Introduction

In this course we introduce the subject of statistical mechanics. This is a thermodynamic theory in which account is taken of the microscopic properties of individual atoms or molecules analysed in a statistical fashion. Statistical mechanics allows macroscopic properties to be calculated from the statistical distribution of the microscopic behaviour of individual atoms and molecules.

Synopsis

Boltzmann factor. Partition function and its relation to internal energy, entropy, Helmholtz energy, heat capacities and equations of state. Quantum states and the Gibbs hypothesis (non-examinable). Density of states. Application to: the spin-half paramagnet; simple harmonic oscillator (Einstein model of a solid); perfect gas; vibrational excitations of a diatomic gas; rotational excitations of a heteronuclear diatomic gas. Equipartition of energy. Bosons and fermions: Fermi-Dirac and Bose-Einstein distribution functions for non-interacting, indistinguishable particles. Partition function for bosons and fermions when the particle number is not restricted and when it is: microcanonical, canonical and grand canonical ensemble (non-examinable). Chemical potential. High-temperature limit and the Maxwell-Boltzmann distribution. Simple treatment of fluctuations (non-examinable). Low-temperature limit for fermions: Fermi energy and low-temperature limit of the heat capacity; application to electrons in metals and degenerate stars. Low-temperature limit for boson gas: Bose-Einstein condensation: calculation of the critical temperature of the phase transition; heat capacity; relevance to superfluidity in helium. The photon gas: Planck distribution, Stefan-Boltzmann law. Kirchhoff's law (non-examinable)