2024 Faculty Courses School of Materials and Chemical Technology Undergraduate major in Materials Science and Engineering
Statistical Mechanics(M)
- Academic unit or major
- Undergraduate major in Materials Science and Engineering
- Instructor(s)
- Yoshihiro Gohda / Kan Nakatsuji / Masaki Tahara
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 7-8 Tue / 7-8 Fri
- Class
- -
- Course Code
- MAT.M202
- Number of credits
- 200
- Course offered
- 2024
- Offered quarter
- 3Q
- Syllabus updated
- Mar 17, 2025
- Language
- Japanese
Syllabus
Course overview and goals
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.
Course description and aims
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
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Exercises with problems are assigned.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| 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. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
Instructions will be given in class. Documents will be distributed if necessary.
Reference books, course materials, etc.
Kittel & Kroemer: Thermal Physics, etc.
Evaluation methods and criteria
Learning achievement is evaluated by report assignments and examinations.
Related courses
- MAT.A203 ： Quantum Mechanics of Materials
- MAT.A204 ： Thermodynamics of Materials
- MAT.M206 ： Electronic Structure and Physical Properties of Metals
Prerequisites
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.
Contact information (e-mail and phone) Notice : Please replace from ”[at]” to ”@”(half-width character).
Yoshihiro Gohda (gohda.y.ab[at]m.titech.ac.jp)
Kan Nakatsuji (nakatsuji.k.aa[at]m.titech.ac.jp)
Masaki Tahara (tahara.m.aa[at]m.titech.ac.jp)
K.N. is responsible for 2024.
Office hours
Contact by e-mail in advance to schedule an appointment.