Introductory description

Any macroscopic object we meet contains a large number of particles, each of which moves according to the laws of mechanics (which can be classical or quantum). Yet we can often ignore the details of this microscopic motion and use a few average quantities such as temperature and pressure to describe and predict the behaviour of the object. Why we can do this, when we can do this and how to do it are the subject of this module.

The most important idea in the field is due to Boltzmann, who identified the connection between entropy and disorder. The empirical laws of thermodynamics required the existence of entropy, but there was no microscopic definition for it. The module shows how the structure of equilibrium thermodynamics follows from Boltzmann's definition of the entropy and shows how, in principle, any observable equilibrium quantity can be computed. This microscopic theory (now called statistical mechanics) provides the basis for predicting and explaining all thermodynamic properties of matter.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1. Introduction to Statistical Mechanics: Systems and states: microstates. Fundamental assumptions of stat. mech.
2. Equilibrium State: Definition of entropy for closed system in equilibrium. Maximization of entropy of a closed system in equilibrium. Fluctuations and Large Systems
3. Boltzmann distribution and Lagrange multipliers: Partition function, Z. Evaluation of Z for a spin-half system in a magnetic field and harmonic oscillator and system with degeneracy. Relationship of Z to thermodynamic quantities E, S and F=E-TS. Minimization of F in equilibrium for systems at fixed T and V. Microscopic basis for thermodynamics and relation to statistical mechanics.
4. Classical Thermodynamics of Gases: Thermal equilibrium, quasistatic and reversible changes. Statistical Mechanics of Classical Gases. Thermodynamic potentials G and H. The ideal gas law, the Gibbs paradox.
5. Grand-Canonical ensembles: system not closed (possibility of particle exchange between systems). Bose- Einstein and Fermi- Dirac distribution functions. Density of states. Chemical potential. Fermi energy. Relevance of Fermi-Dirac and Bose-Einstein to matter. Phonons: Einstein model, Debye model and dispersive phonons, role of elastic modulus, phonon heat capacity, thermal expansion. Thermal properties of the free Fermi gas.

Learning outcomes

By the end of the module, students should be able to:

• Give the definitions of thermal equilibrium, the ergodic hypothesis and the various ensembles
• Explain importance of the partition function and calculate thermodynamic averages from it (this includes the Fermi-Dirac and Bose-Einstein distributions)
• Discuss the structure of statistical mechanics and explain its relation to classical thermodynamics
• Explain the notion of degeneracy and the density of states.

Indicative reading list

Concepts in Thermal Physics by S. J. Blundell and K. M. Blundell (OUP, 2010).
Further reading: Statistical mechanics: a survival guide by A. M. Glazer and J. S. Wark (OUP, 2001);
Statistical Physics by A. M. Guenault (Springer, 2007).

View reading list on Talis Aspire

Subject specific skills

Knowledge of mathematics and physics. Skills in modelling, reasoning, thinking.

Transferable skills

Analytical, communication, problem-solving, self-study