# MMath&Phys Mathematics and Physics

Year of entry: 2024

## Course unit details:Statistical Mechanics

Unit code PHYS20352 10 Level 2 Semester 2 No

### Overview

Statistical Mechanics

### Pre/co-requisites

Unit title Unit code Requirement type Description
Properties of Matter PHYS10352 Pre-Requisite Compulsory
Introduction to Quantum Mechanics PHYS20101 Pre-Requisite Compulsory

### Aims

• To develop the statistical basis of classical thermodynamics

• To deepen the appreciation of the link between the microscopic properties of individual atoms or other particles and the macroscopic properties of many-body systems formed from them

• To demonstrate the power of statistical methods in different areas of physics

• To use the methods of quantum mechanics and statistical physics to calculate the behaviour of gases of identical particles, and to apply the results to a set of important physical system.

### Learning outcomes

On completion successful students will be able to:

1. Explain the basic concepts of statistical mechanics, including entropy, its statistical interpretation and relation to disorder, and the statistical origin of the second law of thermodynamics;

2. Construct the canonical and grand-canonical partition functions for systems in thermal equilibrium, and use them to obtain thermodynamic quantities of interest.

3. Demonstrate an understanding of the implications of the indistinguishability of particles for systems of non-interacting quantum particles

4. Write down the Bose-Einstein and Fermi-Dirac distribution functions, and apply them to calculate the properties of Bose and Fermi gases, for example in the context of White Dwarf stars and black-body radiation.

5. Explain the physical origin of Bose-Einstein condensation, to characterize it quantitatively, and to explain the experiments confirming Bose-Einstein condensation

### Syllabus

1.The statistical theory of thermodynamics (approximately 5 lectures)

Basic of probability theory; microstates and macrostates; the concept of ensembles; the statistical interpretation of entropy and temperature; isolated systems and the microcanonical ensemble

2. Statistical physics of non-isolated systems (approximately 8 lectures)

Derivation of the Boltzmann distribution and the canonical ensemble; the independent-particle approximation; the partition function and its connection with thermodynamics; examples of non-interacting systems (paramagnet set of harmonic oscillators – quantum and classical , ideal gas, classical and quantum rotors). Equipartition theorem; Density of states. Grand-canonical ensemble and chemical potential.

3. Quantum gases (approximately 10 lectures)

Fermi-Dirac and Bose-Einstein distributions. The ideal Fermi gas: Fermi energy. Electronic heat capacity. White Dwarf stars. The ideal Bose gas: Photon gas blackbody radiation (Stefan’s Law and the Planck formula). Bose-Einstein condensation.

### Assessment methods

Method Weight
Other 10%
Written exam 90%

* Other 10% Tutorial Work/attendance

### Feedback methods

Feedback is through weekly tutorials and marked tutorial work.

Mandl, F., Statistical Physics, 2nd edition (Wiley)

Bowley, R. & Sanchez, M. Introductory Statistical Mechanics, 2nd edition (Oxford)

Zemansky, M.W. & Dittman, R.H., Heat and Thermodynamics, 7th edition (McGraw Hill)

Steane, A.M., A complete undergraduate course Thermodynamics (Oxford University Press)

Blundell, S.J.  Blundell, K.M. Concepts in Thermal Physics (Oxford University Press)

### Study hours

Scheduled activity hours
Assessment written exam 1.5
Lectures 24
Tutorials 4
Independent study hours
Independent study 70.5

### Teaching staff

Staff member Role
Judith McGovern Unit coordinator