| Preface |
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xv | |
| Acknowledgments |
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xix | |
| Author |
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xxi | |
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Section I Probability Theory |
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1 Probability and Its Applications |
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3 | (30) |
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3 | (1) |
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1.2 Experimental Probability |
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3 | (1) |
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1.3 The Sample Space Is Related to the Experiment |
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4 | (1) |
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1.4 Elementary Probability Space |
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5 | (1) |
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6 | (3) |
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6 | (1) |
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7 | (2) |
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1.6 Product Probability Spaces |
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9 | (3) |
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1.6.1 The Binomial Distribution |
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11 | (1) |
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11 | (1) |
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1.7 Dependent and Independent Events |
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12 | (1) |
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12 | (1) |
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1.8 Discrete Probability--Summary |
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13 | (1) |
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1.9 One-Dimensional Discrete Random Variables |
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13 | (1) |
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1.9.1 The Cumulative Distribution Function |
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14 | (1) |
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1.9.2 The Random Variable of the Poisson Distribution |
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14 | (1) |
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1.10 Continuous Random Variables |
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14 | (2) |
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1.10.1 The Normal Random Variable |
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15 | (1) |
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1.10.2 The Uniform Random Variable |
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15 | (1) |
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1.11 The Expectation Value |
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16 | (1) |
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16 | (1) |
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17 | (2) |
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1.12.1 The Variance of the Poisson Distribution |
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18 | (1) |
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1.12.2 The Variance of the Normal Distribution |
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18 | (1) |
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1.13 Independent and Uncorrelated Random Variables |
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19 | (1) |
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19 | (1) |
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1.14 The Arithmetic Average |
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20 | (1) |
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1.15 The Central Limit Theorem |
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21 | (2) |
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23 | (1) |
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1.17 Stochastic Processes--Markov Chains |
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23 | (3) |
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1.17.1 The Stationary Probabilities |
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25 | (1) |
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26 | (1) |
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1.19 Autocorrelation Functions |
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27 | (1) |
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1.19.1 Stationary Stochastic Processes |
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28 | (1) |
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28 | (1) |
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A Comment about Notations |
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28 | (1) |
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29 | (4) |
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Section II Equilibrium Thermodynamics and Statistical Mechanics |
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2 Classical Thermodynamics |
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33 | (18) |
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33 | (1) |
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2.2 Macroscopic Mechanical Systems versus Thermodynamic Systems |
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33 | (1) |
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2.3 Equilibrium and Reversible Transformations |
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34 | (1) |
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2.4 Ideal Gas Mechanical Work and Reversibility |
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34 | (2) |
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2.5 The First Law of Thermodynamics |
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36 | (1) |
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37 | (2) |
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39 | (1) |
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2.8 The Second Law of Thermodynamics |
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40 | (3) |
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2.8.1 Maximal Entropy in an Isolated System |
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41 | (1) |
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2.8.2 Spontaneous Expansion of an Ideal Gas and Probability |
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42 | (1) |
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2.8.3 Reversible and Irreversible Processes Including Work |
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42 | (1) |
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2.9 The Third Law of Thermodynamics |
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43 | (1) |
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2.10 Thermodynamic Potentials |
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43 | (4) |
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2.10.1 The Gibbs Relation |
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43 | (1) |
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2.10.2 The Entropy as the Main Potential |
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44 | (1) |
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45 | (1) |
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2.10.4 The Helmholtz Free Energy |
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45 | (1) |
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2.10.5 The Gibbs Free Energy |
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45 | (1) |
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2.10.6 The Free Energy, H (T, μ) |
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46 | (1) |
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2.11 Maximal Work in Isothermal and Isobaric Transformations |
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47 | (1) |
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2.12 Euler's Theorem and Additional Relations for the Free Energies |
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48 | (1) |
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2.12.1 Gibbs-Duhem Equation |
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49 | (1) |
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49 | (1) |
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49 | (1) |
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49 | (1) |
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49 | (2) |
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3 From Thermodynamics to Statistical Mechanics |
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51 | (8) |
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3.1 Phase Space as a Probability Space |
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51 | (1) |
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3.2 Derivation of the Boltzmann Probability |
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52 | (2) |
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3.3 Statistical Mechanics Averages |
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54 | (2) |
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54 | (1) |
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3.3.2 The Average Entropy |
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54 | (1) |
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3.3.3 The Helmholtz Free Energy |
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55 | (1) |
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3.4 Various Approaches for Calculating Thermodynamic Parameters |
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56 | (1) |
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3.4.1 Thermodynamic Approach |
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56 | (1) |
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3.4.2 Probabilistic Approach |
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56 | (1) |
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3.5 The Helmholtz Free Energy of a Simple Fluid |
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57 | (1) |
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58 | (1) |
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58 | (1) |
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4 Ideal Gas and the Harmonic Oscillator |
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59 | (14) |
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4.1 From a Free Particle in a Box to an Ideal Gas |
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59 | (1) |
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4.2 Properties of an Ideal Gas by the Thermodynamic Approach |
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60 | (2) |
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4.3 The chemical potential of an Ideal Gas |
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62 | (1) |
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4.4 Treating an Ideal Gas by the Probability Approach |
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63 | (1) |
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4.5 The Macroscopic Harmonic Oscillator |
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64 | (1) |
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4.6 The Microscopic Oscillator |
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65 | (3) |
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4.6.1 Partition Function and Thermodynamic Properties |
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66 | (2) |
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4.7 The Quantum Mechanical Oscillator |
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68 | (3) |
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4.8 Entropy and Information in Statistical Mechanics |
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71 | (1) |
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4.9 The Configurational Partition Function |
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71 | (1) |
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72 | (1) |
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72 | (1) |
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72 | (1) |
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5 Fluctuations and the Most Probable Energy |
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73 | (10) |
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5.1 The Variances of the Energy and the Free Energy |
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73 | (1) |
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5.2 The Most Contributing Energy E* |
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74 | (2) |
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5.3 Solving Problems in Statistical Mechanics |
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76 | (5) |
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5.3.1 The Thermodynamic Approach |
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77 | (1) |
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5.3.2 The Probabilistic Approach |
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78 | (1) |
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5.3.3 Calculating the Most Probable Energy Term |
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79 | (1) |
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5.3.4 The Change of Energy and Entropy with Temperature |
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80 | (1) |
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81 | (2) |
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83 | (10) |
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6.1 The Microcanonical (petit) Ensemble |
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83 | (1) |
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6.2 The Canonical (NVT) Ensemble |
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84 | (1) |
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6.3 The Gibbs (NpT) Ensemble |
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85 | (3) |
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6.4 The Grand Canonical (uVT) Ensemble |
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88 | (2) |
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6.5 Averages and Variances in Different Ensembles |
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90 | (2) |
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6.5.1 A Canonical Ensemble Solution (Maximal Term Method) |
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90 | (1) |
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6.5.2 A Grand-Canonical Ensemble Solution |
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91 | (1) |
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6.5.3 Fluctuations in Different Ensembles |
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91 | (1) |
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92 | (1) |
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92 | (1) |
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93 | (6) |
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7.1 Finite Systems versus the Thermodynamic Limit |
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93 | (1) |
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7.2 First-Order Phase Transitions |
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94 | (1) |
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7.3 Second-Order Phase Transitions |
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95 | (3) |
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98 | (1) |
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99 | (12) |
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8.1 Models of Macromolecules |
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99 | (1) |
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8.2 Statistical Mechanics of an Ideal Chain |
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99 | (2) |
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8.2.1 Partition Function and Thermodynamic Averages |
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100 | (1) |
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8.3 Entropic Forces in an One-Dimensional Ideal Chain |
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101 | (3) |
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8.4 The Radius of Gyration |
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104 | (1) |
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8.5 The Critical Exponent v |
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105 | (1) |
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8.6 Distribution of the End-to-End Distance |
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106 | (2) |
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8.6.1 Entropic Forces Derived from the Gaussian Distribution |
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107 | (1) |
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8.7 The Distribution of the End-to-End Distance Obtained from the Central Limit Theorem |
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108 | (1) |
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8.8 Ideal Chains and the Random Walk |
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109 | (1) |
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8.9 Ideal Chain as a Model of Reality |
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110 | (1) |
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110 | (1) |
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9 Chains with Excluded Volume |
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111 | (12) |
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9.1 The Shape Exponent v for Self-avoiding Walks |
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111 | (1) |
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9.2 The Partition Function |
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112 | (1) |
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9.3 Polymer Chain as a Critical System |
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113 | (1) |
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9.4 Distribution of the End-to-End Distance |
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114 | (1) |
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9.5 The Effect of Solvent and Temperature on the Chain Size |
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115 | (4) |
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116 | (1) |
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116 | (1) |
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9.5.3 The Crossover Behavior Around G |
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117 | (1) |
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118 | (1) |
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119 | (1) |
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119 | (4) |
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Section III Topics in Non-Equilibrium Thermodynamics and Statistical Mechanics |
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10 Basic Simulation Techniques: Metropolis Monte Carlo and Molecular Dynamics |
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123 | (16) |
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123 | (1) |
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10.2 Sampling the Energy and Entropy and New Notations |
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124 | (1) |
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10.3 More About Importance Sampling |
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125 | (1) |
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10.4 The Metropolis Monte Carlo Method |
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126 | (3) |
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10.4.1 Symmetric and Asymmetric MC Procedures |
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127 | (1) |
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10.4.2 A Grand-Canonical MC Procedure |
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128 | (1) |
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10.5 Efficiency of Metropolis MC |
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129 | (2) |
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10.6 Molecular Dynamics in the Microcanonical Ensemble |
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131 | (3) |
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10.7 MD Simulations in the Canonical Ensemble |
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134 | (1) |
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10.8 Dynamic MD Calculations |
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135 | (1) |
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135 | (2) |
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10.9.1 Periodic Boundary Conditions and Ewald Sums |
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136 | (1) |
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10.9.2 A Comment About MD Simulations and Entropy |
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136 | (1) |
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137 | (2) |
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11 Non-Equilibrium Thermodynamics--Onsager Theory |
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139 | (14) |
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139 | (1) |
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11.2 The Local-Equilibrium Hypothesis |
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139 | (1) |
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11.3 Entropy Production Due to Heat Flow in a Closed System |
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140 | (1) |
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11.4 Entropy Production in an Isolated System |
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141 | (1) |
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11.5 Extra Hypothesis: A Linear Relation Between Rates and Affinities |
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142 | (2) |
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11.5.1 Entropy of an Ideal Linear Chain Close to Equilibrium |
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143 | (1) |
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11.6 Fourier's Law--A Continuum Example of Linearity |
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144 | (1) |
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11.7 Statistical Mechanics Picture of Irreversibility |
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145 | (2) |
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11.8 Time Reversal, Microscopic Reversibility, and the Principle of Detailed Balance |
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147 | (2) |
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11.9 Onsager's Reciprocal Relations |
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149 | (1) |
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150 | (1) |
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11.11 Steady States and the Principle of Minimum Entropy Production |
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151 | (1) |
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152 | (1) |
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152 | (1) |
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12 Non-equilibrium Statistical Mechanics |
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153 | (24) |
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12.1 Fick's Laws for Diffusion |
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153 | (5) |
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153 | (1) |
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12.1.2 Calculation of the Flux from Thermodynamic Considerations |
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154 | (1) |
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12.1.3 The Continuity Equation |
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155 | (1) |
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12.1.4 Second Fick's Law--The Diffusion Equation |
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156 | (1) |
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12.1.5 Diffusion of Particles Through a Membrane |
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156 | (1) |
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156 | (2) |
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12.2 Brownian Motion: Einstein's Derivation of the Diffusion Equation |
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158 | (2) |
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160 | (9) |
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12.3.1 The Average Velocity and the Fluctuation-Dissipation Theorem |
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162 | (1) |
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12.3.2 Correlation Functions |
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163 | (1) |
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12.3.3 The Displacement of a Langevin Particle |
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164 | (2) |
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12.3.4 The Probability Distributions of the Velocity and the Displacement |
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166 | (2) |
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12.3.5 Langevin Equation with a Charge in an Electric Field |
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168 | (1) |
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12.3.6 Langevin Equation with an External Force--The Strong Damping Velocity |
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168 | (1) |
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12.4 Stochastic Dynamics Simulations |
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169 | (2) |
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12.4.1 Generating Numbers from a Gaussian Distribution by CLT |
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170 | (1) |
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12.4.2 Stochastic Dynamics versus Molecular Dynamics |
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171 | (1) |
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12.5 The Fokker-Planck Equation |
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171 | (3) |
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12.6 Smoluchowski Equation |
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174 | (1) |
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12.7 The Fokker-Planck Equation for a Full Langevin Equation with a Force |
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175 | (1) |
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12.8 Summary of Pairs of Equations |
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175 | (1) |
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176 | (1) |
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177 | (12) |
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13.1 Master Equation in a Microcanonical System |
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177 | (1) |
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13.2 Master Equation in the Canonical Ensemble |
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178 | (2) |
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13.3 An Example from Magnetic Resonance |
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180 | (5) |
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13.3.1 Relaxation Processes Under Various Conditions |
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181 | (3) |
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13.3.2 Steady State and the Rate of Entropy Production |
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184 | (1) |
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13.4 The Principle of Minimum Entropy Production--Statistical Mechanics Example |
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185 | (1) |
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186 | (3) |
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Section IV Advanced Simulation Methods: Polymers and Biological Macromolecules |
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14 Growth Simulation Methods for Polymers |
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189 | (22) |
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14.1 Simple Sampling of Ideal Chains |
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189 | (1) |
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14.2 Simple Sampling of SAWs |
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190 | (2) |
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14.3 The Enrichment Method |
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192 | (1) |
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14.4 The Rosenbluth and Rosenbluth Method |
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193 | (2) |
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195 | (11) |
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14.5.1 The Complete Scanning Method |
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195 | (1) |
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14.5.2 The Partial Scanning Method |
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196 | (1) |
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14.5.3 Treating SAWs with Finite Interactions |
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197 | (1) |
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14.5.4 A Lower Bound for the Entropy |
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197 | (1) |
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14.5.5 A Mean-Field Parameter |
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198 | (1) |
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14.5.6 Eliminating the Bias by Schmidt's Procedure |
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199 | (1) |
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14.5.7 Correlations in the Accepted Sample |
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200 | (1) |
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14.5.8 Criteria for Efficiency |
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201 | (1) |
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14.5.9 Locating Transition Temperatures |
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202 | (1) |
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14.5.10 The Scanning Method versus Other Techniques |
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203 | (1) |
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14.5.11 The Stochastic Double Scanning Method |
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204 | (1) |
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14.5.12 Future Scanning by Monte Carlo |
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204 | (1) |
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14.5.13 The Scanning Method for the Ising Model and Bulk Systems |
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205 | (1) |
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14.6 The Dimerization Method |
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206 | (2) |
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208 | (3) |
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15 The Pivot Algorithm and Hybrid Techniques |
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211 | (6) |
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15.1 The Pivot Algorithm--Historical Notes |
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211 | (1) |
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15.2 Ergodicity and Efficiency |
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211 | (1) |
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212 | (1) |
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15.4 Hybrid and Grand-Canonical Simulation Methods |
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213 | (1) |
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214 | (1) |
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214 | (3) |
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217 | (14) |
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16.1 Biological Macromolecules versus Polymers |
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217 | (1) |
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16.2 Definition of a Protein Chain |
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217 | (1) |
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16.3 The Force Field of a Protein |
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218 | (1) |
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16.4 Implicit Solvation Models |
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219 | (1) |
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16.5 A Protein in an Explicit Solvent |
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220 | (1) |
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16.6 Potential Energy Surface of a Protein |
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221 | (1) |
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16.7 The Problem of Protein Folding |
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222 | (1) |
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16.8 Methods for a Conformational Search |
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222 | (3) |
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16.8.1 Local Minimization--The Steepest Descents Method |
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223 | (1) |
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16.8.2 Monte Carlo Minimization |
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224 | (1) |
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16.8.3 Simulated Annealing |
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225 | (1) |
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16.9 Monte Carlo and Molecular Dynamics Applied to Proteins |
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225 | (1) |
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16.10 Microstates and Intermediate Flexibility |
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226 | (1) |
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16.10.1 On the Practical Definition of a Microstate |
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227 | (1) |
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227 | (4) |
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17 Calculation of the Entropy and the Free Energy by Thermodynamic Integration |
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231 | (12) |
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17.1 "Calorimetric" Thermodynamic Integration |
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232 | (1) |
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17.2 The Free Energy Perturbation Formula |
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232 | (2) |
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17.3 The Thermodynamic Integration Formula of Kirkwood |
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234 | (1) |
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235 | (2) |
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17.4.1 Absolute Entropy of a SAW Integrated from an Ideal Chain Reference State |
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235 | (2) |
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17.4.2 Harmonic Reference State of a Peptide |
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237 | (1) |
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17.5 Thermodynamic Cycles |
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237 | (4) |
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240 | (1) |
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17.5.2 Problems of TI and FEP Applied to Proteins |
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240 | (1) |
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241 | (2) |
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18 Direct Calculation of the Absolute Entropy and Free Energy |
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243 | (8) |
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18.1 Absolute Free Energy from <exp[ +ElkT]> |
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243 | (1) |
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18.2 The Harmonic Approximation |
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244 | (1) |
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245 | (1) |
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18.4 The Quasi-Harmonic Approximation |
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246 | (1) |
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18.5 The Mutual Information Expansion |
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247 | (1) |
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18.6 The Nearest Neighbor Technique |
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248 | (1) |
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249 | (1) |
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249 | (1) |
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249 | (2) |
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19 Calculation of the Absolute Entropy from a Single Monte Carlo Sample |
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251 | (26) |
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19.1 The Hypothetical Scanning (HS) Method for SAWs |
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251 | (2) |
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19.1.1 An Exact HS Method |
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251 | (1) |
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19.1.2 Approximate HS Method |
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252 | (1) |
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19.2 The HS Monte Carlo (HSMC) Method |
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253 | (2) |
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19.3 Upper Bounds and Exact Functionals for the Free Energy |
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255 | (5) |
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19.3.1 The Upper Bound FB |
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255 | (1) |
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19.3.2 FB Calculated by the Reversed Schmidt Procedure |
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256 | (1) |
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19.3.3 A Gaussian Estimation of FB |
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257 | (1) |
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19.3.4 Exact Expression for the Free Energy |
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258 | (1) |
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19.3.5 The Correlation Between oA and FA |
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258 | (1) |
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19.3.6 Entropy Results for SAWs on a Square Lattice |
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259 | (1) |
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19.4 HS and HSMC Applied to the Ising Model |
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260 | (1) |
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19.5 The HS and HSMC Methods for a Continuum Fluid |
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261 | (5) |
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261 | (1) |
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262 | (2) |
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19.5.3 Results for Argon and Water |
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264 | (1) |
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19.5.3.1 Results for Argon |
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264 | (2) |
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19.5.3.2 Results for Water |
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|
266 | (1) |
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19.6 HSMD Applied to a Peptide |
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266 | (3) |
|
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|
269 | (1) |
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|
269 | (1) |
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|
270 | (4) |
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19.8.1 The LS Method Applied to the Ising Model |
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270 | (2) |
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19.8.2 The LS Method Applied to a Peptide |
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272 | (2) |
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274 | (3) |
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20 The Potential of Mean Force, Umbrella Sampling, and Related Techniques |
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277 | (24) |
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|
277 | (1) |
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20.2 Bennett's Acceptance Ratio |
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278 | (3) |
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20.3 The Potential of Mean Force |
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281 | (4) |
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|
284 | (1) |
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20.4 The Self-Consistent Histogram Method |
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285 | (4) |
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20.4.1 Free Energy from a Single Simulation |
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|
286 | (1) |
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20.4.2 Multiple Simulations and The Self-Consistent Procedure |
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286 | (3) |
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20.5 The Weighted Histogram Analysis Method |
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289 | (8) |
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20.5.1 The Single Histogram Equations |
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|
290 | (1) |
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20.5.2 The WHAM Equations |
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|
291 | (2) |
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20.5.3 Enhancements of WHAM |
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293 | (2) |
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20.5.4 The Basic MBAR Equation |
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295 | (1) |
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|
296 | (1) |
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|
296 | (1) |
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|
297 | (4) |
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21 Advanced Simulation Methods and Free Energy Techniques |
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|
301 | (30) |
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|
301 | (7) |
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21.1.1 Temperature-Based REM |
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|
301 | (4) |
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21.1.2 Hamiltonian-Dependent Replica Exchange |
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|
305 | (3) |
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21.2 The Multicanonical Method |
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|
308 | (4) |
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|
311 | (1) |
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|
312 | (1) |
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21.3 The Method of Wang and Landau |
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|
312 | (3) |
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21.3.1 The Wang and Landau Method-Applications |
|
|
314 | (1) |
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21.4 The Method of Expanded Ensembles |
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|
315 | (2) |
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21.4.1 The Method of Expanded Ensembles-Appl ications |
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|
317 | (1) |
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21.5 The Adaptive Integration Method |
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|
317 | (2) |
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21.6 Methods Based on Jarzynski's Identity |
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319 | (5) |
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21.6.1 Jarzynski's Identity versus Other Methods for Calculating AF |
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323 | (1) |
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324 | (1) |
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324 | (7) |
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22 Simulation of the Chemical Potential |
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|
331 | (12) |
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22.1 The Widom Insertion Method |
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|
331 | (1) |
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22.2 The Deletion Procedure |
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332 | (2) |
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22.3 Personage's Method for Treating Deletion |
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334 | (2) |
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22.4 Introduction of a Hard Sphere |
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336 | (1) |
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22.5 The Ideal Gas Gauge Method |
|
|
337 | (1) |
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22.6 Calculation of the Chemical Potential of a Polymer by the Scanning Method |
|
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338 | (2) |
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22.7 The Incremental Chemical Potential Method for Polymers |
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|
340 | (1) |
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22.8 Calculation of u by Thermodynamic Integration |
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|
341 | (1) |
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341 | (2) |
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23 The Absolute Free Energy of Binding |
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|
343 | (24) |
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23.1 The Law of Mass Action |
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|
343 | (1) |
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23.2 Chemical Potential, Fugacity, and Activity of an Ideal Gas |
|
|
344 | (1) |
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|
344 | (1) |
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23.2.2 Canonical Ensemble |
|
|
344 | (1) |
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|
345 | (1) |
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23.3 Chemical Potential in Ideal Solutions: Raoult's and Henry's Laws |
|
|
345 | (1) |
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346 | (1) |
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346 | (1) |
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23.4 Chemical Potential in Non-ideal Solutions |
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346 | (1) |
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346 | (1) |
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347 | (1) |
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23.5 Thermodynamic Treatment of Chemical Equilibrium |
|
|
347 | (1) |
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23.6 Chemical Equilibrium in Ideal Gas Mixtures: Statistical Mechanics |
|
|
348 | (1) |
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23.7 Pressure-Dependent Equilibrium Constant of Ideal Gas Mixtures |
|
|
349 | (1) |
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23.8 Protein-Ligand Binding |
|
|
350 | (12) |
|
23.8.1 Standard Methods for Calculating δA° |
|
|
352 | (2) |
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23.8.2 Calculating δA° by HSMD-TI |
|
|
354 | (2) |
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23.8.3 HSMD-TI Applied to the FKBP12-FK506 Complex: Equilibration |
|
|
356 | (1) |
|
23.8.4 The Internal and External Entropies |
|
|
357 | (2) |
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23.8.5 TI Results for FKBP12-FK506 |
|
|
359 | (1) |
|
23.8.6 δA° Results for FKBP12-FK506 |
|
|
359 | (3) |
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|
|
362 | (1) |
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|
362 | (5) |
| Appendix |
|
367 | (2) |
| Index |
|
369 | |