424 Index. Eigenvalue in quantum mechanics, 174 eigenvector in quantum mechanics, 174 Einstein equation, 334, 342, 393

Transcription

1 Index After-effect function, 368, 369 anthropic principle, 232 assumptions nature of, 242 autocorrelation function, 292 average, 18 definition of, 17 ensemble, see ensemble average ideal,23 operational, 23 time, see time average weighted, 16 Axilrod-Teller potential, 228 Barometer equation, 342, 346 baseline, 384 basis vector, 171 change of, 172 choice of, 180 expansion in, 171 bath, 29, 305 black-body radiation, 197,200,202 Rayleigh-Jeans form, 204 Boltzmann, L., 75, 114, 119, 204 Born approximation, 378 Bose-Einstein statistics, 190, 193 Boussinesq equation for drag on sphere, 329 Bragg scattering, 378 Brownian motion, 5, 320, 328, 388, 392, 418 correlations in, 334 mean-square displacement in, 332 Brownian velocity, 415 Brush, S. G., 10 Canonical ensemble, vii, 30, 33, 49, Ill, 120,233,305 stability of, 302, 304, 306 statistical weight. 31 time evolution of, 302 canonical partition function bound on, 231 with chemical reaction, 217 in energy eigenstates, 209 interacting particles, 234 momentum eigenstates, 210 photon gas, 198 Carnot. Sadi, 4 Central Limit Theorem, 320, 321, 382 derivation of, 322 central moment, 67 characteristic function, 322 charging process, 276, 278

2 424 Index chemical potential, 130 from distribution function, 276 of reacting species, 220 Clausius, 4 cluster expansions, 233 cluster integral, 291 evaluation of, 250 from numerical integration, 253 collective modes and hydrodynamics, 329 collision three-body, 237 conductivity electrical, 368 configuration in phase, 36, 111 configuration integral, 229, 404 correlation function, 291 dynamic, 292 higher-order, 298, 301 as scalar product, 350 static, 292 two-point, 294 of vectors, 298 correlation volume dynamic, 383 static, 383 corresponding states law of, 260 Coulomb potential, 282 screened, 287 counterexample reasoning from, 297, 341 cross-correlation function, 292, 298 cumulant expansion using, 390 of scattering spectrum, 394, 414 Debye frequency, 164 Debye length, 284 Debye Model, 160 average energy, 165 specific heat, 163 Debye screening, 287 Debye temperature, 166 Debye--Huckel model, 282 Debye-Huckel potential, 227 Debye-Huckel model, 125 degree of freedom, 100 density of states, 48 density operator classical, 293 one-point, 293 diatomic gas classical, 144 quantum, 146 specific heat of, 141 dice, 18, 23, 25 diffusion coefficient, 321, 333, 339, 388 concentration dependence of, 343 of dimer, 420 experimental of hard spheres, 409 mutual, 344, 389 probe, 389 self, 389 self-,344 translational, 344 direct interaction, 392 direct interactions, 344 distinct terms, 63 distribution function, 22, 264, 291, 402 applications of, 270, 276 average energy from, 271 for Brownian displacements, 336 definition of, 267 energy, 72 generic, 268 joint Gaussian, 382 pair, 295 pressure from, 271 radial, 403 reduced, 46 specific, 267 speed, 52 temperature, 94, 95 time dependent, 366 for total energy, 66 drift velocity, 415, 418 Dulong and Petit Law of, 141, 157, 158, 166 dynamic structure factor, 390, 398, 413 incoherent, 398 Eigenvalue in quantum mechanics, 174 eigenvector in quantum mechanics, 174 Einstein equation, 334, 342, 393

3 Index 425 Einstein Model. 159 partition function of. 160 specific heat. 160 Einstein model. 205 electrolyte solutions. 282 electroneutrality. 283 element chemical. 215 of an ensemble energy average ideal gas. 62 noninteracting spins. 152 single atom. 42 two-atom. 57 mean-square ideal gas. 62 single atom. 43 ensemble canonical. see canonical ensemble constraints on. 29 definition of element of. see element isobaric poly thermal representative surface ensemble average ideal gas. 59 parametric derivatives of. 107 quantum double nature of. 181 quantum corrections to. 186 ensemble variable definition of. 29 entropy. 115 Boltzmann form. 119 is a constant of the motion. 311 of a crystal. 118 Gibbs eorm. 116 Gibbs form. 119 of ideal gas. 118 of mixing. 114 physical nature of. 121 equilibrium chemical physical phase. 73 equilibrium constant calculation of. 220 equilibrium state. 45 equipartition theorem FirstGeneral~ generalized. 104 Second Generalized. 102 ergodic. 317 expectation value. 177 experiment counting. 71 first kind fluctuations in. 64 multiple in quantum mechanics. 175 noise in. 325 null. 71 repeated in quantum mechanics. 175 second kind third kind I function and physical clusters. 251 I-symbol fast variables. 360 Fermi number. 196 Fermi-Dirac statistics Fick's Law of Diffusion Fick's Law of diffusion. 337 field correlation function. 384 fluctuation in displacement of harmonic oscillator. 100 energy and thermodynamic derivatives. 108 light scattering intensity. 378 link to thermodynamic derivatives. 135 number. 134 particle number. 131 photon number. 201 potential energy. 73 pressure. 13 temperature. see temperature fluctuation. 95 time evolution of. 113 fluctuation-dissipation theorem forces long-range. see potential. long-range

4 426 Index Fourier component, 295, 297, 301, 321, 323,349,384 Fourier transform, 254 free energy Helmholtz, see Helmholtz free energy freezing out, 149 friction factor, 331, 341 dynamic, 397, 420 fugacity, 130 g-bond,259 gas diatomic, see diatomic gas nearly ideal, 234 Gaussian integrals, 51 Gibbs paradox, vi, 77,111,114 Gibbs, 1. W., 4, 5, 9, 30, 34, 35, 77,104, 111, 187,213 Gibbs-Duhem relation, 252 glass, 168,421 grand canonical ensemble, 34,114,123, 233,240 fluctuations in, 133 interacting particles, 238 grand canonical partition function, 131, 192,236,240 cluster expansion for, 248 grand canonical potential derivation of, 126 grand potential, 131 graph for B2, 245 for B3, 245 for B4, 246 in cluster expansion, 244 gravity, 32, 228, 282, 288 Hamilton's equations of motion, 304, 308, 310,348 harmonic oscillator classical average energy of, 100 coupled, 161 quantum, 145 Heisenberg uncertainty principle, 187 Heisenberg uncertainty relation, 32, 213 Helmholtz free energy, 31, 111, 130 bound on, 231 extensive, 232 of interacting particles, 260 photon gas, 199 hidden correlations, 388, 413, 418 homodyne coincidence spectroscopy, 409 hydrodynamic drag, 329 hydrodynamic interaction, 392,395 Oseen tensor form, 408 hydrodynamic interactions, 344, 362, 398 Ideal gas, 24 chemically reacting, 220 entropy of, 118 kinetic model, 87 N-atom,59 pressure of, 80 and adsorption, 83 from atom-wall potential, 80 from grand canonical ensemble, 136 from kinetic theory, 84 from partition function, 83 single molecule, 39 static model, 88 two-molecule, 55 ideal gas law from grand canonical ensemble, 239 identical particles, 114 ideon, 203, 207 inner product, 170 integration Monte Carlo, 254. interaction three-particle, 404 invariance inversion, 299 with magnetic field, 299 for pseudovectors, 299 reflection, 361, 416 rotational, 299 time-reversal, 357, 362 lack of, 419 translational, 295, 297, 352, 384 irreversible process, 112, 113 Ising model, 153 isothermal compressibility, 135 Jacobian, 48, 106,308,310,340,346 Katsura's method for ring diagrams, 254

5 Index 427 kinetic model of a liquid, 53 kinetic theory, 84 Kirkwood superposition approximation, 276,280 exact to c l, 280 Kirkwood, J. G., 6, 208, 213, 280, 354 Kirkwood-Wigner theorem, 77, 145,208 Kirkwood-Wigner theorem, 187 Kubo relation, 415 Kubo transform, 370 Kulm, T., 4, 6,10,37,157,187,189,203, 207 kurtosis, 67 Langevin equation, 321, 328, 329, 340, 354,360,391 assumptions in, 330 need for, 339 solution via Central Limit Theorem, 335 Langevin model, see Langevin equation Laplace transform, 349 Law of Equal a Priori Probabilities, 32, 34,93 least-squares fit, 322, 325 light properties of, 377 scattering of, see scattering linear response equation for, 368 linear response theory, 342, 365 quantum, 368 Liouville operator, 348, 366 projected, 356, 361 Liouville theorem, 35 liquid, interactions in, 264 Matrix element quantum, 177 Maxwell-Boltzmann distribution, 43, 86, 273,330,354 McQuarrie, D. A., 9 measurement quantum, 174 mechanical variable, 46, 65, 91,117,352 definition of, 29 fluctuating part, 65 forms vector space, 350 ideal part, 65 memory kernel, 341, 358, 360, 362 micelle, 267 microcanonical ensemble, vii, 32, 33, 93, 120,304 molecular dynamics, see simulation moment, 22 central, 22 momentum eigenstates expansion in, 208 momentum space, 36 Monte Carlo simulation, see simulation Mori-Zwanzig formalism, 342, 347, 371 aftereffect function, 355 assumptions in, 359 frequency matrix, 355 Nonequilibrium state, 45 nonideal gas pressure of, 247 normalization, 17, 170 normalizing factor, 21 Observables, 173 occupation number, 192 operator quantum, 173 oscillator harmonic, 102, 103 quartic, 102, 103 Oseen tensor, 345, 408 Pair approximation, 226 paradox, 413 particles conserved, 190 distinguishable, 129 identical, 75, 76,121,127, 159, 186 quantum noninteracting, 180, 190 interacting, 225 diffusion of, 401 nonconserved, 190 partition function, 87, 111, 291 canonical, 31, 41 ideal gas, 61 if energy separable, 142 manipulations of, 98 quantum diatomic molecule, 146 quantum rotational, 148

6 428 Index partition (continued) quantum vibrational, 147 phase statistical-mechanical, 29 transitions between, 265 phase space, 111, 348 definition, 36 like incompressible fluid, 310 notation, 36 phase space volume conservation of, 311 photon gas, 190, 197 chemical potential of, 199 number fluctuations in, 201 specific heat of, 200 physical cluster, 251 Planck, M., 189,202 Poisson bracket, 348 Poisson process, 327 Poisson-Boltzmann equation nonlinear, 283, 284 analytic solution of, 285 Poisson-Boltzmann equation linearized, 285 position space, 36 potential chemical, see chemical potential Gaussian, 227 hard-sphere, 227, 405 Lennani-Jones, 228 long-range, 38, 132 saturating, 230 saturation of, 132 screened, 125 three-body, 228,244 Yukawa, 227 potential energy, 226 gravity, 125 intennolecular, 124 long-range, 125 pair, 124 screened, 282 three-body, 124 potential of average force, 268, 280, 284 and n-particle potential, 268 power series equality of, 242 power spectrum, 385 pressure average, 13 density expansion, 241 as an ensemble variable, 248 fugacity expansion, 241 gas, 11, 12 as a mechanical variable, 248 osmotic, 247, 251 thennodynamic, 13 probability, 18,20 conditional, 21 density,20 and statistical weight, 18 probe diffusion, see diffusion coefficient, probe process irreversible, 203, 313, 315 absent in a canonical ensemble, 314 reversible, 203, 315 projection, 170 projection operator, 347, 350 definition of, 353 Quantum mechanics, 4 assumptions of, 7 principles of, 169 Random force, 328, 340, 341, 354 Rayleigh scattering, 381 reaction chemical, 216 reaction coordinate, 216 recursion solution by, 212 reduced distribution function, see distribution function, reduced reference frame in hydrodynamics, 406 Reichl, L. E., 10 reversible process, 112 rigid rotator classical, 100 ring diagram, 254 rotator quantized, 145 Sackur-Tetrode equation, 118 saturation, 132, 230 scattering,71,281,295,375 intensity of, 381

7 Index 429 quasielastic, 388, 390 quasielastic light, 376 Rayleigh,381 static, 72 static light, 376 scattering vector, 380 self terms, 63 series nonanalytic, 241 Silverstone-Moats expansion, 256 simple fluid, 227 simulation computer, 23, 26, 30, 156 skewness, 67 sky brightness of, 73 slow variables, 360 Smoluchowski equation, 362 solid crystalline, 157 classical model of, 158 Einstein Model, see Einstein Model specific heat, 72 of classical crystal, 158 Debye Model, 163, 165 Einstein Model of a crystal, 160 noninteracting spins, 152 spectrum light scattering, 363 spin systems, 150 partition function of, 150 state complete set multiple meanings of, 183 complete set of, 171 counting of, 74, 76 most likely, 219 overcounting of, 77 pure and mixed, 184 quantum, 146, 169 complete set of, 186 time dependence of, 176 in statistical mechanics, 29 of a system, 17 states density of, see density of states static model of the gas, 53 stationarity, 303, 316, 331, 414 statistical mechanics assumptions of, 8 quantum, 180 statistical weight, 17,22,29 non-normalized, 17 normalized, 17 in the polythermal ensemble, 93 statistics, see Fermi-Dirac statistics, Bose-Einstein statistics robust, 326 stoichiometric coefficient, 216 Stokes Law, 329, 343, 392 Stokes-Einstein equation, 343 sum rule, 407 superposition approximation Kirkwood, see Kirkwood superposition approximation symmetry, 295, 296, see invariance system, 28 quantum, 142 separable, 225 small,70 spin, see spin systems Temperature average, 95 temperature fluctuation, 91, 92 thermal momentum, 45 thermal wavelength, 229 thermodynamic derivatives from spectroscopic parameters, 145 thermodynamic limit, 72,129,266 thermodynamics, 4 assumptions of, 6 laws of, 115 Third Law of, 120 thermometer, 46, 68, 90, 94 three-body potential, 226 time average, 11, 14,30,316 time correlation function, 315, 355 from ensemble average, 350 intensity, 382 Tolman,R.C.,8,9,32,101,102 trace, 210 in quantum mechanics, 177 translational invariance, 296 Ultraviolet catastrophe, 189,202 Variables

8 430 Index variables (continued) random,18 variance, 67 virial coefficient, 243 hard-sphere, 251 Lennard-Jones potential, 252 virial coefficients, 234 Wavelets, 171,350 Wiener-Khintchine theorem, 386

9 Graduate Texts in Contemporary Physics (continued from page ii) F.T. Vasko and A.V. Kuznetsov: Electronic States and Optical Transitions in Semiconductor Heterostructures A.M. Zagoskin: Quantum Theory of Many-Body Systems: Techniques and Applications