The canonical ensemble describes systems which can exchange energy with their surroundings, which may be modelled as a heat bath.The statistical mechanical quantity that characterizes systems in the canonical ensemble is the partition function, which is shown to be related to the Helmholtz free energy.The connections between statistical mechanics and the laws of thermodynamics are discussed.The application of the canonical ensemble is illustrated through a variety of examples: two-level systems, the quantum and classical simple harmonic oscillator, rigid rotors and a particle in a box.The differences in the statistical properties of distinguishable and indistinguishable particles are considered and used to derive the thermodynamic properties of ideal and non-ideal gases, including the ideal gas equation, the Sackur--Tetrode equation and the Van der Waals equation.The chapter concludes with a discussion of the equipartition theorem and its application to the Dulong--Petit Law.

There are no restrictions on how many bosons can occupy a single particle state, which has important consequences for their thermodynamic behaviour.Photons, quanta of the electromagnetic field, can be viewed as bosons with zero chemical potential, which allows the derivation of the thermodynamic properties of blackbody radiation, including the Stefan--Boltzmann Law.Non-interacting bosons with non-zero chemical potential can exhibit Bose--Einstein condensation at low temperatures, and interacting bosons may form a superfluid state.Low energy excitations in materials -- lattice vibrations (phonons) and spin waves (magnons) -- also behave as bosons, and are important for understanding the specific heat of materials at low temperatures.Of particular note is the Debye model which gives a simple account of the contributions of phonons to specific heat.

Fermi--Dirac statistics lead to specific thermodynamic consequences at low temperatures.A key quantity is the Fermi energy, which is equal to the chemical potential at zero temperature, and can be used to define a temperature scale, the Fermi temperature.At temperatures that are small compared to the Fermi temperature, thermodynamic quantities may be calculated using the Sommerfeld expansion.The properties of metals and the existence of compact stars such as white dwarfs and neutron stars are a direct consequence of Fermi--Dirac statistics.

Kinetic theory is a framework for calculating macroscopic physical properties of systems from their microscopic degrees of freedom.This idea is applied to an ideal gas to derive the Maxwell--Boltzmann velocity distribution, which is demonstrated to be compatible with the ideal gas law and is used to calculate the rate of effusion of an ideal gas.When molecular collisions are important, the mean free path and collision time are quantities that can characterize these collisions.Situations in which collisions are important, such as Brownian motion and diffusion, are presented, along with relevant equations: the Langevin equation and Fick's Law.

Statistical ensembles provide a conceptual framework within whichto obtain the average behaviour of physical systems.The extent to which a system interacts with its environment determines the appropriate statistical ensemble with which to describe its properties.Isolated systems are described by the microcanonical ensemble.The expression for the Boltzmann entropy acts as a bridge equation relating the thermodynamic quantity, the entropy, to a statistical mechanical quantity, the multiplicity of microstates.Other forms of entropy, the Gibbs and Shannon entropies, and the relation of entropy to irreversibility are discussed.

Thermalization in classical systems takes place through the spreading of an intial probability distribution in phase space.Liouville's Theorem implies that the phase space volume occupied by the probability distribution is preserved by Hamiltonian systems.These observations motivate the hypothesis that temporal averages of physical quantities are equivalent to ensemble averages.

The grand canonical ensemble applies to open systems that can exchange both energy and particles with their environment.The grand canonical partition function and its relation to the grand potential are derived, with an emphasis on the chemical potential.Examples in which the grand canonical ensemble applies are presented, including two-level systems, Langmuir adsorption isotherms, chemical equilibrium and the law of mass action.

Unlike classical particles, quantum particles are indistinguishable.Fermions and bosons differ in their quantum statistics, and the consequences of this for their statistical mechanics are explored in the grand canonical ensemble.The Fermi--Dirac and Bose--Einstein distribution functions are derived, and utilized to write thermal averages using the density of states.

The statistical basis of statistical mechanics is introduced using probability and probability distributions, including the binomial and Gaussian distributions.The example of a random walk is used to illustrate the relationship between these distributions and to introduce the central limit theorem.Microstates of quantum and classical systems are defined, along with the multiplicity function, which counts the number of macroscopically identical microstates in a given macrostate.The enumeration of microstates leads to the idea that ignorance of the exact microstate of a system in a macrostate can be quantified with the entropy.

Many different phases of matter can be characterized by the symmetries that they break.The Ising model for interacting spins illustrates this idea.In the absence of a magnetic field, there is a critical temperature, below which there is ferromagnetic ordering, and above which there is not.The magnetization is the order parameter for this transition: it is non-zero only when there is ferromagnetic ordering.The ferromagnetic phase transition in the Ising model is explored using the approximate method of mean field theory.Exact solutions are known for the Ising model in one and two dimensions and are discussed, along with numerical solutions using Monte Carlo simulations.Finally, the ideas of broken symmetry and their relationship to phase transitions are placed in the general framework of Landau theory and compared to results from mean field theory.