General Prerequisites: A familiarity with classical mechanics, probability and fluid mechanics will be helpful.
Course Overview: Statistical mechanics is a subject which has fundamental and powerful connections with probability, mechanics, stochastic processes, fluid mechanics, thermodynamics, quantum mechanics (though we avoid this), and even philosophy. It is also notoriously inaccessible to applied mathematicians. This course will endeavour to trace a rational path towards classical statistical mechanics, beginning with classical mechanics, and then developing the concepts of thermodynamics through study of the Boltzmann equation. In passing, we derive the Navier-Stokes equations, before developing a mechanically-based formulation of thermodynamics and its famous second law concerning entropy. The latter parts of the course develop a variety of applications of current interest.
Lecturer(s):
Prof. Andrew Fowler
Learning Outcomes: Students will have developed a sound knowledge and appreciation of some of the tools, concepts, and computations used in the study of statistical mechanics. They will also get some exposure to some modern research topics in the field.
Course Synopsis: Classical mechanics: Newton's second law, D'Alembert's principle, Lagrange's equations, Hamilton's equations. Chaos. Probability: probability density functions, moment generating function, central limit theorem. Fluid mechanics: material derivative, Euler and Navier-Stokes equations, energy equation. Random walks, Brownian motion, diffusion equation. Loschmidt's paradox.
Liouville equation, BBGKY hierarchy, Boltzmann equation. The collision integral for a hard sphere gas. Boltzmann H theorem. Maxwellian distribution. Definition of entropy and temperature. Gibbs and Helmholtz free energies. Thermodynamic relations.
Classical statistical mechanics. Ergodic theorem, equiprobability. Microcanonical ensemble for the hard sphere gas, entropy. Canonical ensemble. Grand canonical ensemble.
Selected applications and extensions: for example, chemical potential, phase change, binary alloys, surface energy, radiative transfer, polymer solution theory, Arrhenius kinetics, nucleation theory, percolation theory, renormalisation.