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Department of Mathematical Sciences

# MATH3351/4231 Statistical Mechanics III/IV

Statistical mechanics seeks to use statistical techniques to develop mathematical methods necessary to deal with systems with large number of constituent entities, like a box of gas with which comprises of a large number of individual gas molecules. It provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed in everyday life, therefore explaining thermodynamics as a natural result of statistics and mechanics (classical and quantum) at the microscopic level.

We will start by discussing basic concepts in thermodynamics and then proceed to see how these ideas can be derived from a statistical viewpoint. As we go along, we will learn to apply the concepts to various interesting physical phenomena, such as neutron stars, Bose-Einstein condensation, phase transitions, etc.

## Outline of Course

Aim: To develop a basic understanding of dynamics and behaviour of systems with a large number of constituents.

### Term 1

• Thermodynamics: Thermal equlibrium, the laws of thermodynamics. Equations of state, ideal gas law.
• Classical statistical mechanics: Statistical basis of thermodynamics: microstates, macrostates and the thermodynamics limit. Ideal gas. Gibbs paradox and entropy. Microcanonical, canonical and grand-canonical ensembles.

### Term 2

• Distributions and identical particles: Maxwell-Boltzmann distribution. Bose and Fermi distributions, parastatistics.
• Physical phenomena: Ideal Bose and Fermi gases. Black-body radiation, magnetisation, neutron stars (Chandrashekar limit), superfluidity, Bose-Einstein condensation.
• Phase transitions: Mean field theory, Landau-Ginzburg theory of phase transitions.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook: MATH3351, MATH4231.