2016 Statistical Mechanics(M)

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Academic unit or major
Undergraduate major in Materials Science and Engineering
Nakatsuji Kan  Kajihara Masanori  Gohda Yoshihiro 
Class Format
Media-enhanced courses
Day/Period(Room No.)
Tue7-8(S8-101)  Fri7-8(S8-101)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
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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.


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

Competencies that will be developed

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

Class flow

At the beginning of the class (not each class), solutions to exercise problems assigned during the previous class are reviewed.

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.


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".

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