This exercise class gives students a training to understand statistical mechanics and to develop ability of calculation in concrete physical systems.
Binomial distribution, Poisson distribution and normal distribution are explained as a first step to theory of probability. Importing the concepts such as energy level, number of states and density of states, the theory of micro canonical ensemble is built including the definitions of temperature and entropy in the language of statistical mechanics. From there, the theory of canonical ensemble, which is suitable for practical calculations, is built. The theory of canonical ensemble is applied to the two level system and the harmonic oscillator system as most basic systems. Furthermore, theory of lattice vibration in solid and theory of black body radiation are explained.
Elementary theory of probability. Theory of micro canonical ensemble, which is the basis of whole statistical mechanics. In particular, what has equal certainty is each energy eigenstate of the physical system considered. The central tool in the theory of canonical ensemble, which is convenient for practical calculations, is partition function. Once partition function is calculated, through Helmholtz free energy, every thermodynamic quantity can be calculated within thermodynamics. By applying the theory of canonical ensemble to two level system, harmonic oscillator system, system of lattice vibration in solid and system of black body radiation, the ability of practical calculations in these systems is developed.
Binomial distribution, Poisson distribution, normal distribution. Equal certainty. Micro canonical ensemble, energy, temperature, entropy, Boltzmann formula. Canonical distribution, partition function. Helmholtz free energy, Gibbs free energy, enthalpy. Two level system, Schottky type specific heat. Heat of harmonic oscillator system. Lattice vibration in solid, Einstein model, Debye model. Black body radiation, Planck formula.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
Exercise materials are distributed in advance. Students are expected to solve problems at home and to present answers on blackboard. Presented answers are discussed.
Course schedule | Required learning | |
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Class 1 | Preparation of statistical mechanics (What is statistical mechanics? Review of thermodynamics) | Understand division of roles between statistical mechanics and thermodynamics. |
Class 2 | Preparation of statistical mechanics (basics of theory of probability) | Essence of each theory of probability is what has equal certainty. Learn basic distributions (Binomial, Poisson and normal distributions). |
Class 3 | Micro canonical ensemble (energy level, number of states, density of states) | In statistical mechanics, what has equal certainty is each energy eigenstate of object system. |
Class 4 | Micro canonical ensemble II (principle of equal a priori probabilities, entropy, condition for thermal equilibrium and temperature) | According to principle of equal weight, how temperature and entropy are defined in statistical mechanics? |
Class 5 | Micro canonical distribution III (applications of micro canonical ensemble) | By using micro canonical ensemble, calculate thermodynamic properties of several simple systems. |
Class 6 | Canonical distribution I (introduction to canonical distribution) | How is canonical ensemble defined? |
Class 7 | Canonical distribution I (partition function, Helmholtz free energy) | The central quantity in the theory of canonical ensemble is partition function. From it, Helmholtz free energy is calculated. |
Class 8 | Canonical distribution III (Applications of canonical ensemble to simple systems) | Apply theory of canonical ensemble to the same examples that have been analyzed by theory of micro canonical ensemble. Taste how canonical ensemble simplifies calculations. |
Class 9 | Canonical distribution IV (partition function from classical statistical mechanics, phase space) | By using the correspondence principle, classical statistical mechanics is derived from quantum statistical mechanics so far constructed. |
Class 10 | Canonical ensemble V (Simple application of classical statistical mechanics: ideal gases of mono atom and diatom molecule) | The difference of specific heats of monoatomic and diatomic molecules can be understood by classical statistical mechanics. |
Class 11 | Canonical distribution VI (third law of thermodynamics, low temperature limit and high temperature limit) | When temperature decreases to absolute zero, entropy decreases to zero (the third law of thermodynamics). |
Class 12 | Lattice vibration in solid and specific heat I (Einstein model and its limits, coupled oscillations) | In low temperature limit, specific heat of lattice vibration in solid tends to zero. Einstein model explains the limiting value zero but fails to explain how to approach the zero. |
Class 13 | Lattice vibration in solid and specific heat II (normal mode of oscillation, Debye model, specific heat) | To overcome Einstein model, Debye model is constructed. Debye model is based upon the normal modes of lattice vibration. This time, the limiting value zero of specific heat and also how to approach it are explained well. |
Class 14 | Black body radiation I (electromagnetic filed and harmonic oscillator) | Electromagnetic field in vacuum is equivalent to infinite number of harmonic oscillators. Thereby, statistical mechanics of black body radiation can be constructed from that of harmonic oscillator. |
Class 15 | Black body radiation II (quantum theory of black body radiation) | Historically, creation of quantum theory began from Planck formula that explains the spectrum of black body radiation. We derive the formula here. |
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Based on blackboard presentation, report and examination.
It is assumed that students have learned analytical mechanics and electromagnetism and can understand elementary part of quantum mechanics.