# Statistical Physics 2 (PHYS*4240)

**Code and section:** PHYS*4240*01

**Term:** Fall 2018

**Instructor:** Robert Wickham

## Details

## Course Information

### Course Objectives

Two years ago, PHYS*2240 introduced you to the central role that entropy and the second law play in thermodynamics. The entropy concept originates from our lack of precise knowledge about which of the many, many microscopic states a system is actually in, despite our imposition of constraints on the system at a macroscopic level. This statistical perspective on entropy motivated the second law. We explored the consequences of the second law for equilibrium, for phase transitions, and for various applications and measurable quantities. PHYS*4240 continues the discussion begun two years ago.

We will follow the topic selection and structure in the textbook Thermal Physics by Daniel V. Schroeder. PHYS*4240 begins with a brief review of thermodynamics. This covers Chapters 1-3 and selected topics from Chapter 5. We will generalize the thermodynamic formalism from the isolated systems we considered in PHYS*2240, which have constant energy, volume, and particle number, to systems with other constraints, such as fixed temperature, fixed pressure, and/or fixed chemical potential. Free energies and extremum principles, which are ultimately connected to the entropy and the second law, are key concepts in these situations. The overall structure, and universal nature, of thermodynamics will be emphasized.

The central portion of the course (Chapter 6) develops convenient statistical methods for calculating free energies and other thermal-average properties of materials by counting ensembles of appropriately-weighted microstates. Systems we will consider include the ideal gas, the van der Waals fluid, the paramagnet, and the Einstein solid.

The final part of the course is devoted to the statistical mechanics of systems that are described by the laws of quantum mechanics (Chapter 7). We will resolve several issues with the classical treatment of the ideal gas that are related to the counting of microstates when the particles are indistinguishable. We will discuss the distinct statistics of fermions and of bosons. Topics include the quantum ideal gas, the heat capacity of the free electron gas in metals, blackbody radiation and the photon gas, Bose-Einstein condensation, and finally the treatment of quantized lattice vibrations (phonons) of a solid.

You will refine your analytical and problem-solving skills through regular written assignments.

### Class Schedule and Location

Day | Time | Location |
---|---|---|

Monday, Wednesday, and Friday | 11:30 am - 12:20 pm | MCKN 305 |

First Lecture: Friday, September 7th

Last Lecture: Friday, November 30th

The course runs for 12 weeks (36 lectures); there is no lecture on Thanksgiving (Monday, October 8th). Friday, November 30th is a Thanksgiving make-up lecture.

### Instruction

Lecturer | Office | Extension | |
---|---|---|---|

Rob Wickham | MacNaughton 448 | 53704 | rwickham@uoguelph.ca |

#### Office Hours

Monday, 2:00 pm - 3:00 pm and Tuesday, 1:00 pm - 3:00 pm. Assignments will be due on (alternate) Wednesdays. Please send me an email if you can't find me and wish to schedule a meeting.

### Course Materials

#### Course Website

CourseLink: Login via https://courselink.uoguelph.ca/

#### Required Textbook

An Introduction to Thermal Physics, by D. V. Schroeder (Addison Wesley Longman, 2000).

#### Other, optional resources

A typed set of related notes by Eric Poisson which can be found on his web site (see faculty link on the departmental web site).

#### Some of the Classic References

- F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill,1965,QC 175.R43).
- F. Mandl, Statistical Physics, Second Edition (Wiley,1988,QC174.8.M27).
- D.L. Goodstein, States of Matter (Prentice Hall, 1975; Dover, 1985, QC 173.3.G66).
- K. Huang, Statistical Mechanics, Second Edition (Wiley,1987,QC174.8.H83).
- C. Kittel and H. Kroemer, Thermal Physics, Second Edition (Freeman, 1980, QC 311.5.K52).
- L.D. Landau and E.M. Lifshitz, Statistical Physics, Third Edition, Part 1 (Pergamon, 1980, QC 175.L32).
- R.K. Pathria, Statistical Mechanics (Pergamon,1972,QC175.P35).

At this stage of your education, you should be consulting more than one text to enhance your learning and understanding of the material. No particular book is perfect in all respects and scientists regularly refer to several books and papers to understand a concept.

### Evaluation

Assessment | Weight | Due Date |
---|---|---|

Assignments (5) | 30% | Sept. 26, Oct. 10, Oct. 24, Nov. 14, Nov. 28 |

Midterm Test | 30% | Monday, October 29th, 7-9 pm, place TBD |

Final Exam | 40% | December 7h, 7:00 - 9:00 pm, place TBD |

A medical certificate is required if the exam is missed.

Assignments are due at the beginning of class; late assignments will receive a grade of zero.

Physics is not done in a vacuum. (OK, sometimes it is...) Students may discuss assignments amongst themselves but their written solutions must not be shared with anyone (this would be an example of plagiarism).

Plagiarism is the act of appropriating the ``...composition of another, or parts or passages of his [or her] writings, or the ideas or language of the same, and passing them off as the product of one's own mind...'' (Black's Law Dictionary). A student found to have plagiarized will receive zero for the work concerned. Collaborators shown to be culpable will be subject to the same penalties.

### Course Outline

I. Thermodynamics: Review and unfinished business [Chapters 1 to 3, 5.1, 5.2]

1. Equation of state, state variables, constraints, equilibrium

2. van der Waals equation of state, limiting cases, isothermal compressibility, phase coexistence, thermal, mechanical and diffusive equilibrium

3. First law, energy, heat, work, quasi-static processes, adiabatic processes

4. Isolated systems (U, V, N fixed) and the second law, entropy, example: van der Waals model, properties of entropy

5. Structure of thermodynamics: derivatives of entropy are equations of state, second derivatives are response functions, examples: equipartition, thermodynamic identity

6. Equivalent representations of thermodynamics, free energies, geometrical interpretation of the Legendre transform, extremum principles [5.1, 5.2]

7. Derivative relations and thermodynamic identities arising from free energies, Maxwell relations

8. Examples involving Maxwell relations: heat capacities and compressibilities

II. Calculating thermodynamic potentials: ensembles and examples (mainly Ch. 6)

9. Macrostates and microstates, multiplicity of the two-state paramagnet, fundamental assumption

10. Classical entropy of the ideal gas in the microcanonical ensemble [Chapter 2]

11. Two-state paramagnet in thermal contact with a reservoir, the Boltzmann factor [6.1]

12. General theory for systems in contact with a reservoir, partition function, averages in the canonical ensemble [6.2]

13. Examples, paramagnet, Einstein solid

--Thanksgiving--

14. Partition function and Helmholtz free energy, composite systems

15. Classical partition function for the ideal gas, and thermodynamics [8.1]

16. Equipartition of energy among degrees of freedom [1.3, 6.3]

17. Energy fluctuations in the canonical ensemble

18. Weakly interacting gases [8.1]

19. Identical particles, indistinguishability, and mixing

20. Issues with the classical model for ideal gas thermodynamics

21. Partition function for a quantum particle in a 1D box, 3D case, N non-interacting particles, internal degrees of freedom, heat capacity, rotation of diatomic molecules.

Midterm evening of Monday, October 29th (following lecture 22)

22. Chemical potential of an ideal gas, diffusive equilibrium in an external field: isothermal atmosphere, mobile magnetic particles

23. The Gibbs factor, grand partition function, averages in the grand canonical ensemble [7.1]

24. Example: adsorption of oxygen in the blood, particle number fluctuations

40th class day Friday, November 2nd

III. Quantum statistical mechanics (Chapter 7)

25. Fermions and bosons, microstates of N ideal, indistinguishable quantum particles, occupation numbers, quantum statistics, quantum volume

26. Fermi-Dirac and Bose-Einstein distribution functions, classical limit

27. Degenerate ideal Fermi gas, ground state of a Fermi gas (T=0)

28. Thermodynamic properties of the ground state of a Fermi gas

29. Non-zero temperatures, density of states, Sommerfeld expansion

30. Heat capacity of a degenerate ideal Fermi gas, electrons in metals

31. Chemical potential of Bose gas, ground and excited state occupancies

32. Bose-Einstein condensation, examples: liquid 4He, dilute gas

33. Gas of photons in thermal equilibrium, Planck distribution

34. Planck spectrum for blackbody radiation, thermodynamics

35. Debye theory of lattice vibrations in solids: phonons

36. Phonon thermodynamics, phonon contribution to the heat capacity of a solid

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