Statistical Mechanics is a fundamental bedrock of both classical and quantum physics. We cannot possibly hope to know all the details of what happens to all 10²³ atoms or molecules making up a sample of macroscopic size - and why should we try to garner so much information? Many of the measurable properties of the world outside us are really averages of microscopic events. The pressure of a gas on the walls of a container is made up of the impact of a vast number of collisions of individual molecules with the wall and we observe the average effect of these collisions as the pressure. In this module we shall study how such an averaging is carried out by means of both time averaging and ensemble averaging. Clearly there may be no point in calculating an average value to represent a typical value of a measurable physical quantity if fluctuations about that average are very large, and so we shall also study the fluctuations of the system about these averages and see how the fluctuations themselves can be important in determining measurable quantities such as specific heats. There are subtle differences between classical systems and those  governed by the laws of quantum mechanics and this module will explore these differences and also discuss how the quantum statistics of the particles making up a macroscopic system influence the thermal properties of the system. Many applications of these ideas from condensed matter physics, astrophysics and radiation physics will be given. Finally, if time permits, we shall mention the phenomenon of Bose-Einstein condensation which lies beneath an understanding of superfluidity and superconductivity and which has only recently been realised experimentally in atom traps and has led to the award of the Nobel Prize to Wolfgang Ketterle, Carl Wieman and Eric Cornell in 2001.

By the end of this module you will be able to :

• demonstrate an understanding of the problems in going from a microscopic to a macroscopic description of the world
• know and use the Boltzmann definition of Entropy in the microcanonical ensemble
• calculate the macroscopic thermodynamic properties (equation of state, specific heat etc) of an isolated system using the microcanonical ensemble
• demonstrate an understanding of the statistical mechanics of an open system of fixed particle number and in contact with a heat bath at constant temperture (the canonical ensemble)
• use the partition function Z_{N} in the canonical ensemble to calculate thermodynamic quantities such as the internal energy, specific heat and equation of state of simple non interacting systems.
• demonstrate and use the relation between and energy fluctuations and the specific heat
• reproduce the definitions of the Helmholtz and Gibbs Free Energies and the Enthalpy, and deduce the appropriate four Maxwell relations
• prove the minimal property of the Helmholtz Free Energy and use it to deduce the equilibrium properties of a system
• demonstrate an understanding of the classical limit of quantum statistics and of the importance of indistinguishability and distinguishability in a non interacting system of particles
• demonstrate an understanding of the statistical mechanics of an open system whose particle number is not fixed and which is in thermal contact with a heat bath (the grand canonical ensemble)
• demonstrate the use and understanding of the chemical potential µ
• use and calculate the grand partition function Z for simple systems and derive the relation between the fluctuations in particle number and derivatives of Z with respect to µ
• use the grand canonical ensemble to derive the mean occupation number of a single particle state in a system of non interacting fermions (the Fermi-Dirac distribution) n_{i}^{F}
• use the grand canonical ensemble to derive the mean occupation number of a single particle state in a system of non interacting bosons (the Bose-Einstein distribution) n_{i}^{B}
• use the Fermi-Dirac distribution to understand the low temperature behaviour of a set of non interacting fermions and understand how to use this to model the low temperature behaviour of electrons in a metal and understand the idea of Fermi pressure in solids and stars
• use the Bose-Einstein distribution to understand the low temperature behaviour of a set of non interacting bosons and understand why the chemical potential of a set of non conserved bosons must vanish
• use the Bose Einstein distribution to derive the thermodynamic properties of black body radiation and certain excitations in solids