Statistical Mechanics for Chemistry and Materials Science

581 Pages·2018·54.096 MB·English

by  Biman Bagchi

Statistical Mechanics for
Chemistry and Materials
Science
http://taylorandfrancis.com
Statistical Mechanics for
Chemistry and Materials
Science
Biman Bagchi
CRC Press
Taylor & Francis Group
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To my brother Bidyut K. Bagchi (“Dada”) who has remained a constant source of
joy and strength in the turbulence of life, a precious gift by my parents ….
My two sons Kaushik and Kushal for their care, love, sense of humor, and criticisms that
kept me going, and in whom I find a chemistry similar to that between me and my brother.
Let the brotherly love continue.
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Contents
Preface ...........................................................................................................................................xix
Author ...........................................................................................................................................xxi
1. Preliminaries ...........................................................................................................................1
1.1 Why Study Statistical Mechanics? ..............................................................................1
1.2 Statistical Mechanics Explains Why and How: Scope with Examples .................3
1.3 Objectives of Statistical Mechanics ............................................................................4
1.4 Essentials of Thermodynamics ...................................................................................5
1.4.1 Laws of Thermodynamics ..............................................................................6
1.5 Thermodynamic Response Functions and Stability Conditions ...........................8
1.6 Thermodynamics: Configuration Space Description ..............................................9
1.7 From Intermolecular Potential to Thermodynamic Observables:
Shortcomings of Classical Mechanics ......................................................................10
1.8 Towards a Probabilistic Description ........................................................................11
1.9 Summary ......................................................................................................................12
Suggested Reading ................................................................................................................12
Problem Set .............................................................................................................................13
2. Probability and Statistics ....................................................................................................17
2.1 Introduction .................................................................................................................17
2.2 Basic Ideas ....................................................................................................................18
2.2.1 Random Variables ..........................................................................................18
2.2.2 Sample Space ..................................................................................................19
2.2.3 Probability Distribution ................................................................................19
2.2.4 Joint Probability Distribution .......................................................................20
2.2.5 Conditional Probability.................................................................................20
2.2.6 Conditional Probability and Correlation....................................................21
2.3 Central Limit Theorem (CLT) ....................................................................................21
2.4 Applications of Probability Theory in Statistical Mechanics ...............................23
2.5 Summary ......................................................................................................................23
Suggested Reading ................................................................................................................24
Problem Set .............................................................................................................................24
3. Fundamental Concepts and Postulates of Statistical Mechanics ...............................27
3.1 Introduction .................................................................................................................28
3.2 Basic Ideas and Tools of Statistical Mechanics .......................................................28
3.2.1 Phase Space and Trajectory ..........................................................................28
3.2.2 Rationale behind the Postulates ..................................................................30
3.3 Ensemble ......................................................................................................................31
3.4 The First Postulate: Time Average Equals Ensemble Average .............................33
3.5 The Second Postulate: Equal A Priori Probability .................................................33
vii
viii Contents
3.6 Ergodic Hypothesis ....................................................................................................35
3.7 Measure of Ergodicity ................................................................................................35
3.8 Diffusion in Deterministic Systems .........................................................................37
3.9 Periodic Lorentz Gas ..................................................................................................37
3.10 Summary ......................................................................................................................39
Appendix-A ............................................................................................................................39
A.1 Diffusion in Triangular Lattice Potential ...................................................39
References ...............................................................................................................................40
Suggested Reading ................................................................................................................40
Problem Set .............................................................................................................................41
4. Liouville Theorem and Liouville Equation ....................................................................43
4.1 Introduction .................................................................................................................43
4.2 Definition of Phase Space Density ............................................................................44
4.3 Hamilton’s Equation of Motion .................................................................................45
4.4 Phase Space Density and Liouville Theorem .........................................................46
4.5 Liouville Equation (LE) ..............................................................................................46
4.6 Liouville Equation for Dynamical Variables ...........................................................49
4.7 Quantum Liouville Equation (QLE) .........................................................................49
4.8 A Few Useful Comments about the Liouville Equation .......................................51
4.9 Applicability of Liouville Equation ..........................................................................52
4.10 From Liouville Equation to BBGKY Hierarchy ......................................................53
4.11 Summary ......................................................................................................................53
References ...............................................................................................................................54
Suggested Reading ................................................................................................................54
Problem Set .............................................................................................................................54
5. Ensembles and Partition Functions: From Postulates to Formulation ......................57
5.1 Introduction .................................................................................................................57
5.2 Microcanonical Ensemble ..........................................................................................59
5.2.1 Boltzmann’s Entropy Formula: A Few Comments ...................................61
5.2.2 Relationship with Thermodynamics in the (NVE) Ensemble .................62
5.3 Canonical Ensemble ...................................................................................................63
5.4 Microscopic Derivation of Boltzmann Distribution ..............................................66
5.4.1 An Alternative Derivation of Boltzmann Distribution ............................67
5.5 Thermodynamic Potential for Canonical Ensemble ..............................................68
5.6 Relationship between Canonical Partition Function and Thermodynamic
Functions ......................................................................................................................70
5.7 Grand Canonical Ensemble .......................................................................................71
5.7.1 Thermodynamic Potential for Grand Canonical Partition Function .....72
5.7.2 Relationship between Thermodynamic Functions and Grand
Canonical Partition Function .......................................................................74
5.8 Isothermal Isobaric Ensemble (NPT Ensemble) .....................................................74
5.9 Physical Interpretation of Partition Function (PF) in Different Ensembles ........76
5.10 Summary ......................................................................................................................77
References ...............................................................................................................................78
Suggested Reading ................................................................................................................78
Problem Set .............................................................................................................................78
Contents ix
6. Fluctuations and Response Functions ..............................................................................81
6.1 Introduction .................................................................................................................81
6.2 Energy Fluctuations in Canonical Ensemble: Specific Heat at Constant
Volume (C ) ..................................................................................................................83
V
6.3 Fluctuation Formulae for Other Response Functions ...........................................85
6.3.1 Fluctuation Formulae for Specific Heat at Constant Pressure (C )
P
and Isothermal Compressibility (κ) .............................................................85
6.3.2 Fluctuation Formulae in Grand-Canonical (µVT) Ensemble:
Isothermal Compressibility (κ) .....................................................................87
6.4 System Size Dependence of Fluctuations ................................................................87
6.5 Probability Distribution Function of Fluctuations .................................................88
6.6 Stability Conditions ....................................................................................................90
6.7 A Discussion on Specific Heat and Thermal Conductivity ..................................90
6.7.1 Specific Heat ...................................................................................................90
6.7.2 Thermal Conductivity ...................................................................................91
6.8 Fluctuation, Free Energy Expansion, and Response Functions ...........................92
6.9 Digression on “Color of Water” ................................................................................93
6.9.1 What Is the Color of Water? ..........................................................................93
6.10 Summary ......................................................................................................................94
References ...............................................................................................................................95
Suggested Reading ................................................................................................................95
Problem Set .............................................................................................................................95
7. Ideal Monatomic Gas: Microscopic Expression of Translational Entropy ...............97
7.1 Introduction .................................................................................................................97
7.2 Partition Function of a Classical Monatomic Gas ..................................................97
7.3 Free Energy ..................................................................................................................98
7.4 Equation of State..........................................................................................................99
7.5 Entropy of an Ideal Gas: Sackur-Tetrode Equation ................................................99
7.5.1 Application of Sackur-Tetrode Equation ....................................................99
7.6 Specific Heat...............................................................................................................100
7.7 Numerical Values of Entropy of Ideal Monatomic Gases ...................................100
7.8 Partition Function of an Ideal Quantum Gas .......................................................100
7.9 Ideal Monatomic Gas in Grand Canonical Ensemble ..........................................101
7.10 Density of States ........................................................................................................102
7.11 Summary ....................................................................................................................103
Suggested Reading ..............................................................................................................104
Problem Set ...........................................................................................................................104
8. Ideal Gas of Diatomic Molecules: Microscopic Expressions for Rotational
and Vibrational Entropy and Specific Heat ..................................................................107
8.1 Introduction ...............................................................................................................107
8.2 Vibrational Partition Function ................................................................................108
8.2.1 Free Energy ...................................................................................................109
8.2.2 Vibrational Entropy and Specific Heat .....................................................110
8.3 Rotational Partition Function ..................................................................................111
8.3.1 Free Energy and Rotational Entropy ........................................................112
8.4 Ideal Polyatomic Gas ................................................................................................113
8.5 Entropy of Polyatomic Molecules ...........................................................................115