Statistical Mechanics (PHYS7021)

Course overview

Theoretical understanding of the physical properties of samples of material of macroscopic size, on the basis of the known quantum mechanical behaviour of the constituent (microscopic) particles; micro-canonical, canonical, and grand-canonical ensembles; classical and quantum gases; photons and phonons; Planck distribution and black-body radiation; fermions and bosons; Fermi-Dirac distribution and Fermi energy; Bose-Einstein distribution and Bose condensation.

The course will cover the microscopic (Statistical Mechanics) approaches to thermal physics, including: micro-canonical, canonical, and grand-canonical ensembles; classical and quantum gases; photons and phonons; Planck distribution and black-body radiation; fermions and bosons; Fermi-Dirac distribution and Fermi energy; Bose-Einstein distribution and Bose condensation.

Statistical mechanics is the incredibly successful theory which allows us to describe macroscopic or thermodynamic properties of materials in terms of the known dynamics (either classical or quantum) of the constituent microscopic particles (such as atoms and molecules). Statistical mechanics gives a microscopic understanding of phenomena as diverse as entropy and the second law of thermodynamics, the ideal gas law, the cosmic microwave background radiation, neutron stars, superfluidity, the denaturation of proteins, carbon monoxide poisoning, magnetic phase transitions, and the electronic properties of metals. These diverse applications will be illustrated in the tutorial problems.

The subject provides part of a comprehensive, complete and coherent program of education in Physics intended for students aiming to become professional physicists. It is a compulsory subject for entry into Physics Postgraduate Honours.

The course would also be beneficial to students in biochemistry, chemistry, computer science, materials science, and mechanical and chemical engineering.

Course requirements

Assumed background

Essential:

MATH2000/2001/2901 – Calculus and Linear Algebra II

PHYS2020 – Thermodynamics and Condensed Matter Physics

PHYS2041 – Quantum Physics

Desirable:

MATH2100 –ﾠApplied Mathematical Analysis

PHYS3040 –ﾠQuantum Physics

Intending students need to be competent in calculus, particularly including the theory of first order partial derivatives. The course is presented on the assumption that the student is familiar with second year Thermodynamics as outlined, for example, in the textbook by Herbert Callen or in the first few chapters of Daniel Schroeder's "Thermal Physics" and covered in PHYS2020ﾠThe development proceeds from an assumed background that includes all the basic results of a course in elementary quantum mechanics such as PHYS2041. A knowledge of basic statistics and probability theory is also desirable.ﾠ

More specifically, the expected capabilities on entering the course are as follows:

In addition to the general first year capabilities, such as

• basic differentiation and integration
• finding extrema of a function
• integration as area under a curve
• visualise and sketch simple functions
• visualise a function of 2 variables (eg contour plot)
• logarithms, exponentials and trigonometric functions
• definite vs indefinite integrals
• Taylor series of elementary functions, especially of exp(x) and ln(1+x).
• conversion between commonly used units
• keeping track of units in a calculation
• using simple checks to assess whether an answer makes sense (e.g. dimensions, orders-of-magnitude, limiting behaviours, consistency with fundamental principles and axioms)

students are expected to have the following core skills from 2nd year physics and mathematics courses:

ﾠ

1. Conceptual understanding

From second-year thermodynamics (PHYS2020):

• basic understanding of temperature, entropy, heat versus work, heat capacity
• familiarity with the laws of thermodynamics and the ideal gas law
• practical application of partial derivatives, for e.g. calculating the heat capacity of various physical systems

From second-year quantum (PHYS2040):

Statistical mechanics makes use of a variety of results from quantum mechanics. Students should have basic knowledge or familiarity with:

• the quantum mechanical wavefunction
• quantum numbers and energy levels for a particle in a rectangular potential well (finite box potential) and a particle in a harmonic potential (harmonic oscillator problem); familiarity with the respective quantum wavefunctions
• quantum mechanics of angular momentum and addition of angular momenta;
• concept of spin;
• wave optics, characteristics of light – wavelengths/frequency
• photons and the ideas behind quantisation of electromagnetic field;
• quantised energy levels of a mode of electromagnetic field;
• black-body radiation.

Although they are covered in the course, it would be an advantage for students to be familiar with basic ideas behind quantum mechanical treatment of systems of identical particles:

• symmetric versus anti-symmetric wavefunctions;
• fermions and bosons;
• Pauli exclusion principle.

ﾠ

2. Mathematics skills

• Knowledge, understanding, and manipulation of elementary functions such as powers, exponentials, logarithms (including manipulations involving log(AB) and log(A/B)), trigonometric functions, as well functions of complex variables such as exp(ix), and hyperbolic functions (sinh(x), cosh(x), tanh(x) etc.);
• Differentiation: practical knowledge of derivatives of elementary functions, application of the chain rule, and knowledge of partial derivatives;
• Integration: distinction between definite and indefinite integrals, integrals of elementary functions, integration by parts; understanding of line, surface, volume, and multiple integrals;
• Even and odd functions, integrals of even and odd functions between symmetric boundaries;
• Gaussian function – mathematical form and properties; ability to evaluate the mean and the dispersion of a Gaussian, ability to illustrate these quantities graphically;
• Elementary vector algebra; addition of vectors, evaluation of their length;
• Basic knowledge of the theory of probabilities, statistics, and combinatorics; understanding of permutations, factorials, binomial coefficients etc.;
• Knowledge of the geometric series and binomial expansion.

ﾠ

3. Experimental

• design and conduct simple measurements using electrical multimeters, oscilloscopes, thermometers, rulers, etc.;
• estimate uncertainties and propagate uncertainties through calculations;
• write a structured laboratory report, including literature exploration and appropriate citation.

ﾠ

Prerequisites

You'll need to complete the following courses before enrolling in this one:

PHYS2020 + [MATH2000 or MATH2001 or MATH2901] + [PHYS2041 or PHYS2941]

Recommended prerequisites

We recommend completing the following courses before enrolling in this one:

MATH2100

Incompatible

You can't enrol in this course if you've already completed the following:

PHYS3020 (co-taught), PHYS3920 (co-taught)

Course contact

Course staff

Timetable

The timetable for this course is available on the UQ Public Timetable.

Additional timetable information

Wednesday 14 August: Ekka show day - No lecture. No tutorial in week 4.

All classes will be conducted on campus. Consult your personal timetable for times and locations. Students are expected to attend these sessions in person unless they have a valid reason for being unable to attend (such as illness).

Important: if you are ill, then do not attend any classes in person. Alternative arrangements can be organised – consult Blackboard for details. 

Aims and outcomes

This course provides an introduction to the description of many-particle systems using statistical methods. It introduces a few simple, fundamental concepts, and proceeds with their logical application to a wide variety of physical systems. In this manner it builds an understanding of macroscopic system behaviour (thermodynamics) from the knowledge of the microscopic behaviour of the components. The broad aim of the course is to provide the students with an excellent understanding of these fundamentals and develop their ability to apply the methods learned to a wide range of physical systems.

Learning outcomes

After successfully completing this course you should be able to:

Assessment

Assessment summary

A hurdle is an assessment requirement that must be satisfied in order to receive a specific grade for the course. Check the assessment details for more information about hurdle requirements.

Assessment details

Laboratory written report (x 2)

• Hurdle
• Team or group-based

Mode

Written

Category

Paper/ Report/ Annotation

Weight

20% (2 x 10%)

Due date

Due at 2:00 pm on days noted on lecture schedule available on Blackboard.

Learning outcomes

L01, L03, L04

Task description

Hurdle requirements

Submission guidelines

The laboratory component is assessed based on the written reports submitted by the laboratory group on the two experiments carried out. The final reports must be submitted within two weeks of the second afternoon scheduled for each experiment. Submission is via the Blackboard submission portal as a single PDF file. Document must be uploaded by 2:00 pm on the indicated day. The written reports will be marked for correctness and each report will receive a grade that carries 10% weight of the final grade.

Deferral or extension

You may be able to apply for an extension.

Problem solving

Mode

Written

Category

Tutorial/ Problem Set

Weight

20%

Due date

12/08/2024 2:00 pm

2/09/2024 2:00 pm

30/09/2024 2:00 pm

21/10/2024 2:00 pm

Learning outcomes

L01, L02, L04

Task description

Submission guidelines

Solutions to the problems for each module must be submitted as a single PDF file via the Blackboard portal by 2:00 pm as stated in the lecture schedule.

Deferral or extension

You may be able to apply for an extension.

As solutions are released on Blackboard 2 days after submission, no extensions beyond the solution release can be granted. See ADDITIONAL ASSESSMENT INFORMATION for extension and deferred examination information relating to this assessment item.

Late submission

A penalty of 10% of the maximum possible mark will be deducted per 24 hours from time submission is due for up to 7 days. After 7 days, you will receive a mark of 0.

Computational Project and Report

Mode

Written

Category

Project

Weight

20%

Due date

14/10/2024 2:00 pm

Due at 2:00 pm on date noted on lecture schedule available on Blackboard.

Learning outcomes

L01, L02, L03, L04

Task description

Submission guidelines

A single PDF document will be uploaded to the submission portal on Blackboard by 2:00 pm on the indicated day.

Deferral or extension

You may be able to apply for an extension.

The maximum extension allowed is 7 days. Extensions are given in multiples of 24 hours.

Final Examination

• Hurdle

Mode

Written

Category

Examination

Weight

40%

Due date

End of Semester Exam Period

2/11/2024 - 16/11/2024

Learning outcomes

L01, L02

Task description

Hurdle requirements

Exam details

Submission guidelines

Deferral or extension

You may be able to defer this exam.

See ADDITIONAL ASSESSMENT INFORMATION for extension and deferred examination information relating to this assessment item.

Course grading

Full criteria for each grade is available in the Assessment Procedure.

Additional course grading information

Non-integer grade scores will be rounded up to the next integer value (e.g. 74.4% rounds up to 75% and counts as a Grade 6).

In order to receive a passing grade, students must pass the following hurdles:

1. Score at least 40% of the marks for the final exam;

2. Hand in at least one lab report (receiving 30% of the marks or more) to their assigned laboratory experiment.

Supplementary assessment

Supplementary assessment is not available for some items in this course.

Should you fail a course with a grade of 3 you may be eligible for supplementary assessment. Refer to my.UQ for information on supplementary assessment and how to apply. 

Supplementary assessment provides an additional opportunity to demonstrate you have achieved all the required learning outcomes for a course.  

Supplementary assessment is not available if you have failed the lab component of this course. The learning objects of the laboratory components are demonstrated throughout the semester and it is not possible to validly reassess these learning objectives in the supplementary assessment period.  

If you apply and are granted supplementary assessment, the type of supplementary assessment set will consider which learning outcome(s) have not been met.  

Supplementary assessment in this course will be a 2-hour examination similar in style to the end-of-semester examination. To receive a passing grade of 3S4, you must obtain a mark of 50% or more on the supplementary assessment. 

Additional assessment information

Artificial Intelligence

The assessment tasks in this course evaluate students’ abilities, skills and knowledge without the aid of Artificial Intelligence (AI). Students are advised that the use of AI technologies to develop responses is strictly prohibited and may constitute misconduct under the Student Code of Conduct.

Applications for Extensions to Assessment Due Dates

Extension requests are submitted online via my.UQ – applying for an extension. Extension requests received in any other way will not be approved. Additional details associated with extension requests, including acceptable and unacceptable reasons, may be found at my.UQ.

Please note:

• Requests for an extension to an assessment due date must be submitted through your my.UQ portal and you must provide documentation of your circumstances, as soon as it becomes evident that an extension is needed. Your application must be submitted on or before the assessment item's due date and time.
• Applications for extension can take time to be processed so you should continue to work on your assessment item while awaiting a decision. We recommend that you submit any completed work by the due date, and this will be marked if your application is not approved. Should your application be approved, then you will be able to resubmit by the agreed revised due date.
• If an extension is approved, you will be notified via your my.UQ portal and the new date and time for submission provided. It is important that you check the revised date as it may differ from the date that you requested.
• If the basis of the application is a medical condition, applications should be accompanied by a medical certificate dated prior to the assignment due date. If you are unable to provide documentation to support your application by the due date and time you must still submit your application on time and attach a written statement (Word document) outlining why you cannot provide the documentation. You must then upload the documentation to the portal within 24 hours.
• If an extension is being sought on the basis of exceptional circumstances, it must be accompanied by supporting documentation (eg. Statutory declaration).
• For extensions based on a SAP you may be granted a maximum of 7 days (if no earlier maximum timeframe applies). See the Extension or Deferral availability section of each assessment for details. Your SAP is all that is required as documentation to support your application. However, additional extension requests for the assessment item will require the submission of additional supporting documentation e.g., a medical certificate. All extension requests must be received by the assessment due date and time.
• An extension for an assessment item due within the teaching period in which the course is offered, must not exceed four weeks in total. If you are incapacitated for a period exceeding four weeks of the teaching period, you are advised to apply for Removal of Course.
• If you have been ill or unable to attend class for more than 14 days, you are advised to carefully consider whether you are capable of successfully completing your courses this semester. You might be eligible to withdraw without academic penalty - seek advice from the Faculty that administers your program.
• Students may be asked to submit evidence of work completed to date. Lack of adequate progress on your assessment item may result in an extension being denied.
• There are no provisions for exemption from an assessment item within UQ rules. If you are unable to submit an assessment piece then, under special circumstances, you may be granted an exemption, but may be required to submit alternative assessment to ensure all learning outcomes are met.

Applications to defer an exam

In certain circumstances you can apply to take a deferred examination for in-semester and end-of-semester exams. You'll need to demonstrate through supporting documentation how unavoidable circumstances prevented you from sitting your exam. If you can’t, you can apply for a one-off discretionary deferred exam.

Deferred Exam requests are submitted online via mySi-net. Requests received in any other way will not be approved. Additional details associated with deferred examinations, including acceptable and unacceptable reasons may be found at my.UQ.

Please note:

• Applications can be submitted no later than 5 calendar days after the date of the original exam.
• There are no provisions to defer a deferred exam. You need to be available to sit your deferred examination.
• Your deferred examination request(s) must have a status of "submitted" in mySI-net to be assessed.
• All applications for deferred in-semester examinations are assessed by the relevant school. Applications for deferred end-of-semester examinations are assessed by the Academic Services Division.
• You’ll receive an email to your student email account when the status of your application is updated.
• If you have a medical condition, mental health condition or disability and require alternative arrangements for your deferred exam you’ll need to complete the online alternative exam arrangements through my.UQ. This is in addition to your deferred examinations request. You need to submit this request on the same day as your request for a deferred exam or supplementary assessment. Contact Student Services if you need assistance completing your alternative exam arrangements request.

Learning resources

You'll need the following resources to successfully complete the course. We've indicated below if you need a personal copy of the reading materials or your own item.

Library resources

Find the required and recommended resources for this course on the UQ Library website.

Additional learning resources information

All lectureﾠnotes, lab notes, tutorial problems, their solutions, and other relevant material will be made available on Blackboard.
ﾠ

Learning activities

The learning activities for this course are outlined below. Learn more about the learning outcomes that apply to this course.

Additional learning activity information

Readings are from the following main texts (either text can be used):ﾠ

1. Daniel V. Schroeder, An introduction to thermal physics [Schroeder].ﾠ

2. Charles Kittel and Herbert Kroemer, Thermal Physics [Kittel].

ﾠ

Indicative schedule of lectures (subject to change). Final schedule available on the course Blackboard page.

Lecture 1: Introduction; An overview of Thermal Physics.

[Schroeder, pp. 1-20].

L2: Multiplicity; States of model binary systems.

[Schroeder, pp. 49-66, 68-74] or [Kittel, pp. 5-26].

L3: Micro-canonical ensemble; Fundamental assumption.

[Schroeder, pp. 56-59, 74-94] or [Kittel, pp. 29-45].

L4: Entropy; Temperature.

[Schroeder, pp. 74-94, 98-105] or [Kittel, pp. 29-45].

L5: Laws of thermodynamics.

[Kittel, pp. 45-52] or [Schroeder 17-20, 74-98].

Project Background and Info Session

L6: Canonical ensemble formalism: Boltzmann Factor; Partition Function.

[Schroeder, pp. 220-229] or [Kittel, pp. 58-64].

L7: Canonical ensemble formalism: Free energy, Pressure, & Thermodynamic Identity.

[Schroeder, pp. 108-114, 149-152, 156-164] or [Kittel, pp. 64-72].

L8: Ideal gas: a first look; Gibbs Paradox; Equipartition Theorem.ﾠ

[Schroeder, pp. 229-255] or [Kittel, pp. 72-81].

L8 (cont.): Ideal gas: a first look; Gibbs Paradox; Equipartition Theorem.

[Schroeder, pp. 229-255] or [Kittel, pp. 72-81].

L9: Grand-canonical ensemble formalism: Chemical potential; Grand-Canonical Partition Function.

[Schroeder, pp. 115-121, 257-261] or [Kittel, pp. 117-122, 131-140].

Revision Lecture

L10: Identity of particles in Quantum Mechanics; Bosons and Fermions; Bose-Einstein and Fermi-Dirac distributions.

[Schroeder, pp. 262-271,379-380] or [Kittel, pp. 151-161].

L11: Quantum and classical regimes of an ideal gas; Grand canonical description of the ideal gas.

[Schroeder, pp. 262-271,379-380] and [Kittel, pp. 160-166].

L12: Spin multiplicity; Quantum states and quantum density of states.

[Schroeder, pp. 262-271,279-282] or [Kittel, pp. 181-188].

L13: Fermi gases.

[Schroeder, pp.271-282] or [Kittel, pp. 183-189].

L15: Bose gases.

[Schroeder, pp.315-326] or [Kittel, pp. 199-206].

Semester 2 Mid-Semester Break

L16: Finite temperature Bose gases.

[Schroeder, pp.315-326] or [Kittel, pp. 202-210].

Extra-curriculum lecture.

L17: Black-body radiation, Part I: Planck distribution.

[Schroeder, pp.288-292] or [Kittel, pp. 89-94].

L18: Black-body radiation, Part II: Planck radiation law.

[Schroeder, pp. 292-306] or [Kittel, pp. 94-98].

L19: Debye model of a solid.

[Schroeder, pp. 307-314] and [Kittel, pp. 102-110].

Extra-curricular lecture – Two-level systems and non-crystalline solids

L20: The Ising model of a ferromagnet.

[Schroeder, pp. 339-346].

Revision lecture

Policies and procedures

University policies and procedures apply to all aspects of student life. As a UQ student, you must comply with University-wide and program-specific requirements, including the:

• Student Code of Conduct Policy
• Student Integrity and Misconduct Policy and Procedure
• Assessment Procedure
• Examinations Procedure
• Reasonable Adjustments - Students Policy and Procedure

Learn more about UQ policies on my.UQ and the Policy and Procedure Library.