Statistical mechanics is a formalism that aims at explaining the physical properties of matter at the macroscopic level in terms of the dynamical behavior of its microscopic constituents. The scope of the formalism is almost unlimited as it is applicable to systems in any state of aggregation, ranging from plasmas and liquids, to polymers, solids and matter in equilibrium with radiation. The aim of this course is to introduce the principles and methods of equilibrium statistical mechanics and apply them to a number of model systems in order to illustrate the use and extent of the theory in a systematic manner.


 

  1. Bullet Fundamentals of thermodynamics: Systems, phases and state variables. Equilibrium states and temperature. Reversible and irreversible processes. Equations of state. Work and heat. The laws of thermodynamics. Entropy and the second law. Homogeneous functions: Euler’s equation and Gibbs-Duhem relations. Thermodynamic potentials and Legendre transformations. Internal constraints, extremum principles and second law. Maxwell relations. Jacobi transformations. Phase and chemical equilibrium.  Thermodynamic stability and response functions.

  2. Bullet Equilibrium statistical mechanics: Quantum states and phase space. Ensemble theory. The microcanonical ensemble. Connection with thermodynamics: density of states and entropy. The canonical ensemble. Canonical partition function and free energy. Internal energy and energy fluctuations. Microscopic description of heat and mechanical work. Illustration: the statistical mechanics of paramagnetism. The grand canonical ensemble. Grand canonical partition function and grand potential. Density and energy fluctuations. Entropy maximisation: a general method to derive distribution functions. Fluctuations and response functions. Equivalence between ensembles. Final considerations on Boltzmann statistics. Classical statistical mechanics: phase-space and classical partition functions. Maxwell-Boltzmann distribution. “Equipartition” and “virial” theorems. 

  3. Bullet Applications of Boltzmann statistics to ideal systems: Factorisation approximation. Mono-atomic gases. Gases with internal degrees of freedom: vibrations and rotations. Chemical equilibrium and Saha’s ionisation formula.

  4. BulletQuantum statistical mechanics: Indistinguishable quantum particles and symmetry requirements. Fermi-Dirac and Bose-Einstein statistics: derivation of partition functions. Recovering the classical limit.   Statistical mechanics of quasi-particles: phonons and photons. Ideal Fermi gas: low and high-density limits. Electrons in metals. Ideal Bose gas: low and high-density limits. Bose-Einstein condensation.

  5. Bullet Statistical mechanics of interacting systems: Interaction potentials. Perturbation theory using a control parameter. Cluster and virial expansions. The van der Waals equation of state and the liquid-vapor phase transition. Models of Ising and Heisenberg. Mean field theories.  Introduction to Monte Carlo methods in statistical physics.   

  1. Bullet Essential reading:

  Thermodynamics and Statistical Mechanics. Greiner, W; Neise, L. and Stoker, H.,  Springer-Verlag, 2000. ISBN 0387942998. QUB Library Shelf Mark QC311                

  1. Bullet Recommended reading:

   Statistical Mechanics, 2nd ed. Pathria, R.K. Elsevier, 1996. ISBN 0750624698. QUB Library   Shelf Mark Q175/PATH

   Statistical Mechanics. McQuarrie, D. A., Harper and Row, 1976. ISBN 0060443669. QUB Library Shelf Mark QC174.8 MCQA

   Introduction to Modern Statistical Mechanics. Chandler, D. Oxford University Press, 1987. ISBN 0195042778. QUB Library Shelf Mark QC174.8 CHAN

  1. Bullet Complementary reading:

  Introduction to Statistical Physics. Huang, K.  Taylor and Francis, 2001. ISBN 0748409424. QUB Library Shelf Mark QC174.8 HUAN