## Lecturer: Paul Goddard

## Weighting: 7.5 CATS

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.

**Aims:**

The module introduces statistical mechanics and its central role in physics. It should give an appreciation of Boltzmann's insights into the nature and role of entropy. Many of the ideas introduced here will be used in the description of the properties of matter in solid state, nuclear and astrophysics modules.

**Objectives:**

At the end of the module you should

- Be familiar with the definition of thermal equilibrium, the ergodic hypothesis and the various ensembles.
- Know the definition and importance of the partition function and be able to calculate thermodynamic averages from it. This includes the Fermi-Dirac and Bose-Einstein distributions.
- Understand the structure of statistical mechanics and its relation to classical thermodynamics
- Be familiar with the notion of degeneracy and the density of states

**Syllabus:**

- Introduction to Statistical Mechanics: Systems and states: microstates. Fundamental assumptions of stat. mech.
- Equilibrium State: Definition of entropy for closed system in equilibrium. Maximization of entropy of a closed system in equilibrium. Fluctuations and Large Systems
- 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. - 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. - 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.

**Commitment:** about 18 Lectures + 4 problems classes

There will be four examples sheets associated with the module with some problems to be done for assessment.

**Assessment:** 1 hour examination (85%) + assessed work (15%).

This module has a home page.

**Recommended Texts: **S. Blundell and K. Blundell, Concepts in Thermal Physics, OUP; M. Glazer and J. Wark, Statistical Mechanics - a survival guide, OUP; A.M. Guenault, Statistical Physics, Springer.

**Leads from:** PX121 Thermal Physics I

**Leads to:** PX366 Statistical Physics