Modern Thermodynamics and Statistical Mechanics: A Comprehensive Foundation (Undergraduate Lecture Notes in Physics)

✍ Scribed by Ravinder R. Puri

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 406
• Edition: 2024
• Category: Library

✦ Synopsis

This undergraduate-level textbook offers a unique and in-depth approach to the study of thermodynamics and statistical mechanics. It covers the fundamentals of thermodynamics using both traditional and postulatory approaches, including origin of the concept of thermodynamic entropy, Euler’s equation, Gibbs-Duhem relations, stability of equilibrium, and the concept of thermodynamic potentials, and that of independent thermodynamic observables. The book then delves into the microscopic foundation of thermodynamics, starting with the kinetic theory and highlighting its historical development. Boltzmann's concept of entropy is explored, along with its applications in deriving Planck’s, Bose’s, Bose-Einstein, and Fermi-Dirac distribution functions. The formal structure of classical and quantum statistical mechanics is built based on the concept of statistical entropy and the maximum entropy principle and used to investigate in detail the thermodynamic properties of ideal classical and quantum systems. The book also covers phase transitions, simple theory of critical phenomena, and the theory of interacting van der Waals gases. Throughout the text, the book provides historical context, enriching the reader's understanding. This textbook is a valuable resource for undergraduate physics students, offering comprehensive coverage, including overlooked topics, and a historical perspective on thermodynamics and statistical mechanics.

✦ Table of Contents

Preface
Contents
1 Fundamentals of Thermodynamics-I
1.1 Brief History
1.2 Carnot Engine
1.2.1 Absolute Temperature
1.2.2 Entropy
1.3 Laws of Thermodynamics
1.3.1 Zeroth Law
1.3.2 First Law
1.3.3 Second Law
1.3.4 Third Law
1.4 Ideal Gas Equations of State
1.4.1 Internal Energy of Ideal Gas
1.5 A Cyclic Process to Realize Carnot Engine
1.5.1 Entropy of Ideal Gas
1.6 Van der Waals Equation of State
References
2 Fundamentals of Thermodynamics-II
2.1 Postulates of Thermodynamics
2.1.1 First Postulate
2.1.2 Second Postulate
2.1.3 Third Postulate
2.1.4 Fourth Postulate
2.1.5 Examining Admissible Forms of upper S left parenthesis upper U comma upper V comma upper N right parenthesisS(U, V, N)
2.1.6 Connection with the Laws of Thermodynamics
2.2 Justification of Definitions of Intensive Parameters
2.2.1 Temperature
2.2.2 Pressure
2.2.3 Chemical Potential
2.3 Equations of State
2.4 Euler Equation
2.5 Gibbs–Duhem Relation
2.6 Thermodynamic Potentials
2.6.1 Helmholtz Potential
2.6.2 Gibbs Potential
2.6.3 Enthalpy
2.6.4 Grand Potential
2.7 Massieu Functions
2.8 Maxwell Relations
2.9 Independent Thermodynamic Observables
2.10 Stability from Maximum Entropy Principle
2.11 Stability from Minimum Energy Principle
2.12 Stability in Terms of Thermodynamic Potentials
2.12.1 Helmholtz Potential
2.12.2 Enthalpy
2.12.3 Gibbs Potential
2.13 Thermodynamic Potentials: Alternative Formulation
2.13.1 System Interacting with Heat Reservoir
2.13.2 System Interacting with Heat and Pressure Reservoirs
2.13.3 System Interacting with Pressure Reservoir
2.13.4 Exergy
2.14 Second Equation of State
2.14.1 upper PP Linear in upper TT
2.15 Joule–Thomson Process
References
3 Kinetic Theory
3.1 Early Kinetic Theory
3.2 Maxwell Distribution
3.3 Phase Space Distribution Function
3.3.1 Liouville's Theorem
3.3.2 Reduced Distribution Functions
3.4 Boltzmann Equation: Single-Particle Phase Space Approach
3.5 Scattering
3.6 BBGKY Hierarchy
3.7 Boltzmann Equation from BBGKY Hierarchy
3.8 The H-Theorem
3.9 Equilibrium Distribution
References
4 Boltzmann Entropy
4.1 Discrete Energy Levels
4.1.1 Connection of Discrete Model with Thermodynamics
4.2 Continuous Distribution of Energy
4.3 Distribution of Velocities
4.4 Relation Between Thermodynamic and Boltzmann Entropies
4.5 Planck's Distribution
4.6 Bose Statistics
4.7 Einstein's Quantum Theory of Ideal Gas
4.7.1 Indistinguishable Molecules
4.7.2 Distinguishable Molecules
4.7.3 Distribution of Light Quanta
4.8 Distribution of Particles Obeying Exclusion Principle
References
5 Shannon and Statistical Entropies
5.1 Shannon Entropy
5.1.1 Discrete Probabilities
5.1.2 Continuous Probabilities
5.2 Maximum Entropy
5.2.1 Discrete Variables
5.2.2 Continuous Variable
5.3 Statistical Entropy
5.3.1 Classical Systems
5.3.2 Quantum Systems
References
6 Equilibrium Distributions
6.1 Principle of Maximum Entropy
6.2 Systems Having Fixed Number of Particles
6.2.1 Microcanonical Ensemble
6.2.2 Canonical Ensemble
6.3 Grand Canonical Ensemble
6.4 Relation with Thermodynamics
6.4.1 Zeroth Law of Thermodynamics
6.4.2 First Law of Thermodynamics
6.4.3 Second Law of Thermodynamics
6.4.4 Third Law of Thermodynamics
6.5 Thermodynamic Potentials in Terms of Partition Functions
7 Non-interacting Classical Gas
7.1 Thermodynamics Using Canonical Ensemble
7.1.1 Position Distribution Function
7.1.2 Free Non-interacting Particles: Ideal Gas
7.2 Thermodynamics Using Grand Canonical Ensemble
7.3 Equipartition Theorem
7.4 Internal Motion
7.4.1 Rotational Motion
7.4.2 Vibrational Motion
7.5 Gas in Gravitational Field
7.5.1 Temperature Gradient
7.6 Gibbs Paradox
References
8 Ideal Quantum Gases
8.1 Canonical Partition Function
8.1.1 Configurational Degeneracy
8.2 Distinguishable Particles
8.2.1 Classical Ideal Gas
8.3 Quantum Particles
8.4 Grand Canonical Partition Function
8.4.1 Classical Particles
8.4.2 Quantum Particles
8.4.3 Classical Limit of Quantum Distributions
8.5 Single Particle Energy Levels in Free Space
8.5.1 Determining Density of States
8.6 Thermodynamics of Ideal Classical Gas
8.6.1 Translational Motion
8.6.2 Internal Motion
8.7 Thermodynamics of Quantum Gases
8.8 Quantum Corrections to the Classical Limit
References
9 Ideal Fermi Gas
9.1 Fermi Gas at Zero Temperature
9.2 Fermi Gas at Low Temperature
10 Ideal Bose Gas
10.1 Bose Gas
10.1.1 Conditions for BEC
10.1.2 Thermodynamic Properties
10.1.3 BEC as a Phenomenon of Phase Transition
10.2 Gas of Photons
10.2.1 Thermodynamic Properties
References
11 Phase Transitions and Critical Phenomena
11.1 Phase Equilibrium
11.1.1 Triple Point
11.1.2 upper P minus vP- v Isotherms in Coexistence Region
11.1.3 Lever Rule
11.2 Equation of State in Coexistence Region
11.3 First-Order Phase Transition
11.3.1 Entropy Discontinuity
11.3.2 Energy Discontinuity
11.3.3 Latent Heat
11.3.4 Clapeyron–Clausius Equation
11.4 Critical Phenomenon
11.4.1 Critical Exponents
12 Interacting Classical Gas
12.1 Virial Expansion
12.2 Van der Waals Equation of State
12.3 Critical Point
12.3.1 Law of Corresponding States
12.4 P-v Isotherms
12.5 Gas–Liquid Transition
12.6 Critical Exponents for van der Waals Fluid
References
13 Density Operator Formalism
13.1 Density Matrix
13.2 Quantum Entropy
13.3 Equilibrium Density Matrix
13.4 Standard Distributions
13.4.1 Microcanonical Ensemble
13.4.2 Canonical Ensemble
13.4.3 Grand Canonical Ensemble
13.5 Equilibrium Density Matrix of Harmonic Oscillators
13.6 Time Evolution of Entropy
Reference
14 Quantum Master Equation
14.1 Master Equation
14.2 Steady State Solution
14.3 Harmonic Oscillator Interacting with Reservoir of Harmonic …
References
Appendix A Some Relations Involving Partial Derivatives
Appendix B Legendre Transform
B.1 Relations Between Second Derivatives of a Function and Its Legendre Transform: Single Variable Transform
B.2 Relations Between Second Derivatives of a Function and Its Legendre Transform: Two Variables Transformation
Reference
Appendix C Concave and Convex Functions
Appendix D Some Combinatorics Formulas
Appendix E Cubic Equation
Appendix F Thermodynamic Properties of Blackbody Radiation
Appendix G Harmonic Oscillator Number and Coherent States
Appendix H Some Mathematical Formulas
Appendix Bibliography
Index