# C5.3 Statistical Mechanics - Archived material for the year 2016-2017

A familiarity with classical mechanics, probability and fluid mechanics will be helpful.

16 lectures

### Assessment type:

- Written Examination

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.

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.

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.

- David Chandler,
*Introduction to Modern Statistical Mechanics*(Oxford University Press 1987) - M. Kardar,
*Statistical Physics of Particles*(Cambridge University Press 2007) - F. Schwabl,
*Statistical Mechanics*2nd ed. (Springer-Verlag 2006) - J.P. Sethna,
*Statistical Mechanics: Entropy, Order Parameters, and Complexity*(Oxford University Press 2006) [available online at http://pages.physics.cornell.edu/sethna/StatMech/ ]