### 2019　Statistical Mechanics(M)

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Undergraduate major in Materials Science and Engineering
Instructor(s)
Kajihara Masanori  Gohda Yoshihiro  Nakatsuji Kan
Course component(s)
Lecture
Day/Period(Room No.)
Tue7-8(S8-101)  Fri7-8(S8-101)
Group
-
Course number
MAT.M202
Credits
2
2019
Offered quarter
3Q
Syllabus updated
2019/3/18
Lecture notes updated
-
Language used
Japanese
Access Index ### Course description and aims

The course teaches the partition function of the microcanonical ensemble which is mathematically derived using the method of Lagrange multiplier based on the Boltzmann's equation. The partition functions of canonical and grand canonical ensembles and their relations to the Helmholtz energy and grand potential are also derived.
Students learn how to derive the entropy of lattice vibration in solid, free-electron gas, magnetic spin system, etc. They also learn Einstein model and Debye model which analytically describe the specific heat of solid.

### Student learning outcomes

By completing this course, students will be able to:
1) Understand the relation between thermodynamics and statistical mechanics.
2) Understand the difference between microcanonical, canonical and grand canonical ensembles.
3) Understand the difference of partition functions for the ensembles in which the particles in the system are distinguishable and non-distinguishable.

### Keywords

Boltzmann's equation, Canonical ensemble, Partition function, Thermodynamic functions, Einstein model, Debye model, Fermion, Boson

### Competencies that will be developed

Intercultural skills Communication skills Specialist skills Critical thinking skills Practical and/or problem-solving skills
- - -

### Class flow

Exercises with problems are assigned.

### Course schedule/Required learning

Course schedule Required learning
Class 1 Boltzmann's equation Understand that the Boltzmann's equation connects the thermodynamics and statistical mechanics.
Class 2 Method of Lagrange multiplier Understand how to derive the partition function of microcanonical ensemble using the method of Lagrange multiplier based on principle of equal weight.
Class 3 Partition functions Students must be able to derive internal energy, Helmholtz energy and entropy.
Class 4 Basic laws of thermodynamics Explain the first, second and third laws of thermodynamics.
Class 5 Thermodynamic functions and Legendre transformation Students must be able to derive thermodynamic functions using Legendre transformation.
Class 6 Specific heat -Einstein model- Students must be able to derive specific heat of solid using Einstein model from the knowledge of quantum harmonic oscillator.
Class 7 Specific heat -Debye model- Understand how to derive specific heat of solid using Debye model and its difference from Einstein model.
Class 8 Perfect gas Students must be able to derive the partition function of the perfect gas.
Class 9 Fermion Explain Fermi-Dirac statistics and the free-electron gas.
Class 10 Boson Explain Bose-Einstein statistics and the Bose-Einstein condensation.
Class 11 Magnetization in localized spin system Students must be able to derive specific heats, magnetization, etc. from the partition function for the localized spin system.
Class 12 Phase equilibrium of gas Understand the law of mass action and the Clausius-Clapeyron relation.
Class 13 Canonical ensemble and Helmholtz free energy Understand the difference of canonical ensemble from microcanonical ensemble and derive Helmholtz free energy from the partition function.
Class 14 Grand canonical ensemble and grand potential Understand the difference of grand canonical ensemble from canonical ensemble and derive grand potential from the partition function.
Class 15 Solution model Understand the entropy of configuration and the solubility gap.

### Textbook(s)

Lecture notes are distributed.

### Reference books, course materials, etc.

Kittel: Thermal Physics,
Rushbrooke: Introduction to Statistical Mechanics

### Assessment criteria and methods

Learning achievement is evaluated by examinations.

### Related courses

• MAT.A203 ： Quantum Mechanics of Materials
• MAT.A204 ： Thermodynamics of Materials
• MAT.M206 ： Electronic Structure and Physical Properties of Metals

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

It is desirable that the students have successfully completed "Quantum Mechanics of Materials" as well as " Thermodynamics of Materials".
Enrollment of only one of "Statistical Mechanics (M)" "Physical Chemistry (Statistical Mechanics)" and "Statistical Mechanics (C)" is permitted. 