学部・大学院区分
Undergraduate / Graduate | | 理学部 | | 時間割コード
Registration Code | | 0680490 | | 科目区分
Course Category | | 専門科目
Specialized Courses | | 科目名 【日本語】
Course Title | | 物理学演習４ａ | | 科目名 【英語】
Course Title | | Physics Tutorial IVa | | コースナンバリングコード
Course Numbering Code | | | | 担当教員 【日本語】
Instructor | | 山川 洋一 ○ | | 担当教員 【英語】
Instructor | | YAMAKAWA Youichi ○ | | 単位数
Credits | | 0.5 | | 開講期・開講時間帯
Term / Day / Period | | 春 金曜日 ３時限
春 金曜日 ４時限
Spring Fri 3
Spring Fri 4 | | 授業形態
Course style | | 演習
Seminar | | 学科・専攻
Department / Program | | | | 必修・選択
Compulsory / Selected | | | See the "Course List and Graduation Requirements" for your program for your enrolment year. |
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授業の目的 【日本語】
Goals of the Course(JPN) | | | | 授業の目的 【英語】
Goals of the Course | | This tutorial course is a companion course to the Statistical Physics III lecture course. The two should be taken together.
The purpose of this course is to cultivate students’ calculation skills and deepen their understanding of statistical physics.
Statistical Mechanics is one of the major fields of physics: around 30% of Physics Nobel Prizes have been awarded for discoveries directly or indirectly related to Statistical Mechanics, particularly phase transitions and field theories. The principles and methods are applicable in many fields of physics, including condensed matter physics (e.g. Bose Einstein Condensation, superconductivity, materials science) and high energy physics (spontaneous symmetry breaking, lattice gauge field theories, the Higgs Mechanism (which is a type of phase transition) and so on) as well as astrophysics (neutron stars, simulations of galaxy evolution, and so on). The principles and methods are also applicable in a very wide variety of fields outside physics, such as biology, neuroscience, modelling of pandemics, network theory, machine learning and artificial intelligence. G30 students in the past have shown that students in chemistry, chemical engineering and materials science with a thorough grounding in the principles of Statistical Mechanics have a significant advantage over their peers who do not possess this grounding.
This is the second part of a full-year, intermediate-advanced Year 3 course in statistical mechanics and thermodynamics. |
| | 到達目標 【日本語】
Objectives of the Course(JPN)) | | | | 到達目標 【英語】
Objectives of the Course | | At the end of this course, students will have mastered basic aspects of quantum statistics of ideal gases, statistical mechanics of systems of interacting particles, and the theory of phase transitions and critical phenomena, including modern topics such as the scaling hypothesis, an introduction to renormalization group theory (the spatial renormalization group), and the Bogolyubov Variational Theorem and its application to constructing an optimal Mean Field Theory.
In this course, students learn quantum statistics of ideal gases, introductory statistical mechanics of systems of interacting particles, introductory theory of phase transitions and critical phenomena, Mean Field Theory, and some modern theory such as the scaling hypothesis, an introduction to renormalization group theory (the spatial renormalization group), and the Bogolyubov Variational Theorem and its application to constructing an optimal Mean Field Theory. Students will encounter the ideas of spontaneous symmetry breaking, universality, critical exponents, transformation between -- proof of equivalence of -- various models such as the Two-state Ising Model, Lattice Gas Model, Binary Alloy Model and Two-state Potts Model.
At the end of Statistical Physics III students will be adequately prepared with regards to their knowledge of statistical mechanics and thermodynamics to undertake further studies in Sc-lab, St-lab, R-lab, TB-lab, E-lab, H-lab, QG-lab and other, including experimental, labs in both the Department of Physics and Department of Applied Physics, as well as chemistry and computational biology labs at Nagoya University. For the high energy physics labs and theoretical condensed matter physics labs students should also have studied Quantum Mechanics II and Quantum Mechanics III. |
| | 授業の内容や構成
Course Content / Plan | | The tutorial course contents is defined by the lecture course contents, which is outlined below. Some topics are covered in assignments. The precise order and content of the lectures might vary slightly.
Lecture 1. Revision of quantum statistical mechanics. Quantum states of a single particle. Reflecting boundary conditions, periodic boundary conditions. Density of states in 3, 2 and 1 dimensions, for linear and quadratic dispersion relations. Turning sums into integrals. Example: EM radiation. The quantum distribution functions: Fermi-Dirac, Bose-Einstein distributions. Photon statistics: Planck distribution. Systems with varying number of particles: the Grand Canonical ensemble and partition function. Occupation number formalism: mean occupation number and dispersion. Role of the chemical potential.
Lecture 2. Examples. Vapour pressure of a solid. Diatomic molecules. Grand Canonical partition function and probability of a many-body state at temperature T. Example: adsorption of a gas onto a 2D surface.
Lecture 3. The ideal Fermi fluid: conduction electrons in metals. Specific heat and ground state energy in 3D, 2D, 1D. Sommerfeld expansion.
Lecture 4. The ideal Bose fluid: Bose-Einstein condensation in 3D. What about in 2D or 1D? Critical temperature. Mean energy, specific heat.
Lecture 5. Relativistic Quantum Gas: the Photon Gas (Black body radiation). Stefan-Boltzmann Law; Wien’s Displacement Law; radiation pressure; mechanical equation of state. NONEXAMINABLE: (if time allows) Classical theory of screening: the Debye-Hueckel Model.
Lecture 6. Introduction to Non-Ideal Systems (1): The Debye Model of solids. The Harmonic Approximation. Classical Theory. Quantized Theory. Normal modes. Phonons. The Debye Approximation. Specific heat. Rundown of main points in Ashcroft and Mermin Chapts 22,23 placing Debye Theory in perspective: Classical Theory of the Harmonic Crystal; Quantum Theory of the Harmonic Crystal.
Lecture 7. Introduction to Non-Ideal Systems (2). Weakly nonideal gases: virial expansion; 2nd virial coefficient and resulting equation of state. Derivation of the Van der Waals equation of state for a weakly non-ideal gas; derivation for a fluid using a self-consistent mean field approach. Derivation of 2nd virial coefficient and van der Waals Equation again, this time using Mayer f function. NONEXAMINABLE: The Cluster Expansion.
Lecture 8. Stability of thermodynamic systems. Concavity/convexity of thermodynamic potentials. Le Chatelier’s Principle. First Order phase transitions, features of the free energy. Discontinuity in the entropy: latent heat. Slope of the coexistence curves: Clausius-Clapeyron Equation. A Clausius-Clapeyron Equation for Magnetic Systems: Coexistence Curve of Superconducting and Normal Phases in a metal.
Lecture 9. Van der Waals fluid: unstable isotherms, physical isotherm, Maxwell equal-area rule. Multicomponent systems: Gibbs phase rule. Why does the phase diagram of water not have more than three phases coexisting at the same point?
Lecture 10. The Fluctuation-Dissipation Theorem. Response functions and correlations. Quantitative explanation of critical opalescence.
Lecture 11. Examples of phase transitions (order-disorder transition, which is a structural phase transition). Why do fluctuations get out of control near the critical point? Alben’s Model. Landau Theory: classical theory in the critical region. Order Parameter. Continuous phase transition. Spontaneous symmetry breaking. The critical exponents α,β,γ,δ and their classical values.
Lecture 12. Introduction to interacting magnetic systems: ferromagnetism and models for it. Ising model. 1D Ising chain with free ends. Mean field theory treatment of the 1D Ising chain. Effective field. Critical exponents.
Lecture 13. 1D Ising chain continued. No phase transition in the 1D Ising chain: proof by a simple argument; and by solving the model exactly. Exact solution of 1D Ising chain in zero field. Exact solution of 1D Ising ring with field switched on: transfer matrix. Spin correlation function: exact calculation for the 1D Ising chain. 2D Ising model on a square lattice (just mention): Exact critical exponents, behaviour of the specific heat. Phase diagram of ferromagnetic systems in 3D.
Lecture 14. Breakdown of the classical theory and advent of the modern theory. Cause of the breakdown (qualitative). Derivation of an inequality involving critical exponents – but all experiments suggest equality holds. Scaling hypothesis: ad hoc argument. Justification of the scaling hypothesis using Kadanoff’s block spins. Spatial renormalization group theory and sample calculation for the 1D Ising chain.
Lecture 15. Bogolyubov Variational Theorem. Order-Disorder Transition: constructing the Hamiltonian and deriving the optimal Mean Field Theory for its solution. Mean Field Theory for 1D Ising Model revisited. Transformation between Models and Universality classes: many problems that appear completely different are in fact manifestations of the same problem. Broken Symmetry, Universality Classes, and Goldstone’s Theorem (qualitative).
Lecture 16. NONEXAMINABLE: 2D Ising model on a square lattice: Low-T solution -- Peierls Droplets; High-T solution; Kramers-Wannier Duality. Critical temperature for 2D Ising Model on a Square Lattice. Lee-Yang Zeroes and Phase Transitions. |
| | 履修条件
Course Prerequisites | | Statistical Physics II; or Consent of Instructor
Students must have passed Statistical Physics II to take Statistical Physics III. |
| | 関連する科目
Related Courses | | Quantum Mechanics II; Physics Tutorial IVb; Statistical Physics II.
It is strongly advised that students taking Statistical Physics III concurrently enrol in Physics Tutorial IVa. |
| | 成績評価の方法と基準
Course Evaluation Method and Criteria | | | Tutorial preparation and tutorial performance count for 50% of the grade. Tutorial assignments count for the other 50%. |
| | 不可(F)と欠席(W)の基準
Criteria for "Fail (F)" & "Absent (W)" grades | | The “Absent” grade is reserved for students who withdraw by the official deadline in May. After that day, a letter grade will be awarded based on marks earned from all assessment during the semester.
If Statistical Physics III is NOT A COMPULSORY SUBJECT and the student plans never to take Statistical Physics III in the future, then a late withdrawal request will be considered. |
| | 参考書
Reference Book | | 1. Ashcroft & Mermin, Solid State Physics (Chapters 22,23 only).
2. Yeomans, J.M., Statistical Mechanics of Phase Transitions, Oxford Science Publications, 1992. (Simple, clear overview relevant to the second half of this course.)
3. Cardy, J., Scaling and renormalization in statistical physics, Cambridge Univ. Press, 1996.
4. Huang, K., Statistical Mechanics, Wiley. (Advanced reference.)
5. Kittel, C. and Kroemer, H., Thermal Physics, W.H. Freeman. (Try as alternative.)
6. Landau, L.D. and Lifshitz, E.M., Statistical Physics, Part I, by E.M. Lifshitz and L.P. Pitaevskii, Pergamon Press. (A classic book: thorough, advanced treatment.Highly recommended.) |
| | 教科書・テキスト
Textbook | | 1. Callen, Herbert, Thermodynamics and an Introduction to Thermostatistics, 2nd Ed., Wiley. (The Japanese translation has fewer misprints.)
2. Reif, F., Fundamentals of Statistical and Thermal Physics, McGraw-Hill, 1965.
3. Plischke, M. & Bergersen, B., Equilibrium Statistical Mechanics, 3rd Ed., World Scientific, 2006. |
| | 課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | This course is part of your training to be a professional researcher. You are expected to revise the lecture notes, read and work through the textbook, and solve assignment problems outside lecture hours. You cannot learn physics by only attending lectures. The exams will consist of questions covering both lecture notes and assignments.
Students must be willing to work hard if they want to achieve a good, internationally competitive level. |
| | 注意事項
Notice for Students | | Plagiarism (e.g. copying solutions that you have found on the Internet) is an act of academic dishonesty. Cheating in exams (e.g. having lecture notes, assignment solutions or online references open on your computer screen during an online exam) is a serious offence. Copying other people's solutions and claiming them as your own is also an act of academic dishonesty. Nagoya University has a strict policy towards academic dishonesty:
"Acts of academic dishonesty are prohibited during exams, for reports and assignments. If acts of academic dishonesty are discovered, you may be subject to discipline, which may affect your ability to graduate on time."
The punishment for serious breaches (such as cheating in an exam or repeated plagiarism despite a warning) is the loss of all grades from all subjects during the semester and cancellation of any scholarships received.
Even if your course is difficult and it is to be expected that you'll find it hard to finish assignments, it is far better that you submit an honest effort than take the dishonest path. Remember, to be on course for a "B" you only need to score over 70% in the assignments -- and if you paid attention in the tutorials, you would have seen nearly all the problems done for you (in my subjects, at least).
There's no excuse for cheating. |
| | 他学科聴講の可否
Propriety of Other department student's attendance | | | Students from any department are welcome as long as they have the necessary grounding in mathematics and physics. |
| | 他学科聴講の条件
Conditions for Other department student's attendance | | | | レベル
Level | | | | キーワード
Keyword | | | | 履修の際のアドバイス
Advice | | It is strongly advised that students studying Statistical Physics III concurrently enroll in Physics Tutorial IVa.
No solutions are handed out in class. It pays to come prepared and pay attention during the tutorial.
Students must be willing to work hard if they wish to achieve a good, internationally competitive level. |
| | 授業開講形態等
Lecture format, etc. | | | Face to face lectures and tutorials are compulsory (other than in exceptional circumstances; e.g. COVID infection). However, in order to record a video of the tutorials -- including student interaction with each other and with the lecturer -- the tutorials will simultaneously be carried out online using MS Teams. Students are therefore requested to bring their laptop or tablet to the lecture room. Make sure it has a microphone. Bring an electrical cord. |
| | 遠隔授業（オンデマンド型）で行う場合の追加措置
Additional measures for remote class (on-demand class) | | All tutorials will be live face-to-face and online via MS Teams. Face-to-face attendance is compulsory (barring exceptional circumstances such as COVID infection).
A tutorial video will be available immediately after each tutorial to help with student revision.
The Statistical Physics III lecturer (Professor John Wojdylo) will be available to answer questions via Teams chat. |
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