Unit information: Statistical Mechanics in 2012/13

Unit name	Statistical Mechanics
Unit code	MATH34300
Credit points	20
Level of study	H/6
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Professor. Liverpool
Open unit status	Not open
Pre-requisites	Mechanics 1; cannot be taken with physics units Thermal Physics 203 or Statistical Physics 303 due to overlap
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

The unit begins with a discussion of thermodynamics, the macroscopic (large scale) laws of heat. In contrast to mecanical systems, thermodynamics is fundamentally irreversible, so for example processes like thermal quilibration, combustion, and mixing can occur spontaneously, but the reverse processes never occur without external input. This leads to fixed constraints on the capabilities of (for example) engines, fridges and living organisms. The remainder of the unit ("statistical mechanics") deals with the microscopic basis of thermodynamis, that is, explaining large scale properties from properties of individual molecules. Although the dynamical equations can be solved exactly in only a very few cases, the very large number of particles means that statisitcal assumptions are often justified, making a strongly predictive and irreversible theory from recerisbly mecanics. Both equilibrium and non-equilibrium situations will be covered, ending with a brief discussion of numerical simulation methods.

Aims

Introduction to the mathematical foundations of thermodynamics and statistical mechanics.

Syllabus

Thermodynamics

• State variables, laws, potentials.
• Applications in phase and chemical equilibria, heat engines, fridges, the atmosphere.

Equilibrium statistical mechanics

• Classical ensembles (microcanonical, canonical, grand canonical), entropy of mixing, quantum statistics, derivation of thermodynamic quantities.
• Computations for the ideal gas (classical, Fermi and Bose) and applications.

Dynamical foundations

• Review of Hamiltonian mechanics, Liouville equation, ergodicity, mixing, Poincare recurrence.

Nonequilibrium statistical mechanics

• Boltzmann equation and H-theorem.

Modern Topics

• Brief introduction to computational techniques.

Relation to Other Units

Statistical mechanics is a branch of mathematical physics, along with mechanics, quantum mechanics and relativity. Its molecular treatment of fluids is complementary to the continuum approaches in the fluids units. There are also connections with information theory and chaotic dynamics. Connections with probability and statistics exist, but are not strong. Some parts of the unit are similar to Thermal Physics and Condensed Matter and Statistical Physics offered in physics; the approach here is more mathematical, and more directed towards research interests of the department, including fluids, dynamical systems, biological physics, nonequilibrium systems and computational methods.

Intended Learning Outcomes

By the end of the unit the students should be familiar with the main concepts of thermodynamics, equilibrium and nonequilibrium statistical mechanics, understand thermodynamic limitations of systems, and be able to derive thermodynamic properties of systems of weakly interacting particles.

Transferable Skills:

Clear, logical thinking and an ability to comprehend and solve problems of mathematical physics.

Teaching Information

A standard chalk-and-talk lecture unit of about 30 lectures, with occasional problems classes or informal discussion to meet the needs of individual students.

Assessment Information

The final assessment mark for Statistical Mechanics 3 is constructed as follows:

• 20% is based on specified marked homework assignments;
• 80% is based on the mark in a 2.5 hour examination in May/June, with five questions (calculators are not permitted). The best four answers will be considered for credit.

Different homework assignments will be given out for the level 3 and M versions of this unit.

The exam for the level 3 and M versions of the unit will also be different.

Reading and References

See the unit homepage for advice.

• Statistical Mechanics, R.K. Pathria, Elsevier 2005, 529 pages.
• Introduction to Modern Statistical Mechanics, D. Chandler, Oxford 1987, 274 pages.
• Equilibrium and non-equilibrium statistical thermodynamics, M. LeBellac, F.Mortessagne and George Batronni, Cambridge 2004, 616 pages.
• An introduction to chaos in non-equilibrium statistical mechanics , J.R.Dorfman. Cambridge 1999, 287 pages.