Page 562
14.5 Quantum linear response theory......Page 565
14.6 Approximations to quantum time correlation functions......Page 571
14.7 Problems......Page 581
15.1 The general model of a system plus a bath......Page 585
15.2 Derivation of the generalized Langevin equation......Page 588
15.3 Analytically solvable examples based on the GLE......Page 596
15.4 Vibrational dephasing and energy relaxation in simple fluids......Page 601
15.5 Molecular dynamics with the Langevin equation......Page 604
15.6 Sampling stochastic transition paths......Page 609
15.7 Mori–Zwanzig theory......Page 611
15.8 Problems......Page 617
16.1 Phase transitions and critical points......Page 622
16.2 The critical exponents α, β, γ, and δ......Page 624
16.3 Magnetic systems and the Ising model......Page 625
16.4 Universality classes......Page 630
16.5 Mean-field theory......Page 631
16.6 Ising model in one dimension......Page 637
16.7 Ising model in two dimensions......Page 639
16.8 Spin correlations and their critical exponents......Page 646
16.9 Introduction to the renormalization group......Page 647
16.10 Fixed points of the RG equations in greater than one dimension......Page 654
16.11 General linearized RG theory......Page 656
16.12 Understanding universality from the linearized RG theory......Page 658
16.13 Problems......Page 660
Appendix A: Properties of the Dirac delta-function......Page 666
Appendix B: Evaluation of energies and forces......Page 669
Appendix C: Proof of the Trotter theorem......Page 680
Appendix D: Laplace transforms......Page 683
References......Page 688
C......Page 704
E......Page 705
G......Page 706
H......Page 707
L......Page 708
O......Page 709
P......Page 710
S......Page 711
T......Page 712
Z......Page 713
