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# (NS) Statistical Mechanics - PHY00061H

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• Department: Physics
• Module co-ordinator: Dr. Paul Davies
• Credit value: 10 credits
• Credit level: H
• Academic year of delivery: 2020-21

• None

## Module will run

Occurrence Teaching cycle
A Autumn Term 2020-21

## Module aims

In Statistical Mechanics we will develop formalisms of equilibrium statistical mechanics from fundamental considerations of the microscopic states available to the system, and relate statistical mechanics to the classical thermodynamical descriptions of heat, work, temperature and entropy. Statistical mechanics will be used to derive formulae for the internal energy, entropy, specific heat, free energy and related properties of classical and quantum-mechanical systems, and to apply these formulae to a variety of realistic examples.

## Module learning outcomes

Learning outcomes: at the end of this module successful students will be able to:

• Discuss, at the level of detail given in the lectures, how the results of statistical mechanics may be derived from fundamental statistical considerations and how they are related to classical thermodynamics;

• Apply the definitions and results of statistical mechanics to deduce physical properties of the systems studied in the lectures and other systems of similar complexity, drawing in part on your knowledge of the microstates of simple systems from core courses in quantum mechanics and solid state physics.

## Module content

Please note, students wishing to take this module should take the prerequisite modules listed above (Thermodynamics & Solid State II - PHY00031I; Quantum Physics II - PHY00032O; and Mathematics II - PHY00030I) or the appropriate equivalent modules.

Syllabus

Microstates: microstates (quantum states) and macrostates of a system, degeneracy W, density of states, illustration for a set of N harmonic oscillators, principle of equal equilibrium probability of an isolated system, term “microcanonical ensemble” [1 lecture]

Thermal equilibrium, temperature: statistical nature of equilibrium illustrated for 2 sets of N harmonic oscillators, definition of temperature, Boltzmann distribution, partition function Z, term “canonical ensemble” [2]

Entropy: general statistical definition of entropy S, law of increase of entropy, entropy of isolated system in internal equilibrium (“microcanonical ensemble”), entropy of system in thermal equilibrium with a heat bath (“canonical ensemble”), Helmholtz free energy F; equivalence of classical and statistical entropy [2.5]

Elementary applications: Vacancies in solids; two-level systems (including magnetic susceptibility of dilute paramagnetic salt), simple harmonic oscillator (partition function, heat capacity). [2]

Vibrational heat capacity of solids: Quantisation of phonon modes, labelling of modes using wavevector k; Einstein and Debye models [2]

Ideal gas: Partition function of monatomic gas, classical gas law, Maxwell-Boltzmann speed distribution, molecular gases (rotation and vibration), classical limit of occupation numbers [2]

Systems with variable number of particles: Grand canonical ensemble, chemical potential, Gibbs distribution [1.5]

Identical particles: Fermions and bosons, Fermi and Bose distributions, Bose-Einstein condensation, with applications to free-electron metals and nuclear physics (fermions), and liquid 4He and superconductivity (bosons) [3]

Black body radiation: Energy density, pressure [1]

The classical limit: Phase space, classical equipartition theorem [1]

Comprehensive lecture notes should be taken down from the blackboard during lectures, and will be supplemented by a small number of handouts. These handouts, together with audio recordings of lectures, interactive apps, a record of problems set, and similar information, will also be made available through the VLE.

## Assessment

Task Length % of module mark
Essay/coursework
Solid State II Assignment 1
N/A 32
Essay/coursework
Solid State II Assignment 2
N/A 32
Essay/coursework
Solid State II Assignment 3
N/A 36

None

### Reassessment

Task Length % of module mark
Essay/coursework
Solid State II Assignment 1
N/A 32
Essay/coursework
Solid State II Assignment 2
N/A 32
Essay/coursework
Solid State II Assignment 3
N/A 36

## Module feedback

Physics Practice Questions (PPQs) - You will receive the marked scripts via your pigeon holes. Feedback solutions will be provided on the VLE or by other equivalent means from your lecturer. As feedback solutions are provided, normally detailed comments will not be written on your returned work, although markers will indicate where you have lost marks or made mistakes. You should use your returned scripts in conjunction with the feedback solutions.

Exams - You will receive the marks for the individual exams from eVision. Detailed model answers will be provided on the intranet. You should discuss your performance with your supervisor.

Statistical Mechanics

Waldram JR: The theory of thermodynamics (Cambridge University Press)***

Bowley R and Sánchez M: Introductory statistical mechanics (Oxford University Press)***

Glazer M and Wark J: Statistical Mechanics: A Survival Guide (Oxford University Press)***

Mandl F: Statistical physics (Wiley)**

Reif F: Fundamentals of statistical and thermal physics (McGraw-Hill)**

Blundell SJ and KM: Concepts in Thermal Physics (Oxford University Press)*

Swendsen RJ: An Introduction to Statistical Mechanics and Thermodynamics (Oxford University Press 2012)*

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.

## Coronavirus (COVID-19): changes to courses

The 2020/21 academic year will start in September. We aim to deliver as much face-to-face teaching as we can, supported by high quality online alternatives where we must.

Find details of the measures we're planning to protect our community.