The situation that the number of particles of a physical system varies is described by the grand canonical ensemble, which takes the chemical potential as a parameter. The grand partition function is introduced as a tool of calculation. Many particle quantum mechanics for Bose and Fermi particles is explained. The statistical mechanics for Bose particle and that for Fermi particle are quite different in low temperature and high density (Bose condensation, Fermi degeneracy). Basic exercises on phase transition, critical phenomena, non-equilibrium phenomena are prepared.
This exercise class aims to develop students' ability to solve basic problems on statistical mechanics when the number of particles varies, quantum statistical mechanics and theory of phase transition.
This course facilitates students' understanding and ability of calculation of grand canonical ensemble, statistical mechanics of quantum ideal gases, physical phenomena originated from difference between Fermi and Bose particles, phase transition, critical phenomena, non-equilibrium phenomena.
grand canonical ensemble, chemical potential, quantum ideal gases, Bose distribution, Fermi distribution, Bose condensation, Fermi degeneracy, super-fluid transition of Helium 4, specific heat of metals, phase transition, critical phenomena, critical exponent, mean field approximation, transfer matrix method, Ising model
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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Exercise materials are distributed in advance. Students are expected to solve exercises at home and to present answers on blackboard. Presented answers are discussed.
Course schedule | Required learning | |
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Class 1 | grand canonical ensemble I (review of thermodynamics, basics of granduanonical ensemble) | In grand canonical ensemble, particle number varies, whereas chemical potential is specified. |
Class 2 | grand canonical ensemble II (grand partition function, application of grand canonical ensemble) | Grand partition function is the tool for calculation in the theory of grand canonical ensemble. |
Class 3 | grand canonical ensemble III (application of grand canonical ensemble) | Several examples are calculated by the theory of grand canonical ensemble. |
Class 4 | statistical mechanics of quantum ideal gases I (many particle quantum mechanics) | In many particle quantum mechanics, any state has a definite parity for two particle exchange based on statistics of the particle. |
Class 5 | statistical mechanics of quantum ideal gases II (Fermi particle, Bose particle) | May particle systems of Boson and Fermion obey Fermi and Bose distributions, respectively. |
Class 6 | statistical mechanics of quantum ideal gases III (ideal Fermi case, Sommerfeld expansion) | A strongly degenerate Fermi system can be analyzed by Sommerfeld expansion. |
Class 7 | statistical mechanics of quantum ideal gases IV (specific heat and magnetic susceptivity of ideal Fermi gas) | Specific head and magnetics susceptivity of metals can be explained by Sommerfeld expansion. |
Class 8 | statistical mechanics of quantum ideal gases V (ideal Bose case, black body radiation, introduction of Bose condensation) | When ideal Bose case is placed in low temperature enough, macroscopic number of particles condense in the microscopic ground state. |
Class 9 | statistical mechanics of quantum ideal gases VI (Bose condensation) | The super fluid transition of Helium 4 can be understood as Bose condensation of ideal Bose gas. |
Class 10 | phase transition and critical phenomena I (phase and phase transition, first order transition, second order transition) | Distinction of phases in a phase transition is described by the corresponding order parameter. |
Class 11 | phase transition and critical phenomena II (What is critical phenomena?, critical exponent, Landau theory) | Critical phenomena in a phase transition are described by universal critical exponents irrespective of details of the system. |
Class 12 | phase transition and critical phenomena III (What is mean field approximation?, Bragg-Williams approximation) | The first step to analyze phase transition in a model is to apply the mean field approximation to it. |
Class 13 | phase transition and critical phenomena IV (mean field theory of Ising model) | Most basic model for second order magnetic transition of magnet is Ising model. It can be analyzed by the mean field approximation. |
Class 14 | phase transition and critical phenomena V (one-dimensional Ising model, transfer matrix method) | One-dimensional Ising model can be analyzed exactly by the transfer matrix method. |
Class 15 | non-equilibrium statistical mechanics (linear response theory) | Entrance to investigation of non-equilibrium phenomena is the investigation of transfer coefficients. The linear response theory of Kubo calculates them. |
not yet decided
not yet decided
Based on presentation on blackboard, report, examination, etc..
Students are expected to have learned Thermodynamics and Statistical Mechanics I and Exercises in Thermodynamics and Statistical Mechanics I.