学部・大学院区分
Undergraduate / Graduate | | 理学部 | | 時間割コード
Registration Code | | 0680170 | | 科目区分
Course Category | | 専門科目
Specialized Courses | | 科目名 【日本語】
Course Title | | 統計物理学２ | | 科目名 【英語】
Course Title | | Statistical Physics II | | コースナンバリングコード
Course Numbering Code | | | | 担当教員 【日本語】
Instructor | | WOJDYLO John Andrew ○ | | 担当教員 【英語】
Instructor | | WOJDYLO John Andrew ○ | | 単位数
Credits | | 2 | | 開講期・開講時間帯
Term / Day / Period | | 秋 金曜日 ５時限
Fall Fri 5 | | 授業形態
Course style | | 講義
Lecture | | 学科・専攻
Department / Program | | | | 必修・選択
Compulsory / Selected | | | See "Course List and Graduation Requirements" for your program for your enrollment 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 | | In this course, students will gain a throrough grounding in Thermodynamics and Statistical Mechanics, enabling them to continue to further studies next semester in Statistical Physics III, as well as to solve advanced problems in Thermodynamics and intermediate problems in the Canonical and Microcanonical Formalisms.
Statistical Mechanics is one of the major fields of physics: around 30% of Physics Nobel Prizes have been awarded for discoveries 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 (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 unit is the first half of a full-year course. After learning the mathematical structure of thermodynamics and how thermodynamics works -- with many examples of systems beyond the ideal gas -- students are introduced to equilibrium statistical mechanics, which describes the equilibrium conditions of systems consisting of a large number of particles. Applications are considered in condensed matter physics, solid state physics, cosmology, chemistry, materials science and biology. Problem-solving is an integral part of the course: students should attend fortnightly tutorials (Physics Tutorial IIIa) where they will discuss many of the assignment questions and receive hints for solutions. Weaker students are particularly encouraged to attend tutorials and submit assignments. This semester students are thoroughly prepared for quantum statistical mechanics in SP3 next semester. It is recommended that students take Quantum Mechanics II concurrently.
At the end of Statistical Physics III next semester students will be adequately prepared with regards to their knowledge of statistical mechanics and thermodynamics to undertake further studies in S-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. A knowledge of statistical mechanics (quantum and classical) is essential for students interested in experimental physics, theoretical physics, chemistry and mathematical biology. |
| | 到達目標 【日本語】
Objectives of the Course(JPN)) | | | | 到達目標 【英語】
Objectives of the Course | | After learning the mathematical structure of thermodynamics and why thermodynamics works – with many examples of systems beyond the ideal gas -- students are introduced to equilibrium statistical mechanics, which describes the equilibrium conditions of systems consisting of a large number of particles.
Students will gain a throrough grounding in Thermodynamics and Statistical Mechanics, enabling them to continue to further studies next semester in Statistical Physics III, as well as to solve advanced problems in Thermodynamics and intermediate problems in the Canonical and Microcanonical Formalisms. |
| | 授業の内容や構成
Course Content / Plan | | Course Contents
Callen Chapts 1-8, 15-17, 21 (some parts omitted); Reif Chapts 1-3, 6-7, Appendix A12. Some topics are more fully explored in tutorials. All lectures are recorded and made available in MS Teams immediately after the lecture.
Lecture 1. Fundamental Relation, Entropy Representation; Postulates of the Entropy. Partial derivatives and experiments. Thermodynamic coordinates. Existence of the internal energy thermodynamic potential. Existence of an entropy function of state -- proof from within thermodynamics. Basic postulates of thermodynamics. Nernst Postulate (3rd Law of Thermodynamics.) Fundamental relation in the Entropy representation. Based on Callen Chapter 1; and Zemansky and Dittman Chapter 2.
Lecture 2. Top-Down Approach: Equations of State from the Fundamental Relation. Examples. Extensive parameters are homogeneous order 1. Intensive parameters are homogeneous order 0. Thermal and Mechanical Equilibrium. Euler relation. Gibbs-Duhem relation. Based on Callen Chapter 2.
Lecture 3. Bottom Up Approach: Fundamental Relation from the Equations of State. Mathematical theorems underlying thermodynamics. Examples of applying compatibility condition, Gibbs-Duhem Relation, Euler Relation, 1st Law in molar form. 2nd Equation of State from van der Waals Equation of State. Example: rubber band; photon gas; Fundamental Relation for one-component ideal gas. Ideal gas: "Gibbs Paradox"? Entropy of a mixture: “entropy of mixing”. Molar heat capacity and other derivatives. Based on Callen Chapter 3.
Lecture 4. The Maximum Work Theorem. Possible and impossible processes. Quasistatic and reversible processes – how can temperature be increased reversibly? Heat flow and coupled systems. The maximum work theorem (proof without using Carnot cycle). Carnot Efficiency. Carnot cycle: why is it necessary? Carnot cycle for a photon gas.
Lecture 5. Thermodynamic potentials and their physical interpretation. The Legendre transform. Thermodynamic potentials: internal energy, Helmholtz free energy, Gibbs free energy, enthalpy. Based on Callen Chapter 5.
Lecture 6. The Extremum Principle in the Legendre Transformed Representations. Physical meaning of the potentials: a first look. Minimum principles. Applications of thermodynamics. How to measure the entropy. How to liquefy gases -- throttling (Joule-Thompson process). Based on Callen Chapter 6.
Lecture 7. Maxwell's relations and applications. Algorithm for reducing thermodynamic derivatives to a combination of easily-measurable quantities. Applications: adiabatic compression; isothermal expansion; free expansion. Based on Callen Chapter 7.
Lecture 8. Introduction to Statistical Mechanics: the “flavour” of SM. Intuitive introduction to the SM of “isolated” systems: i.e. entropy representation – microcanonical ensemble (semi-classical treatment). Why are predictions possible at all? Central Limit Theorem. Postulate of equal a priori probability. Importance of the number of accessible states Omega. Some examples of counting Omega: Einstein model of a crystalline solid; the “two-state model” and the Schottky Hump. Entropy and Omega. Equilibrium conditions. Combinatorial methods for counting Omega are usually difficult and of limited use – two ways to overcome this: 1) take advantage of high dimensionality [today]; 2) use Legendre transformed representations [later]. Based on Callen Chapter 15.
Lecture 9. Basic Facts About the Binomial Distribution. How a Gaussian emerges from the binomial distribution; mean, variance, standard deviation, etc. Reif Chapter 1. NONEXAMINABLE: Theory of the Classical Microcanonical Ensemble: Gibbs Ensemble; Classical Liouville Theorem; role of symmetry in Statistical Mechanics. Classical microstates. The density function and thermodynamic averages. Classical Ergodic Theorem. (Huang p. 52-54; 62-65; 127-135). The density function is a conserved quantity, therefore it must be expressible as a linear combination of fundamental additive conserved quantities. (Lifshitz and Pitaevski p. 11.)
Lecture 10. Evolution of an Isolated System to Equilibrium – Entropy. Why do many kinds of isolated systems tend to equilibrium; why do systems evolve out of equilibrium states? Why do systems in an ensemble want to occupy each possible state equally; why do probability distributions want to be flat, thus maximizing our ignorance of the system state? Why does entropy increase in a spontaneous process? Why is entropy maximum at equilibrium? Why S=k Log(Omega)? Why S= -k Sum{pr Log pr}? Principle of Detailed Balance; Boltzmann’s H-Theorem. (Reif A12.)
Lecture 11. Statistical Mechanics in the Helmholtz Representation: Canonical formalism. Canonical partition function. Boltzmann probability distribution derived for a system in contact with a heat reservoir. Connection with thermodynamics. Weakly-interacting systems: additivity of energies and factorizability of the partition function. Basic examples. (Callen 16-1, 16-2.)
Lecture 12. Canonical Partition Function cont’d. Fluctuations. Adiabatic work. Microscopic effect of work and heat. In equilibrium, energy probability distribution is Gaussian: overwhelming probability that the system is within 1 standard deviation of the mean. 3rd Law of Thermodynamics. If the mean energy of a system is known then the canonical ensemble applies (temperature is fixed). Proof: allow energy of the system to fluctuate and use Method of Lagrange Multipliers to find the MOST PROBABLE probability distribution, which is Boltzmann's distribution. (Reif Ch. 6.) Other kinds of partition function: equilibrium distribution for systems in contact with a particle reservoir or volume reservoir; equivalence between all these distributions (away from a phase transition) due to high dimensionality (Hill, Ch. 1,2); a constructive, physical proof of equivalence (van Kampen).
Lecture 13. Paramagnetism. We solve the problem of paramagnetism first for the spin-1/2 case then for the arbitrary spin case. (Reif p. 206-208; 257-262.) Bohr-van Leeuwen Theorem. (Kubo, p. 138.) NONEXAMINABLE: 1st Law of Thermodynamics for Magnetic Systems: role of a pair of intensive and extensive parameters is swapped (Kittel Chapt. 18; Callen Appendix B).
Lecture 14. The Classical Limit. Why is our approach classical even though we write the partition sum as a discrete sum over states (with many-body energy eigenvalues Er)? Pauli Exclusion Principle and the enumeration of states. A combinatorial approach to proper counting: derivation of the quantum distribution functions. The Bose-Einstein and Fermi-Dirac Distributions tend to the Boltzmann Distribution in the “classical limit” – why? What characterizes the classical limit? The essential condition: large quantum numbers; implication for the PEP as well as for simultaneous measurement of canonically conjugate observables. Example: translational and internal modes (vibrational, rotational etc. modes) of a diatomic ideal gas (Callen 16-3). Symmetry number. How the classical partition function emerges from the semi-classical partition function. Classical ideal gas: equations of state, entropy (Reif 7.2, 7.3 p 239-246). The classical equipartition theorem (Reif 7.5 p 248-250).
Lecture 15. 1. Kinetic Theory of Dilute Gases in Equilibrium. Maxwell velocity distribution. Distribution of a component of velocity. Speed distribution: mean, rms, most probable speeds. (Reif 7.9, 7.10). NONEXAMINABLE: 2. Fundamental Postulates of Quantum Statistical Mechanics: in an isolated system, equal a priori probabilities and the postulate of random phase. The operation of taking a statistical average and quantum average simultaneously: the density operator. Properties of the density operator. Canonical partition function in terms of the density operator. Why the off-diagonal entries of the density operator must be zero – this leads to the random phase postulate. Meaning of the de Broglie wavelength of a molecule in an ideal gas in equilibrium at temperature T. |
| | 履修条件
Course Prerequisites | | | Calculus I; Calculus II; or Consent of Instructor |
| | 関連する科目
Related Courses | | Quantum Mechanics II; Physics Tutorial IIIa; Statistical Physics III (next semester).
It is strongly advised that students concurrently enroll in Physics Tutorial IIIa. |
| | 成績評価の方法と基準
Course Evaluation Method and Criteria | | | Attendance and class performance, attitude: 5%; Weekly quizzes or other written assessment: 30%; Midterm exam: 32.5%; Final Exam: 32.5% |
| | 不可(F)と欠席(W)の基準
Criteria for "Fail (F)" & "Absent (W)" grades | | The "Absent (W)" grade is reserved for students who withdraw by November 16. After that day, a letter grade will be awarded based on marks earned from all assessment during the semester.
If Statistical Physics II is NOT A COMPULSORY SUBJECT and the student plans never to take Statistical Physics II in the future, then a late withdrawal request will be considered. |
| | 参考書
Reference Book | | 1. Hill, T., An Introduction to Statistical Thermodynamics, Dover, 1986. (Excellent introduction to Statistical Mechanics at Year 3 level. Alternative textbook. Highly recommended. Cheap to buy.)
2. Zemansky, M.W. and Dittman, R.H., Heat and Thermodynamics, An Intermediate Textbook, McGraw-Hill, 1992. (Excellent for empirical basis of thermodynamics.)
3. Kittel, C., Elementary Statistical Physics, Dover, 2004. (Highly recommended. Cheap to buy.)
4. Kubo R., Statistical Mechanics, North Holland, 1965. (More of a second course on Stat Mech, but contains many examples and worked solutions.)
5. Huang, K., Statistical Mechanics, Wiley. (Advanced reference.)
6. Kittel, C. and Kroemer, H., Thermal Physics, W.H. Freeman. (Try as alternative.)
7. 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.) |
| | 教科書・テキスト
Textbook | | 1. Callen, H., Thermodynamics and an Introduction to Thermostatistics, 2nd ed.,Wiley, 1985.
(The central textbook in this course. Japanese translation has fewer typographical errors.)
2. Reif, F., Fundamentals of Statistical and Thermal Physics, McGraw-Hill, 1965. |
| | 課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | | You are expected to revise the lecture notes, read 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. |
| | 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 | | | | 他学科聴講の条件
Conditions for Other department student's attendance | | | Students from any department are welcome as long as they have the necessary mathematical grounding. |
| | 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 concurrently enrol in Physics Tutorial IIIa.
• To get the full benefit of this unit, students should enrol in Statistical Physics III next semester. To get the full benefit of Statistical Mechanics, students should also enrol in Quantum Mechanics II, and Quantum Mechanics III next semester. The next level up -- Year 4 or graduate level -- is Condensed Matter Field Theory, which has as its starting point the first half of QMIII, parts of QMII, and all of SPIII. |
| | 授業開講形態等
Lecture format, etc. | | | Live lectures via MS Teams (Online only). Before the start of semester students should ensure that they have corerctly installed MS Teams using their Nagoya University email account. |
| | 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 lectures will be live via MS Teams (online only).
A lecture video will be available immediately after the lecture to help with student revision.
The lecturer will be available at most times during the day to answer questions via Teams chat. |
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