696 Index
difference from thermodynamic
integration, 319
difference from WHAM, 346
metadynamics, 357
potential of mean force, 325
profile, 322
surface, alanine dipeptide, 333
van der Waals theory, 175
Hilbert space, 368
Hilbert transform, 555
Holonomic constraint, 32
Homogeneous function, 265
Hybrid Monte Carlo, 297–300
Idempotency, 599
Importance sampling, 287–289
Inertia tensor, 44
Initial conditions
choosing, 101–104
sensitive dependence on, 127–128
Integrator
classical propagator, 108–115
hybrid Monte Carlo, 298–300
isokinetic algorithm, 205–207
Langevin equation, 592–595
Martyna–Tobias–Klein algorithm, 249–254
multiple time step, 118
multiple time-step, 115
nonequilibrium molecular dynamics,
520–524
Nos ́e–Hoover chain algorithm, 197–202
position Verlet, 132
RATTLE, 107–108
shadow Hamiltonian, 121–124
SHAKE, 105–107
Velocity Verlet algorithm, 100–101
Verlet algorithm, 99–100
Interaction picture, 534–535
Ising model
equation of state, 614
Hamiltonian, 614
mean-field treatment, 619–623
one-dimensional, 624–626
phase diagram, 614
renormalization group, 635–641
two-dimensional
graph-theoretic approach, 630–633
transfer matrix approach, 626–630
Isokinetic equations, 203
Jacobi identity, 123
Jarzynski’s equality, 325–333
Ket vector, Dirac, 368
Kramers–Kr ̈onig relations, 555
Lagrange undetermined multipliers, 33, 104,
203, 257, 258, 260, 287, 340, 345
Lagrangian
action integral, 29–32
central potential, 14
constant-pressure molecular dynamics, 233
constrained systems, 32–34, 105
definition, 10
Euclidean, 466
Euler–Lagrange equations, 10
generalized coordinates, 13
Hamiltonian, 19
harmonic oscillator, 11
Legendre transform, 18
momentum, 18
normal mode, 27
path integral, 463
rigid body motion, 39–47, 119
two-body system, 15–16
λfree energy dynamics, 322–325
Langevin equation, 581
generalized, 578
numerical integration, 592–595
Laplace transform, 140, 225, 554, 561, 576,
583, 585, 586, 591, 600, 603,
670–673
convolution theorem, 670
Lattice models, 613
Lees–Edwards boundary conditions, 518
Legendre transform, 16–18
canonical to grand canonical, 267
canonical to isothermal-isobaric, 220
classical Lagrangian, 18
microcanonical to canonical, 136
microcanonical to isoenthalpic-isobaric,
219
Lennard–Jones potential, 126
Lennard–Jones reduced units, 127
Linear response theory
classical, 497–503
Liouville equation
classical, 69–70, 498
perturbative solution, 498–502
equilibrium, 70, 78, 189, 221, 400, 544
generalized, 188
quantum, 399
perturbative solution, 552–554
Liouville operator
adiabatic dynamics, 350
classical, 108, 111, 113, 123, 498, 499, 599
constant-pressure molecular dynamics,
249, 251
hybrid Monte Carlo, 298
isokinetic ensemble, 205
multiple time-scale integration, 116, 201,
253
Nos ́e–Hoover chain, 197, 521
Nos ́e-Hoover chain, 249
quantum, 400
SLLOD, 520
symplectic quaternion, 120
Trotter theorem, 111
Liouville’s theorem, 64–70, 187, 328, 511
Liouville, Joseph, 66