Note: Cover -- Half Title -- Series Page -- Title Page -- Copyright Page -- Dedication -- Table of Contents -- Preface -- Acknowledgments -- Author -- Section I: Probability Theory -- 1: Probability and Its Applications -- 1.1 Introduction -- 1.2 Experimental Probability -- 1.3 The Sample Space Is Related to the Experiment -- 1.4 Elementary Probability Space -- 1.5 Basic Combinatorics -- 1.5.1 Permutations -- 1.5.2 Combinations -- 1.6 Product Probability Spaces -- 1.6.1 The Binomial Distribution -- 1.6.2 Poisson Theorem -- 1.7 Dependent and Independent Events -- 1.7.1 Bayes Formula -- 1.8 Discrete Probability-Summary -- 1.9 One-Dimensional Discrete Random Variables -- 1.9.1 The Cumulative Distribution Function -- 1.9.2 The Random Variable of the Poisson Distribution -- 1.10 Continuous Random Variables -- 1.10.1 The Normal Random Variable -- 1.10.2 The Uniform Random Variable -- 1.11 The Expectation Value -- 1.11.1 Examples -- 1.12 The Variance -- 1.12.1 The Variance of the Poisson Distribution -- 1.12.2 The Variance of the Normal Distribution -- 1.13 Independent and Uncorrelated Random Variables -- 1.13.1 Correlation -- 1.14 The Arithmetic Average -- 1.15 The Central Limit Theorem -- 1.16 Sampling -- 1.17 Stochastic Processes-Markov Chains -- 1.17.1 The Stationary Probabilities -- 1.18 The Ergodic Theorem -- 1.19 Autocorrelation Functions -- 1.19.1 Stationary Stochastic Processes -- Homework for Students -- A Comment about Notations -- References -- Section II: Equilibrium Thermodynamics and Statistical Mechanics -- 2: Classical Thermodynamics -- 2.1 Introduction -- 2.2 Macroscopic Mechanical Systems versus Thermodynamic Systems -- 2.3 Equilibrium and Reversible Transformations -- 2.4 Ideal Gas Mechanical Work and Reversibility -- 2.5 The First Law of Thermodynamics -- 2.6 Joule's Experiment -- 2.7 Entropy -- 2.8 The Second Law of Thermodynamics. , 2.8.1 Maximal Entropy in an Isolated System -- 2.8.2 Spontaneous Expansion of an Ideal Gas and Probability -- 2.8.3 Reversible and Irreversible Processes Including Work -- 2.9 The Third Law of Thermodynamics -- 2.10 Thermodynamic Potentials -- 2.10.1 The Gibbs Relation -- 2.10.2 The Entropy as the Main Potential -- 2.10.3 The Enthalpy -- 2.10.4 The Helmholtz Free Energy -- 2.10.5 The Gibbs Free Energy -- 2.10.6 The Free Energy, , H.(T,µ) -- 2.11 Maximal Work in Isothermal and Isobaric Transformations -- 2.12 Euler's Theorem and Additional Relations for the Free Energies -- 2.12.1 Gibbs-Duhem Equation -- 2.13 Summary -- Homework for Students -- References -- Further Reading -- 3: From Thermodynamics to Statistical Mechanics -- 3.1 Phase Space as a Probability Space -- 3.2 Derivation of the Boltzmann Probability -- 3.3 Statistical Mechanics Averages -- 3.3.1 The Average Energy -- 3.3.2 The Average Entropy -- 3.3.3 The Helmholtz Free Energy -- 3.4 Various Approaches for Calculating Thermodynamic Parameters -- 3.4.1 Thermodynamic Approach -- 3.4.2 Probabilistic Approach -- 3.5 The Helmholtz Free Energy of a Simple Fluid -- Reference -- Further Reading -- 4: Ideal Gas and the Harmonic Oscillator -- 4.1 From a Free Particle in a Box to an Ideal Gas -- 4.2 Properties of an Ideal Gas by the Thermodynamic Approach -- 4.3 The chemical potential of an Ideal Gas -- 4.4 Treating an Ideal Gas by the Probability Approach -- 4.5 The Macroscopic Harmonic Oscillator -- 4.6 The Microscopic Oscillator -- 4.6.1 Partition Function and Thermodynamic Properties -- 4.7 The Quantum Mechanical Oscillator -- 4.8 Entropy and Information in Statistical Mechanics -- 4.9 The Configurational Partition Function -- Homework for Students -- References -- Further Reading -- 5: Fluctuations and the Most Probable Energy -- 5.1 The Variances of the Energy and the Free Energy. , 5.2 The Most Contributing Energy E* -- 5.3 Solving Problems in Statistical Mechanics -- 5.3.1 The Thermodynamic Approach -- 5.3.2 The Probabilistic Approach -- 5.3.3 Calculating the Most Probable Energy Term -- 5.3.4 The Change of Energy and Entropy with Temperature -- References -- 6: Various Ensembles -- 6.1 The Microcanonical (petit) Ensemble -- 6.2 The Canonical (NVT) Ensemble -- 6.3 The Gibbs (NpT) Ensemble -- 6.4 The Grand Canonical (µVT) Ensemble -- 6.5 Averages and Variances in Different Ensembles -- 6.5.1 A Canonical Ensemble Solution (Maximal Term Method) -- 6.5.2 A Grand-Canonical Ensemble Solution -- 6.5.3 Fluctuations in Different Ensembles -- References -- Further Reading -- 7: Phase Transitions -- 7.1 Finite Systems versus the Thermodynamic Limit -- 7.2 First-Order Phase Transitions -- 7.3 Second-Order Phase Transitions -- References -- 8: Ideal Polymer Chains -- 8.1 Models of Macromolecules -- 8.2 Statistical Mechanics of an Ideal Chain -- 8.2.1 Partition Function and Thermodynamic Averages -- 8.3 Entropic Forces in an One-Dimensional Ideal Chain -- 8.4 The Radius of Gyration -- 8.5 The Critical Exponent ν -- 8.6 Distribution of the End-to-End Distance -- 8.6.1 Entropic Forces Derived from the Gaussian Distribution -- 8.7 The Distribution of the End-to-End Distance Obtained from the Central Limit Theorem -- 8.8 Ideal Chains and the Random Walk -- 8.9 Ideal Chain as a Model of Reality -- References -- 9: Chains with Excluded Volume -- 9.1 The Shape Exponent ν for Self-avoiding Walks -- 9.2 The Partition Function -- 9.3 Polymer Chain as a Critical System -- 9.4 Distribution of the End-to-End Distance -- 9.5 The Effect of Solvent and Temperature on the Chain Size -- 9.5.1 θ Chains in d = 3 -- 9.5.2 θ Chains in d = 2 -- 9.5.3 The Crossover Behavior Around -- 9.5.4 The Blob Picture -- 9.6 Summary -- References. , Section III: Topics in Non-Equilibrium Thermodynamics and Statistical Mechanics -- 10: Basic Simulation Techniques: Metropolis Monte Carlo and Molecular Dynamics -- 10.1 Introduction -- 10.2 Sampling the Energy and Entropy and New Notations -- 10.3 More About Importance Sampling -- 10.4 The Metropolis Monte Carlo Method -- 10.4.1 Symmetric and Asymmetric MC Procedures -- 10.4.2 A Grand-Canonical MC Procedure -- 10.5 Efficiency of Metropolis MC -- 10.6 Molecular Dynamics in the Microcanonical Ensemble -- 10.7 MD Simulations in the Canonical Ensemble -- 10.8 Dynamic MD Calculations -- 10.9 Efficiency of MD -- 10.9.1 Periodic Boundary Conditions and Ewald Sums -- 10.9.2 A Comment About MD Simulations and Entropy -- References -- 11: Non-Equilibrium Thermodynamics-Onsager Theory -- 11.1 Introduction -- 11.2 The Local-Equilibrium Hypothesis -- 11.3 Entropy Production Due to Heat Flow in a Closed System -- 11.4 Entropy Production in an Isolated System -- 11.5 Extra Hypothesis: A Linear Relation Between Rates and Affinities -- 11.5.1 Entropy of an Ideal Linear Chain Close to Equilibrium -- 11.6 Fourier's Law-A Continuum Example of Linearity -- 11.7 Statistical Mechanics Picture of Irreversibility -- 11.8 Time Reversal, Microscopic Reversibility, and the Principle of Detailed Balance -- 11.9 Onsager's Reciprocal Relations -- 11.10 Applications -- 11.11 Steady States and the Principle of Minimum Entropy Production -- 11.12 Summary -- References -- 12: Non-equilibrium Statistical Mechanics -- 12.1 Fick's Laws for Diffusion -- 12.1.1 First Fick's Law -- 12.1.2 Calculation of the Flux from Thermodynamic Considerations -- 12.1.3 The Continuity Equation -- 12.1.4 Second Fick's Law-The Diffusion Equation -- 12.1.5 Diffusion of Particles Through a Membrane -- 12.1.6 Self-Diffusion -- 12.2 Brownian Motion: Einstein's Derivation of the Diffusion Equation. , 12.3 Langevin Equation -- 12.3.1 The Average Velocity and the Fluctuation-Dissipation Theorem -- 12.3.2 Correlation Functions -- 12.3.3 The Displacement of a Langevin Particle -- 12.3.4 The Probability Distributions of the Velocity and the Displacement -- 12.3.5 Langevin Equation with a Charge in an Electric Field -- 12.3.6 Langevin Equation with an External Force-The Strong Damping Velocity -- 12.4 Stochastic Dynamics Simulations -- 12.4.1 Generating Numbers from a Gaussian Distribution by CLT -- 12.4.2 Stochastic Dynamics versus Molecular Dynamics -- 12.5 The Fokker-Planck Equation -- 12.6 Smoluchowski Equation -- 12.7 The Fokker-Planck Equation for a Full Langevin Equation with a Force -- 12.8 Summary of Pairs of Equations -- References -- 13: The Master Equation -- 13.1 Master Equation in a Microcanonical System -- 13.2 Master Equation in the Canonical Ensemble -- 13.3 An Example from Magnetic Resonance -- 13.3.1 Relaxation Processes Under Various Conditions -- 13.3.2 Steady State and the Rate of Entropy Production -- 13.4 The Principle of Minimum Entropy Production-Statistical Mechanics Example -- References -- Section IV: Advanced Simulation Methods: Polymers and Biological Macromolecules -- 14: Growth Simulation Methods for Polymers -- 14.1 Simple Sampling of Ideal Chains -- 14.2 Simple Sampling of SAWs -- 14.3 The Enrichment Method -- 14.4 The Rosenbluth and Rosenbluth Method -- 14.5 The Scanning Method -- 14.5.1 The Complete Scanning Method -- 14.5.2 The Partial Scanning Method -- 14.5.3 Treating SAWs with Finite Interactions -- 14.5.4 A Lower Bound for the Entropy -- 14.5.5 A Mean-Field Parameter -- 14.5.6 Eliminating the Bias by Schmidt's Procedure -- 14.5.7 Correlations in the Accepted Sample -- 14.5.8 Criteria for Efficiency -- 14.5.9 Locating Transition Temperatures -- 14.5.10 The Scanning Method versus Other Techniques. , 14.5.11 The Stochastic Double Scanning Method.